Escola Internacional de Doutoramento

Fa¨ı¸calNda¨ırou

TESE DE DOUTORAMENTO

Modelado matem´atico de certas enfermidades relacionadas coa auga Mathematical modelling of some diseases related to water

Dirixida polos doutores: Iv´anCarlos Area Carracedo (Universidade de Vigo) e Delfim Fernando Marado Torres (Universidade de Aveiro, Portugal)

Ano: 2020

Escola Internacional de Doutoramento

Iv´anCarlos Area Carracedo e Delfim Fernando Marado Torres

FAN CONSTAR que o presente traballo titulado “Modelado matem´atico de certas enfermidades relacionadas coa auga” “Mathematical modelling of some diseases related to water”, que presenta Fa¨ı¸calNda¨ırou para a obtenci´ondo t´ıtulo de Doutor/a, foi elaborado baixo a s´ua direcci´onno pro- grama de doutoramento Programa de Doutoramento en Auga, Sustentabilidade e Desenvolvemento (O03D040P06).

Ourense, 17 de setembro de 2020.

Os Directores da tese de doutoramento

Dr. Iv´anCarlos Area Carracedo Dr. Delfim Fernando Marado Torres

Summary

This thesis dissertation focusses on the study of some infectious diseases dynamics from a double point of view: modelization and control. Our main aim is to formulate new mathematical models and combining them with existing ones in order to analyze the dynamics of diseases related to water. We consider compartmental models described by ordinary di↵erential equations and perform rigorous qualitative and quantitative techniques for acquiring insights into the dynamics of these models. My contribution to the material in this thesis is contained in the following original papers: P1) F. Nda¨ırou, I. Area and D. F. M. Torres. Mathematical Modeling of Japanese Encephalitis Under Aquatic Environmental E↵ects.Submitted; P2) F. Nda¨ırou, I. Area, J. J. Nieto, C. J. Silva, and D. F. M. Torres. Mathematical modeling of Zika disease in pregnant women and newborns with microcephaly in Brazil. Mathematical Methods in the Applied Sciences, 41(18):8929–8941, 2018; P3) F. Nda¨ırou, I. Area, J. J. Nieto, and D. F. M. Torres. Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan. Chaos, Solitons & Fractals, 135:109846, 2020; P4) I. Area, F. Nda¨ırou, J. J. Nieto, C. J. Silva, and D. F. M. Torres. Ebola model and opti- mal control with vaccination constraints. Journal of Industrial & Management Optimization, 14(2):427–446, 2018; done in collaboration with my advisors (Professor Iv´anArea and Professor Delfim F. M. Torres) and co-authored with my collaborator and former advisor Professor Juan Jos´eNieto and my collaborator Doctor Cristiana Jo˜aoda Silva. This thesis discusses some recent knowledge and investigation on the transmission dynamics of Ebola disease, Zika disease, Japanese encephalitis disease as well as COVID-19. The following are the main topics: (1) The Ebola virus disease is a severe viral haemorrhagic fever syndrome caused by Ebola virus. This disease is transmitted by direct contact with the body fluids of an infected person and objects contaminated with virus or infected animals, with a death rate close to 90% in humans. Recently, some mathematical models have been presented to analyse the spread of the 2014 Ebola outbreak in West Africa. For this disease, we introduce vaccination of the susceptible population with the aim of controlling the spread of the disease and analyze two optimal control problems related with the transmission of Ebola disease with vaccination. Firstly, we consider the case where the total number of available vaccines in a fixed period of time is limited. Secondly, we analyze the situation where there is a limited supply of vaccines at each instant of time for a fixed interval of time. The optimal control problems have been solved analytically. Finally, we have performed a number of numerical simulations in order to compare the models with vaccination and the model without vaccination, which has recently been shown to fit the real data. Three vaccination scenarios have been considered for our numerical simulations, namely: unlimited supply of vaccines; limited total number of vaccines; and limited supply of vaccines at each instant of time. (2) We propose a compartmental mathematical model for the spread of the COVID-19 disease with special focus on the transmissibility of super-spreaders individuals. We compute the basic reproduction number threshold, we study the local stability of the disease free equilibrium in terms of the basic reproduction number, and we investigate the sensitivity of the model with respect to the variation of each one of its parameters. Numerical simulations show the suitability of the proposed COVID-19 model for the outbreak that occurred in Wuhan, China. (3) We propose a new mathematical model for the spread of Zika virus. Special attention is paid to the transmission of microcephaly. Numerical simulations show the accuracy of the model with respect to the Zika outbreak occurred in Brazil. (4) Also, we propose a mathematical model for the spread of Japanese encephalitis, with emphasis on environmental e↵ects on the aquatic phase of mosquitoes. The model is shown to be biolog- ically well-posed and to have a biologically and ecologically meaningful disease free equilibrium point. Local stability is analyzed in terms of the basic reproduction number and numerical simulations presented and discussed. Resumo

A presente tese c´entrase no estudo dalgunhas din´amicas de enfermidades infecciosas desde un do- bre punto de vista: modelizaci´one control. O noso principal obxectivo ´eformular novos modelos matem´aticos e combinalos cos existentes para analizar a din´amica das enfermidades relacionadas coa auga. Consideramos modelos compartimentais descritos por ecuaci´ons diferenciais ordinarias e reali- zamos t´ecnicas cualitativas e cuantitativas rigorosas para adquirir informaci´onsobre a din´amica destes modelos. As mi˜nas achegas ao material da tese est´arecollida nos seguintes artigos: P1) F. Nda¨ırou, I. Area and D. F. M. Torres. Mathematical Modeling of Japanese Encephalitis Under Aquatic Environmental E↵ects. Enviado a publicaci´on; P2) F. Nda¨ırou, I. Area, J. J. Nieto, C. J. Silva, and D. F. M. Torres. Mathematical modeling of Zika disease in pregnant women and newborns with microcephaly in Brazil. Mathematical Methods in the Applied Sciences, 41(18):8929–8941, 2018; P3) F. Nda¨ırou, I. Area, J. J. Nieto, and D. F. M. Torres. Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan. Chaos, Solitons & Fractals, 135:109846, 2020; P4) I. Area, F. Nda¨ırou, J. J. Nieto, C. J. Silva, and D. F. M. Torres. Ebola model and opti- mal control with vaccination constraints. Journal of Industrial & Management Optimization, 14(2):427–446, 2018; realizados en colaboraci´oncos meus directores de tese (Profesor Iv´anArea e Profesor Delfim F. M. Torres) nos que tam´enparticipou o co-director do meu traballo fin de mestrado, Profesor Juan Jos´e Nieto, e a Doutora Cristiana Jo˜aoda Silva. Esta tese trata de alg´uns co˜necementos e investigaci´ons recentes sobre a din´amica de transmisi´on da enfermidade do ´ebola, a enfermidade de Zika, a enfermidade da encefalite xaponesa e a COVID-19. Os temas principais son os seguintes: (1) En primeiro lugar, a enfermidade do virus do ´ebola ´eunha s´ındrome de febre hemorr´axica viral grave causada polo virus do ´ebola. Esta enfermidade transm´ıtese por contacto directo cos flu´ıdos corporais dunha persoa infectada e obxectos contaminados co virus ou animais in- fectados, cunha taxa de mortalidade pr´oxima ao 90% en humanos. A orixe desta enfermidade non est´aclara. Peter Piot descubriu o ´ebola en 1976 e axudou a conter a primeira crise nese mesmo ano, concretamente en Sud´an(territorio que actualmente est´adividido en dous esta- dos) entre os meses de xu˜no e novembro. Desde ent´onhoubo polo menos 18 crises confirmadas de ´ebola entre 1976 e 2014. Recentemente, present´aronse alg´uns modelos matem´aticos para analizar a propagaci´ondo brote de ´ebola na Africa´ occidental de 2014. Tr´atase dos primeiros pasos para unha seguinte fase do problema, ´edicir, cos modelos existentes na literatura era posible predicir un avance da pandemia no futuro. Posto que non est´aclara a orixe da enfer- midade (se ben boa parte da comunidade cient´ıfica apunta a que poden ser uns determinados morcegos) non se pode predicir cando existir´a,nin onde, un novo surto. Por esta raz´ono problema seguinte ´epensar que facer no caso de ter un surto e como administrar a vacina, no caso de existir. Neste sentido, conv´ensinalar que desde decembro de 2019 existe unha vacina para a enfermidade provocada polo virus do ´ebola, cunha taxa de efectividade suficientemente alta. Pero os problemas non rematan con ter unha vacina, pois ´epreciso ter cantidade sufi- ciente en doses para a s´ua administraci´one, sobre todo, poder contar con persoal m´edico que poida administrar a vacina. Se supo˜nemos un surto do virus do ´ebola non ´edoado contar con persoal sanitario capacitado e disposto a administrar vacina contra esta enfermidade. Por estes motivos e para esta enfermidade, introducimos a vacinaci´onda poboaci´onsusceptible co obxectivo de controlar a propagaci´onda enfermidade e analizamos dous problemas de control ´optimos relacionados coa transmisi´onda enfermidade do ´ebola con vacinaci´on. En primeiro lugar, consideramos o caso onde o n´umero total de vacinas dispo˜nibles nun per´ıodo de tempo fixo ´elimitado. En segundo lugar, analizamos a situaci´onna que hai unha oferta limitada de vacinas en cada instante por un intervalo de tempo fixo. Resolv´eronse os problemas de control ´optimo analiticamente. Finalmente, realizamos unha serie de simulaci´ons num´ericas en para comparar os modelos coa vacinaci´one o modelo sen vacinaci´on, que recentemente demostrou que se axusta aos datos reais. Consider´aronse tres escenarios de vacinaci´ons para as nosas simulaci´ons num´ericas, nomeadamente: subministraci´onilimitada de vacinas; n´umero total limitado de vacinas; e oferta limitada de vacinas en cada instante de tempo. Deste xeito pretendeuse anticipar un problema real con cuesti´ons tan complexas como a vacinaci´on e unha enfermidade altamente letal. Neste ´ultimo aspecto conv´ensinalar que a taxa de mor- talidade do ´ebola var´ıa entre o 50 e o 90%, e estudos recentes demostran que determinadas diferenzas xen´eticas xogan un papel fundamental na taxa de mortalidade. Esta cuesti´onest´a baixo an´alise noutra das enfermidades analizadas nesta tese de doutoramento, a COVID-19. De feito, durante a pandemia do ´ebola do ano 2014 que afectou fundamentalmente a Guinea, Liberia e Serra Leoa p´odense observar diferenzas important´ısimas nas taxas de mortalidade, que foron a orixe do estudo xen´etico. En calquera caso, son cuesti´ons moi complexas porque as recomendaci´ons sanitarias durante a pandemia do Ebola´ de 2014 conti˜nan a incineraci´on das persoas falecidas. Esta recomendaci´onest´arecollida no modelo compartimental que anal- iza a evoluci´ondesta enfermidade ao contemplar persoas susceptibles, expostas, infectadas, recuperadas, hospitalizadas, falecidas, incineradas e completamente recuperadas. (2) Propo˜nemos un modelo matem´atico compartimental para a propagaci´onda enfermidade da COVID-19 con especial ´enfase na transmisibilidade dos individuos super-propagadores. Calcu- lamos o n´umero b´asico de reproduci´on, estudamos a estabilidade local do punto de equilibrio libre da enfermidade en termos do n´umero de reproduci´onb´asica, e investigamos a sensibi- lidade do modelo con respecto ´avariaci´onde cada un dos seus par´ametros. As simulaci´ons num´ericas mostran a idoneidade do modelo proposto para o brote de COVID-19 ocorrido en Wuhan, China. A enfermidade da COVID-19 tivo a s´ua orixe posiblemente en China e tam´en con bastante exactitude a finais do ano 2.019 e comezos do ano 2.020. Desde que a OMS come- zou a publicar datos sobre a existencia desta enfermidade comezamos a analizar os datos para tentar dispor dun modelo matem´atico que permitise explicar o avance da enfermidade. Nesta li˜na houbo distintas reuni´ons de traballo e nos primeiros d´ıas de febreiro de 2020 xa ti˜namos o modelo presentado nesta tese de doutoramento. A principal novidade foi a de incorporar unha nova clase de persoas, as denominadas superprogagadoras, que permit´ıa explicar con bastante exactitude os datos que chegaban da China. En comparaci´oncon outras enfermidades m´ais co˜necidas, foi preciso estimar alg´uns dos par´ametros que determinan a taxa de variaci´onde cada un dos compartimentos nos que se dividiu a poboaci´on. Con este modelo, e cando o d´ıa 4 de marzo houbo o primeiro caso en Galicia, elaborouse un informe que foi remitido a todos os partidos pol´ıticos con representaci´onno Parlamento de Galicia advertindo do avance da pandemia. Segundo as simulaci´ons realizadas, o maior n´umero de persoas novas infectadas ter´ıa lugar o d´ıa 5 de abril, coincidindo coa data inicialmente prevista para as elecci´ons galegas. Esta conxectura foi comprobada como certa posteriormente e as elecci´ons foron adiadas para o domingo d´ıa 12 de xullo. Os traballos realizados foron fundamentais para a predici´ondos n´umeros de persoas infectadas, hospitalizadas e nas unidades de coidados intensivos a nivel galego. Tr´atase de problemas de alt´ısima complexidade pois a experiencia indica que unha predici´oncomo a feita no seu d´ıa ´emoi complexa de realizar con predici´ons aca´ıdas. M´ais a´ında, a predici´onde n´umero de persoas hospitalizadas e nas unidades de coidados intensivos nos hospitais require unha predici´onde casos como a antes descrita para, en base a esa onda, extraer as posibles persoas infectadas para predicir as hospitalizaci´ons. Ter predici´ons cun nivel aceptable de fiabilidade pode permitir pensar en decisi´ons tan complexas como: ´enecesario un novo hospital? ´enecesario cancelar as operaci´ons non vitais en toda Galicia? ´epreciso contar con estudantes de ´ultimos anos para tarefas que deberan ser realizadas por profesion- ais nos hospitais? A soluci´onadoptada para frear a pandemia, en base a confinamentos, ten producido numerosas consecuencias a todos os niveis, inclu´ındo os sanitarios, econ´omicos e sociais, sen deixar de mencionar o educativo. Posiblemente esta pandemia puxo enriba da mesa a necesidade de contar con persoas traballando en modelos epidemiol´oxicos, campo de traballo que conta con numeroso persoal investigador e recursos noutros continentes, con es- pecialistas de primeiro nivel mundial en Africa.´ Tr´atase de traballos interdisciplinares, que deben contar con matem´aticos e, por suposto, persoal sanitario. Durante o tempo de dep´osito da tese de doutoramento estaban baixo ensaio cl´ınico, en distintas fases, distintas vacinas con- tra a enfermidade. Novamente, a existencia de vacina d´alugar a un n´umero importante de problemas matem´aticos que c´ompre resolver desde a teor´ıa de control. Basicamente, tr´atase de maximizar os efectos da vacinaci´onminimizando o n´umero de vacinas aplicadas, pois non haber´avacina para todas as persoas que habitamos actualmente a terra. Se ben se trata de datos que se actualizan diariamente, o n´umero de casos de persoas infectadas pola COVID-19 supera xa os 50 mill´ons de persoas en todo o mundo, con arredor do 2,6% de persoas falecidas. Estas cifras var´ıan dun estado a outro e tam´envar´ıan segundo territorios, o que apunta ´a necesidade de an´alise desde o punto de vista xen´etico para poder saber m´ais sobre por que hai persoas que falecen por esta enfermidade e outras que a superan simplemente cun pouco de febre. As taxas de infecci´ona nivel galego de arredor do 1 por cento da poboaci´onest´an moi por debaixo de p.ex. Chile ou Per´uonde as porcentaxes son superiores ao 3 por cento de persoas infectadas. M´ais a´ında, en Per´uxa faleceu m´ais do un por mil da poboaci´ontotal pola COVID-19, dato que dista moito de Galicia onde as cifras oficiais falan de arredor do tres por dez mil da poboaci´ongalega falecida pola COVID-19. Por suposto, ´ecomplexo comparar cifras pois non hai o mesmo n´umero de probas PCRs realizadas nen tan sequera a nivel do estado espa˜nol, menos a´ında en Europa e en ning´un xeito comparable a nivel mundial, pero as cifras que publica a Organizaci´onMundial da Sa´ude son as que se analizan para as comparativas anteriores. Por outra banda, desde o comezo da pandemia estivo moi presente a denominada “inmunidade de reba˜no”. Esta cuesti´onsemella que non ´eposible sen unha vacina pois xa apareceron os primerios casos de persoas reinfectadas pola COVID-19, demostrados cientifi- camente. M´ais a´ında as distintas mutaci´ons do virus da COVID-19, a ´ultima delas a trav´es dos vis´ons, fai pensar que non ´eunha doenza que permita ese tipo de inmunidade. Os estudos de seroprevalencia tampouco indican que sexa posible atinxir a inmunidade de reba˜no, pois as cifras que se publican son moi baixas. (3) No seguinte cap´ıtulo da tese propo˜nemos un novo modelo matem´atico para a propagaci´ondo virus Zika, prestando especial atenci´on´atransmisi´onda microcefalia. Neste estudo realizamos distintas simulaci´ons num´ericas para amosar a precisi´ondo modelo con respecto ao brote de Zika ocorrido en Brasil. O nome do virus do Zika prov´endo bosque Zika en Uganda, onde o virus foi aillado por primeira vez no ano 1.947. Cinco anos m´ais tarde detect´aronse os primeiros casos en humanos. Desde ent´onesta enfermidade tivo distintos surtos, concentrados en Africa´ e en Asia. O surto do ano 2.015-2.016 tivo unha alt´ısima repercusi´onao afectar aos xogos ol´ımpicos de Rio de Janeiro. Esta enfermidade en humanos transm´ıtese a trav´esda picadura dun mosquito infectado da clase Aedes, tanto o Aedes Aegypti como o Aedes Albopictus. Inicialmente a infecci´onnon ´emoi chamativa ao producir erupci´oncut´anea, febre leve, conx- untivite e dor muscular. Debemos ter presente que estes problemas ocultan un dos grandes problemas, pois o virus do Zika est´arelacionado con determinadas desordes neurol´oxicas e malformaci´ons fetais. En marzo do ano 2.016 un equipo de investigadores apuntou ao virus do Zika como causante de microcefalia en mulleres embarazadas, pois o virus infecta un tipo de c´elulas nai neuronais responsables da cortiza cerebral. Como consecuencia desta infecci´on, as c´elulas nai morren ou perden a capacidade de dividirse, polo que a cortiza cerebral non se forma correctamente ou, chegado o caso, non se rexenera. Esta ´eunha das causas posibles para a mencionada microcefalia e outras enfermidades asociadas a este virus. As consecuencias non rematan aqui e outros especialistas indican que este virus tam´enpode ser causante da s´ındrome de Guillain-Barr´e.S´ofai tres anos que un grupo de investigadores determinou, nun artigo pub- licado na prestixiosa revista Science, a estrutura do virus e o que ´ea´ında m´ais importante: as s´uas diferenzas entre o virus do Zika e outros da clase dos flavivirus, como poden ser o dengue, o virus do Nilo Occidental, a febre amarela, a encefalite xaponesa (obxecto de estudo noutro cap´ıtulo da tese) ou o virus da encefalite transmitida por garrapatas. Desde o punto de vista matem´atico a modelizaci´onde enfermidades transmitidas por mosquitos incrementa a s´ua dificultade. Por unha banda, xa que ´epreciso termos en conta d´uas poboaci´ons (humanos e mosquitos) e, consecuentemente e en segundo lugar, ao incrementar o n´umero de inc´ognitas, o sistema de ecuaci´ons diferenciais ten maiores dificultades para a s´ua an´alise e maior custe computacional para as simulaci´ons num´ericas. C´ompre sinalar que o tipo de mosquito que transmite esta(s) enfermidade(s) est´aa propagarse por territorios onde non existiu nos ´ultimos tempos, de xeito que enfermidades ´asque non se lles prestaba atenci´ondesde determinados grupos humanos comezan agora a preocupar. (4) No ´ultimo cap´ıtulo, propo˜nemos un modelo matem´atico para a propagaci´onda encefalite xaponesa, con ´enfase sobre os efectos ambientais na fase acu´atica dos mosquitos. Probamos que o modelo biol´oxico est´aben plantexado e ten un punto de equilibrio libre de enfermidades biol´oxicas, ecoloxicamente significativo. Analizamos a estabilidade local en termos do n´umero de reproduci´onb´asica. Para conclu´ır, ach´eganse simulaci´ons num´ericas do modelo. Tr´atase dunha enfermidade transmitida por mosquitos, que normalmente habitan en zonas con abun- dante auga. Precisan da auga para a s´ua reproduci´one precisan neste caso de humanos para transmitir a enfermidade. A enfermidade ´etransmitida polo mosquito Culex tritaeniorhynchu. Este tipo de mosquito cr´ıa onde hai auga abundante en zonas agr´ıcolas rurais, como os ar- rozais, e inf´ectanse aliment´andose de hospedadores vertebrados (principalmente porcos e aves lim´ıcolas) infectados polo virus da encefalite xaponesa. Os s´ıntomas da enfermidade duran entre 5 e 15 d´ıas e incl´uen febre, dor de cabez, confusi´one dificultades para moverse. Tr´atase dunha enfermidade que pode causar a morte de persoas. Como xa foi sinalado anteriormente, ´eunha enfermidade transmitida por un virus da clase dos flavivirus, polo que o estudo da s´ua evoluci´onpermite co˜necer m´ais sobre outros virus e enfermidades letais para a especie humana. Na actualidade det´ectanse arredor de 68.000 casos cada ano de persoas infectadas por esta enfermidade, pero o n´umero ten que ser maior e s´ounha parte das persoas infec- tadas son detectadas. Segundo os informes da Organizaci´onMundial da Sa´ude, arredor de 17.000 persoas falecen cada ano por esta enfermidade. Os primeiros casos detectados datan do ano 1.871 e prod´ucense grandes surtos cada 2-15 anos, posiblemente debidos a alg´un tipo de comportamento dos mosquitos que a´ında non se co˜nece. A diferenza das outras enfermidades analizadas, non existen datos de seguimento de surtos con casos diarios, sen´onque os n´umeros publicados son mensuais ou anuais, con grandes dificultades para ter estimaci´ons cl´ınicas dos par´ametros do sistema de ecuaci´ons diferenciais proposto. A diferenza de modelos anteriores publicados na literatura, o modelo matem´atico para explicar a expansi´onda enfermidade incor- pora efectos ambientais na fase acu´atica dos mosquitos, como fonte principal de reproduci´on. O ciclo biol´oxico do mosquito presenta as fases de ovo, larva, pupa e adulto. As fase inmaduras (larva e pupa) son acu´aticas, mentre que a de adulto ´eunha fase de vida a´erea. No mod- elo matem´atico proposto, consideramos os factores ambientais dentro de tres poboaci´ons de hospedaxe diferentes: humanos, mosquitos e animais vertebrados (tale como se sinalou, funda- mentalmente porcos e aves lim´ıcolas). De feito, as condici´ons ambientais non hixi´enicas poden mellorar a presenta e o crecemento de poboaci´ons de vectores (mosquitos) que levan a unha r´apida propagaci´onda enfermidade. Isto d´ebese a varios tipos de residuos dom´esticos e p.ex. vertidos ao medio ambiente en zonas residencias, que proporcionan un ambiente moi propicio para o crecemento dos vectores. A cuesti´onfundamental ´eque este tipo de situaci´onnon se puido modelar con compartimentos epidemiol´oxicos, polo que se recurriu a modelos existentes para tratar ese efecto sobre a enfermidade da encefalite xaponesa. No que respecta ao mod- elado de mosquitos e humanos si que se empregaron compartimentos, baixo certas hip´oteses nomeadamente: non se considerou emigraci´onde persoas humanas infectadas; a poboaci´on humana non ´econstante (taxa de mortalidade inducida por enfermidade); e tam´ense supuxo que o coeficiente de transmisi´ondo virus ´econstante e non var´ıa segundo as estaci´ons; no finalmente, no modelo marcouse que todos os mosquitos nacen como susceptibles de contraer a enfermidade. (5) Finalmente, conv´enter presente o impulso ´aciencia aberta desde distintos organismos e in- stituci´ons, como pode ser a Fundaci´onEspa˜nola para la Ciencia y la Tecnolog´ıa, que aspira a po˜ner en valor o labor das universidades, investigadores/as e emprendadores/as. Tr´atase dun debate moi interesante e intenso onde conv´endistinguir ciencia aberta de publicar en aberto e ter presente a cada vez maior presenza de determinadas revistas e editoriais que non cumpren os est´andares b´asicos de calidade. Consid´erase un novo enfoque do proceso cient´ıfico, xa que a ciencia aberta est´abaseada no traballo colaborativo e na dispo˜nibilidade e intercambio de co˜necemento pola comunidade cient´ıfica, a sociedade e as empresas, coa finalidade de aumentar o seu reco˜necemento e o seu (potencial) impacto social e con´omico. Como sinalaba, a ciencia aberta (e en xeral o co˜necemento aberto) ´ealgo m´ais que o acceso aberto a datos e publicaci´ons, e incl´ue a apertura do proceso cient´ıfico a todos, reforzando o concepto de responsabilidade social e cient´ıfica. Neste sentido, como ap´endices da tese est´anpublicados os c´odigos que per- miten facer as simulaci´ons num´ericas dos distintos modelos propostos cun c´odigo libre, tanto para a COVID como para a encefalite xaponesa. Todos os artigos aceptados para publicaci´on foron publicados no repositorio arxiv.org

Agradecementos – Acknowledgements

I express my gratitude to my parents for their love, prayers and support, I say “may Allah care for you as you cared for me when I was young”. I am also grateful to my entire family especially my sisters and brothers. I would like to express my gratitude to my supervisors, Prof. Iv´anArea and Prof. Delfim F. M. Torres, for their endless support, advices, availability and their work to achieve this essay. I say, may Allah give you long life and good health.

Contents

Chapter 1. Introduction 3 1. A brief review on mathematical models 4 2. A brief review on optimal control measures 5

Chapter 2. Mathematical background 7 1. Mathematical modelling 7 2. Mathematical analysis 9 3. Optimal control 12

Chapter 3. Ebola model and optimal control with vaccination constraints 19 1. Introduction 19 2. Initial mathematical model for Ebola 21 3. Mathematical model for Ebola with vaccination 23 4. Optimal control with an end-point state constraint 24 5. Optimal control with a mixed state control constraint 26 6. Numerical simulations 28 7. Discussion 34

Chapter 4. Mathematical modeling of COVID-19 dynamics with a case study of Wuhan 37 1. Introduction 37 2. The Proposed COVID-19 Compartment Model 37 3. Qualitative Analysis of the Model 38 4. Sensitivity Analysis 41 5. Numerical Simulations: The Case Study of Wuhan 42 6. Discussion 43

Chapter 5. Some models related to mosquitoes, water and environments 45 1. Mathematical modelling of Zika disease 46 2. Modeling of Japanese Encephalitis under Aquatic Environmental E↵ects 58

Appendix A. SageMath code to compute R0 and its sensitivity indexes of Chapter 4 67

Appendix B. Matlab code for the Wuhan case study of Chapter 4 71

Appendix C. Python code for the Japanese encephalitis simulations 75

Appendix. Bibliography 79

1

CHAPTER 1

Introduction

Mathematical or epidemiological modeling can be seen as fundamental techniques that support the understanding of the spread of infectious diseases in populations. It plays an important role in analyzing, predicting, controlling potential outbreaks [9, 10, 32, 105].

In this PhD thesis, we study dynamics of some infectious diseases from a double mathematical point of view: modelization and control. In modelization, we are concerned by formulating new mathematical models and combining them with existing ones in order to analyze the dynamics of diseases, some of them transmitted by mosquitoes, water or environment.

In control theories, we are interested in the study of measures that will help to combat the progression of the diseases. Here, we will emphasize either on the notion of basic reproduction number which quantifies the transmission potential of a disease; or on the notion of optimal control that deals with the problem of finding the best control strategy for a given dynamics of a disease.

It might be interesting to notice that, the su↵ering of people from infectious diseases surpasses the individual problems but goes to collectives, countries and even the whole world. It impacts the social living and thus the economic by keeping children away from school and adults away from work. In fact, nowadays with the novel coronavirus (COVID-19), many countries are in lockdown putting their citizens away from their activities [68, 76]. The dissertation is organized as follows: We begin by chapter 2, mathematical background related to our work. We focus on three points: mathematical modeling, dynamical systems and optimal control theory. Mainly, in order to highlight the novelty of our work and the related mathematical models proposed, we begin by recalling the basic and simple models that are assumed to be the foundations of this field of mathematical epidemiology. Then, we emphasize on the mathematical analysis tools with close view on dynamical systems. We conclude the chapter, by optimal control tools and precisely by stating the Maximum Principle of problems related to our study. In Chapter 3, we firstly consider a mathematical model for the 2014 Ebola outbreak and introduce vaccination of susceptible individuals with the aim of controlling the spread of the disease. Then, analyze, two optimal control problems related to limitation on the total number of available vaccines. This is mathematically known as state and control constraints optimal control problem and represents a public health problem of limited total number of vaccines. Lastly, we present some numerical experiments for the scenarios: unlimited supply of vaccines; limited total number of available vaccines; and limited supply of vaccines at each instant of time. Chapter 4 is devoted to the novel pandemic COVID-19. We propose a compartmental mathematical model for the spread of the COVID-19 disease with special focus on the transmissibility of superspreaders individuals. We compute the basic reproduction number threshold, we study the local stability of the disease free equilibrium in terms of the basic reproduction number, and we investigate the sensitivity of the model with respect to the variation of each one of its parameters. Numerical simulations show the suitability of the proposed COVID-19 model for the outbreak that occurred in Wuhan, China. At the end, we extend the study by considering the reality of Galiza, Spain and Vigo.

3 4 1. INTRODUCTION

We end by chapter 5, in which we emphasize on diseases related to water. Thus, we propose and study two di↵erent models: the first one is about the Zika outbreak occurred in Brazil with a special attention to the transmission of microcephaly to newborn babies. And the second model is on a study of a Japanese encephalitis diseases from multiples source of transmission with a focus on an environmental discharge. For the Zika disease, we propose a model that includes human and mosquito compartments with an aquatic phase of mosquitoes. The novelty of our work is that the human populations is restricted to women individuals with four epidemiological states: susceptible pregnant women; infected pregnant women; women who gave birth to babies without neurological disorder; and women who gave birth to babies with neurological disorder due microcephaly. This restriction on populations appears to be useful to estimate the number of newborn babies with microcephaly as it has been shown in our published paper [92]. Regarding the Japanese encephalitis disease, the proposed model consider environmental factors within three di↵erent host populations: human, mosquito and vertebrate animals (pigs, wading birds) interconnected with the aquatic phase of mosquitoes. Our main contribution is related to considering environmental discharge e↵ect on that aquatic phase of mosquitoes. Our results show that infected populations decrease highlighting the importance of considring environmental e↵ect on the aquatic phase as primarily source of reproduction of mosquitoes. Now we present a brief state of the art on mathematical models and optimal control theories.

1. A brief review on mathematical models Mathematical epidemiology models goes back to the work of Daniel Bernouilli on inoculation against Small pox. In fact, in 1760, he formulated and studied a model for Small pox. Later, at the beginning of twentieth century, mathematical epidemiology based on compartmental models was established, namely by Hamer in 1906. Then this subject developed as follows: on the one hand, by a sequence of three papers by Kermack and McKendrick in late 1920s after a seminal work [59, 60, 61], and on the other hand by a remarkable work of R. Ross in the context of malaria modeling in early 1900s. In the study of compartmental models, the population under study is divided into compartments or classes according to the disease status of each single individual in the population. The simplest and most used model is the so-called SIR (Susceptible class, Infected class, and recovered class) model, an extension is SEIR (Susceptible class, Infected class, Exposed class and recovered class) model. The rates of transfer between compartments are expressed mathematically as derivatives with respect to time of the sizes of the compartments leading to a system of di↵erential equations model. Noticed that modeling infectious diseases dynamics with partial di↵erential equations, specially with age distribution has been anticipated by Lotka in early 1920s. During time many researchers as McKendrick, Norton, Fisher, Leslie, Andrewartha and Birch etc, had considered such techniques of modeling leading to the fundation of this field of studies. Disease transmission process is basically of two types: Direct contact that is between infected individuals and susceptible individuals in the commu- • nity. For example, diseases as Measles, Smallpox, Ebola, etc. Indirect contact that is due to the presence of carriers in the environment. Either through • mosquitoes for vector borne diseases like Zika, Dengue, Chikungunya, Malaria etc, or through water for water-borne diseases like cholera, typhoid, giardiasis, etc, or through air for airborne diseases like Tuberculosis etc. Several mathematical models have been proposed and studied for infectious diseases transmision. For direct contact transmission, we may refer to recent works on 2014 Ebola outbreak [95, 132, 133]. Also, for indirect contact transmission, we may refer to recent works on Zika disease [4, 92, 89]. And very recently a compartmental mathematical model for the spread of the COVID-19 disease with special focus on the transmissibility of super-spreaders individuals has been analyzed [93]. In this thesis, for the 2. A BRIEF REVIEW ON OPTIMAL CONTROL MEASURES 5 diseases related with water we will consider models that involve both human and mosquito populations interconnected by an aquatic phase. Some years ago very few people was considering that mathematical modelling could have so huge interest and relations with economy. Nowadays, it seems that the analysis of infectious diseases, even of not widely spread at this time, could be crucial in future pandemics.

2. A brief review on optimal control measures Optimal control, the theory of mathematics that deals with the problem of finding the best control strategy for a given system, is an extension of the Calculus of Variations, born in 1696 after Johann Bernouilli’s work, proposing solution to the Brachistochrone problem. Furthermore, Newton, Leibniz and L’Hˆopital gave correct solutions to the Brachistochrone problem. However, it is in fact, with the formulation of the celebrated Pontryagin Maximun principle [100] that the theory of Optimal control emerged and gained much interest through use of computer simulations, which are helpful for plotting and analyzing the optimal trajectories. Numerical schemes are indeed valuable tools for testing theories, assessing quantitative conjectures, and we aim through this thesis to investigate numerical method for computing optimal solutions for some given problem and their optimal states. The method of optimal control is to extremize (minimize or maximize) a (cost) functional subject to a state dynamical system and boundaries conditions. In the theory of mathematical models for the spread of infectious disease in populations, when a disease invade a community, it is necessary to look for control measures as well as studying optimal control strategies in order to combat the progression of the disease. there is a threshold quantity denoted universally now by R0, named earlier by Mac Donald [79] as basic reproduction number which quantifies the transmission potential of a disease. If the basic reproduction number falls below one (R0 < 1), i.e. the infective may not pass the infection on during the infectious period, the infection dies out. On the other hand, if R0 > 1 the disease persists in the populations and it is urgent to react by considering control measures. In [11], for the first time on the Ebola literature, they were investigated and analyzed two optimal control problems related with the transmission of the 2014 Ebola outbreak with vaccination constraints. Nowadays, water-borne diseases are of great concern as they are related to water quantity and/or quality issues and could be a↵ected by climate change. Also, everyday there is an estimated of 4500 deaths occurring among children under the age of 14 in the world due to water-borne diseases like Cholera, Typhoid, Giardiasis, Shigella, etc. Recall, the notable work of John Snow since 1855 [119], about the Broad Street Cholera outbreak in London, which suggested that consumption of contaminated water by susceptible individuals is the the main source of Cholera infection, not as was believed at that period through air by fecal-oral transmission. Recent literature about water borne diseases can be found in [81, 88] and references therein. Moreover, optimal control is a beautiful subject of studies, with many applications in epidemic models, see for instance [108, 109, 30, 116].

CHAPTER 2

Mathematical background

In this chapter, we present the mathematical concepts, tools and definitions that are used in this thesis. Mainly, in order to highlight the novelty of our work and the related mathematical models proposed, we begin by recalling the basic and simple models that are assumed to be the foundations of the field of mathematical epidemiology. Then, we emphasize on the mathematical analysis tools with close view on dynamical systems. We conclude the chapter, by optimal control tools and precisely by stating the Maximum Principle of problems related to our study.

1. Mathematical modelling 1.1. The SIR and SEIR models. The classical SIR (Susceptible–Infected–Recovered) and SEIR (Susceptible–Exposed–Infected–Recovered) allow the determination of critical threshold for a disease to occur or persist in a population and during a relatively short period of time. The SIR model is described by the following ODE system:

dS(t) I(t) = S(t), dt N 8 >dI(t) I(t) (1) > = S(t) I(t), > dt N <> dR(t) = I(t), > dt > > where the parameter is the e↵ective: contact rate of infective individuals and susceptible, and stands for recovery rate, and N the constant total population size. The SEIR model extends the basic SIR model in the sense that an extra compartment is introduced, precisely the so-called Exposed compartment E¯. The Exposed compartment E¯ characterize individuals who are infected but not yet capable to transmit the virus to non infected individuals. The ODE system describing the evolution of individuals and transmission of the virus is the following one:

dS¯(t) I¯(t) = S¯(t), dt N 8 ¯ ¯ >dE(t) I(t) ¯ > = S ✏E(t), > dt N (2) > >dI¯(t) <> = ✏E¯(t) I¯(t), dt > >dR¯(t) > = I¯(t). > dt > > Where is the e↵ective contact rate: of infective individuals and susceptible, ✏ > 0 is the rate constant at which exposed individuals become infectious, > 0 the rate constant that infectious individuals become recovered.

7 8 2. MATHEMATICAL BACKGROUND

I(t) In both models the term S(t) represents the force of infection a disease at each instant of N time t. This term known as mass action law had its origin in chemical reaction kinetics and describes the transmission process due to the contacts between susceptible and infected individuals. Moreover, the Law of Mass Action has applicability in many areas of science. In chemistry, it is called Fundamental Law of Chemical Kinetics (the study of rates of chemical reactions), formulated in 1864–1879 by the Norwegian scientists Cato M. Guldberg (1836–1902) and Peter Waage (1833– 1900). The law states that for a homogeneous system, the rate of any simple chemical reaction is proportional to the probability that the reacting molecules will be found together in a small volume (see for instance [128] and references therein). Applied to population processes, if the individuals in population mix homogeneously, the rate of interaction between two di↵erent cohorts of the population is proportional to the product of the numbers in each of the cohorts concerned. If several processes occur simultaneously, then the e↵ects on the numbers in any given cohort from these processes are assumed to be additive. Therefore in case of epidemic modeling, the law is applied to rates of transition of individuals between two interacting compartments of the population (e.g. susceptibles who become infectives after an adequate contact). In brief, the rates of transfer between compartments are expressed mathematically as derivatives with respect to time of the sizes of the compartments leading to a system of di↵erential equations model [21, 52, 53]. However, the complexity in the transmission of a virus due to the existence of several source of infection makes the original SIR model not suitable for complex diseases. Taking this fact into consid- eration, it is clear that for complex diseases, it is necessary to pay more attention on the formulation of the related force of infection. For instance, we have studied in this thesis (see Chapter 3 and Chapter 4), two di↵erent complex models having multiple source of infection: A model for Ebola transmission dynamics consisting of 8 mutually exclusive compartments; • This model consider 4 di↵erent infectives classes as source of infection to formulate the force of infection of ebola. A model for corona disease consisting of 7 mutually exclusive compartments. Its consider 3 • infectives classes as source of infection to formulate the related force of infection. Besides the Law of Mass Action, there exists other form of techniques modeling the transmission dynam- ics of infectious diseases in accordance with the pathogenic agents. Next, we will consider transmission dynamics from two di↵erent species (mosquitoes and human population): mosquito-borne diseases.

1.2. The Ross-MacDonald type SIS model. The Ross-Macdonald model is a basis foundation for a broader theory of mosquito-borne disease transmission and control [65, 79, 83, 111, 117]. This kind of model consider two set of ordinary di↵erential equations: one for human populations and the other for mosquito populations. For example, to highlight both categories of populations, a basic Ross- MacDonald model can be label as ShIhSh-SmIm, where the human populations is referred by using h and the mosquito populations is referred by using m. Thus the model consists of 4 compartments with 2 compartments for humans that is ShIh(susceptible human–infected human), and 2 compartments for mosquitoes SmIm( susceptible mosquito–for infected mosquito). The classical Ross-MacDonald SIS type model is described by the following ordinary di↵erential equations:

dS I h = µ N B m S + r I µ S , dt h h mh N h h h h h 8 m >dIh Im > = Bmh Sh (rh + µh)Ih, > dt Nm (3) > >dSm Ih <> = µmNm Bhm + µm Sm, dt Nh ⇣ ⌘ >dIm I > = Bhm Sm µmIm, > dt N > > ⇣ ⌘ :> 2. MATHEMATICAL ANALYSIS 9 where Nh and Nm denote the total human population and total mosquito population respectively. The rest of the parameters are described in table 1.

Parameter Description

1/µh average lifespan of humans (in days) Proportion of bites that produce infection in human B average number of bites (per day)

mh transmission probability from Im (per bite)

1/rh the average duration of infection (in days)

1/µm average lifespan of adult mosquitoes (in days)

hm transmission probability from Ih (per bite)

Table 1. Parameters description of model (3)

Im Ih Notice that the terms Bmh Sh and Bhm Sm represent the forces of infection from Nm Nh mosquitoes to human and from human to mosquitoes respectively. They are known as incidence terms, and measure the rate at which individuals in the human populations get infected and vice versa. An incidence term refers to the number of new cases per unit time and depends on the biological behavior of mosquitoes as well on the product of the densities of the susceptible and infectives individuals [52]. Mathematical models for mosquito-borne diseases appear to be complex in general, due to complex behavior of mosquitoes. It is shown that in order to obtain their blood meal, mosquitoes are capable to locate humans [85] and can exhibit a varied biting habit: a daily biting (as for Dengue or Zika disease) or a nocturnal biting (as for malaria). In some cases where there is an adequate protective measures through use of impregnated mosquito nets, or application of protective gel or other means of protection to avoid mosquito bites, mosquitoes (in particular malaria vector) may look for alternatives ways to feed from animals such as cattle [77, 90]. In this direction, some mathematical models might incorporate this latter information leading to a more complex models with three di↵erent populations in the transmission process (see for instance [63, 97]). Moreover, the life cycle of a mosquito is usually made up of 4 distinct stages: egg, larva, pupa and adult. The egg, larva and pupa stages are aquatic, while the adult stage is terrestrial. The aquatic phase is usually associated to a breeding site where the female mosquito lays its eggs. Stagnant water reservoir such as bad drainage systems, depressions in the soil are commonly used by the female mosquito [118]. Finally, mathematical models of infectious diseases can be expressed in terms of di↵erential equa- tions. The main reason is that the sizes of compartments change with respect to time (and with respect to space too in some cases), when an individual is transferred from one compartment to another due to the transmission dynamics of a disease. Therefore, a good understanding and interpretation of mathe- matical models might require the theory of dynamical systems as background. This will be addressed in the next section.

2. Mathematical analysis Here we briefly review some properties of non-linear dynamical systems described by di↵erential equations of the form ˙ n (4) X(t)=F (X(t)),X(t0)=0, where X, F R . 2 Notice that due to nonlinearities, many di↵erential equations can not be solved analytically. As conse- quence, we emphasize on the qualitative analysis in order to acquire insights into the dynamical behavior of the system. In general, qualitative analysis includes determination and classification of equilibrium 10 2. MATHEMATICAL BACKGROUND points, study of the stability, appearance and disappearance of these equilibrium points, and transitions between them as a system parameter is varied. 2.1. Well posedness of a system. The notion of well posedness of a system coincide with the notion of positively invariance of a set (see definition 2.2) in the context of applied mathematics. The next theorem introduces the concept of flow of a dynamical systems which is useful to show that solutions of (4) are feasible, that is attracted to a certain invariant region.

n n 1 n Theorem 2.1. Let F : R R be a vector field. For each initial condition X0 R the ! C 2 di↵erential equation X˙ = F (X) has a unique solution, which shall be denoted by X(t). Therefore, X(t0)=X0,andX˙ (t)=F (X(t)).Theflow n n ' : R R R ⇥ ! of X˙ = F (X) —or of F — is defined by

'(t, X0)=X(t). As a consequence, the defining properties of ' are given by d '(t ,X )=X , '(t, X )=F ('(t, X )), t. 0 0 0 dt 0 0 8 The time-t map of the flow is the map n n 't : R R ! defined by

't(X0)='(t, X0).

For any initial condition X0 of the ODE, the time t-map gives the information about the state of the system after t units of time [55]. Remark that, for a linear mapping F (X)=AX,whereA is a n n matrix, the flow is ⇥ 't(X)=exp(tA)X.

And it is clear that, at each time-t, the map 't is a linear map. Definition 2.2. Let ' (X),t 0 be the flow of a di↵erential system X˙ = F (X). We say a set { t > } ⌦ is positively invariant (or the dynamical system (4) is well posed) if for each X ⌦, ' (X)isdefined 2 t and in ⌦ for all t > 0. 2.2. Stability solution of nonlinear dynamical system.

n Definition 2.3. (Stable solution) A solution X⇤(t) R of the system (4) is said to be stable if 2 n given ✏ > 0, (t0, ✏) > 0 such that for any neighboring solution X(t) R ,wehave 9 2 X(t ) X⇤(t ) < X(t) X⇤(t) < ✏, t t , k 0 0 k )k k 8 > 0 otherwise, X⇤(t) is said to unstable [56]. In particular, stability of constant solutions of the system (4) is crucial in the theory of dynamical systems. Indeed, constant solutions of the system (4) are its fixed points solution. Also called equilibrium points solution or steady state solution and can be defined as in the following

n Definition 2.4. (Equilibrium point) A point X⇤ R is called an equilibrium point of (4), if 2 F (X⇤)=0.

Similarly, if X(t0)=X⇤,thenX(t)=X⇤ for all t > t0. Thus solution curves starting at X⇤ remain there and it is possible give an indication of the long-term behaviour of the system [7, 123]. The mathematical properties defining the stability and unstability of an equilibrium can be easily obtained from definition 2.3. 2. MATHEMATICAL ANALYSIS 11

Remark 1. With respect to definition 2.3, we have the following The stability and unstability of equilibrium points are easily obtained from definition 2.3. • Clearly, an equilibrium point X⇤ is said to be locally stable if given ✏ > 0, (✏) > 0 such 9 that X(t ) X⇤ < X(t) X⇤ < ✏, t t , k 0 k )k k 8 > 0 otherwise, X⇤ is said to be unstable; This means that, if the initial values X(t0)issuciently close to X⇤ then the solution X(t) remains close to X⇤ for all t > 0. In addition, an equilibrium point X⇤ is said to be locally asymptotically stable if it is locally • stable and there exists 1 > 0 such that

X(t0) X⇤ < lim X(t)=X⇤. t k k ) !1 As it has been mention, from the study of equilibrium points of a system it is possible to give an indication on the long-term behavior of a system. This simply because, stable solutions tend to remain closer to the equilibrium point with respect to time evolution. In the next paragragh we will recall an important theorem in the theory of nonlinear dynamical system which is give criteria for checking stability. However, for more and extensive study on equilibrium points, their classification and stability we refer to [7, 120].

Theorem 2.5. (Local stability) Let X⇤ be an equilibrium point of the di↵erential system (4).Assume n that the function F is continuously di↵erentiable on a neighborhood of X⇤ denoted by D R . Let the ⇢ Jacobian matrix A at X⇤ be @F (5) A = , @X |X=X⇤ such that the linearized system is du = Au, u = X X⇤. dt Therefore, we have

a) X⇤ is asymptotically stable if ( (A)) < 0 for i =1, ,n; Re i ··· b) X⇤ is locally unstable if ( (A)) > 0 for at least one i. Re i Where ( (A)) designates the real part of the i th eigenvalue of the linearized system A. Re i It’s become clear that the study of local stability through the linearized system defined by the Jacobian matrix (5), is similar to the study of the eigenvalue problem of A. Hence, the Routh Hurwitz [43, p. 226] criteria is fundamental for this purpose.

Theorem 2.6 (Routh-Hurwitz criteria). Given the monic polynomial n n 1 P ()= + a1 + + an 1 + an, ··· where the coecients ai are real constants, i =1,...,n, define the n Hurwitz matrices using the coe- cients ai of the characteristic polynomial a 10 a 1 1 H = a ,H= 1 ,H= a a a , 1 1 2 a a 3 0 3 2 11 ✓ 3 2◆ a a a 5 4 3 @ A and a 100 0 1 ··· a a a 1 0 0 3 2 1 ··· 1 H = a5 a4 a3 a2 0 , n B ··· C B . . . . . C B . . . . . C B ··· C B 0000 anC B ··· C @ A 12 2. MATHEMATICAL BACKGROUND where aj =0if j>n.AlltherootsofthepolynomialP () are negative or have negative real part if and only if the determinants of all Hurwitz matrices are positive

det Hj > 0,j=1, 2,...,n.

In the particular, the Routh-Hurwitz criteria for polynomials of degree n =2, 3, 4, 5 are summarized below.

(1) If n = 2, then a1 > 0 and a2 > 0; (2) If n = 3, then a1 > 0,a3 > 0, and a1a2 >a3; 2 2 (3) If n = 4, then a1 > 0,a3 > 0,a4 > 0, and a1a2a3 >a3 + a1a4; 2 2 (4) If n = 5, then ai > 0 for i =1, 2, 3, 4, 5, we have a1a2a3 >a3 + a1a4, and (a a a )(a a a a2 a2a ) >a (a a a )2 + a a2 1 4 5 1 2 3 3 1 4 5 1 2 3 1 5 . The Routh-Hurwitz conditions can be sometimes very dicult to apply, usually when n > 3. There exists another method which reduces the complexity of the computation in Routh-Hurwitz criteria. It is a corollary known as Li´enard-Chipart test[44, 74].

3. Optimal control Optimal control is a mathematical theory that emerged after the Second World War with the for- mulation of the celebrated Pontryagin maximum principle, responding to practical needs of engineering, particularly in the field of aeronautics and flight dynamics [100]. In the last decade, optimal control has been largely applied to biomedicine, namely to models of cancer chemotherapy (see, e.g., [69]), and recently to epidemiological models [40, 84, 116]. The method of optimal control is to extrem- ize (minimize or maximize) a (cost) functional subject to a control dynamical system and boundaries conditions. In this direction and in particular in the field of epidemiology, large savings in cost could be obtained with small improvements of control strategies to prevent or to combat the progression of disease. However, at first sight on optimal control problems, there is a need to pay much more attention to the mathematical modelization and formulation of the problem.

3.1. Mathematical formulation of optimal control problems. Regarding fixed time optimal control problems, we might distinguish three major mathematical formulations: Bolza form, Lagrange form and Mayer form. We begin by considering the general Bolza form problem:

b Maximize l (x(b),b)+ a L (t, x(t),u(t)) dt, 8 R >subject to, > > > x˙(t)=f (t, x(t),u(t)) ,a.et[a, b], (BP) > 2 > <> x(a)=xa, > u( ) U, > · 2 > > x(b) E. > 2 > n > m Here x( ) R is the state:> variable, u( ) U R is the control. The interval [a, b] is fixed and xa is · 2 · 2 ⇢ the initial state. The function f :[a, b] Rn U Rn describes the system state dynamics with U a ⇥ ⇥ ! closed subset of Rm. It is assumed that the state function x :[a, b] Rn is an absolutely continuous m ! 1 function and the control function u :[a, b] R belong to a certain space ( can be L1,L ,C,PC). ! U U 3. OPTIMAL CONTROL 13

Furthermore, the functional L :[a, b] Rm U R is the running cost (or the Lagrangian), and ⇥ ⇥ ! l : E b R,withE Rn closed, is the terminal cost and specify and end-point constraint. Note ⇥ { } ! ✓ that the functional b (6) J(u)=l (x(b),b)+ L (t, x(t),u(t)) dt Za to be maximize is called the payo↵ or cost functional. The aim of this problem is to find the optimal pair (x⇤,u⇤) solution, that is satisfying the control system and all the constraints of the problem (BP), and minimizing the cost. In particular, any pair (x, u) satisfying the control systemx ˙(t)=f (t, x(t),u(t)) is called a process, and a process satisfying all the constraints of (BP) is called an admissible process. Next, the Lagrange form is obtained as special form of Bolza problem (BP), when the terminal cost is identically zero (l 0). This problem and of course its name come from Calculus of Variations. ⌘ We can reformulate Bolza problem into Lagrange form. Given a terminal cost l, we can write:

b d l (x(b),b)=l(x ,a)+ l (x(t),t) dt a dt Za b = l(xa,a)+ lt (x(t),t)+lx (x(t),t) f (t, x(t),u(t)) dt. a · Z h i Since l(xa,a) is a constant independent of u, we get an equivalent problem in the Lagrange form

b Maximize l (x(t),t)+l (x(t),t) f (t, x(t),u(t)) + L (t, x(t),u(t)) dt + l(x ,a), t x · a 8 Za > h i >subject to, > > > x˙(t)=f (t, x(t),u(t)) ,a.et[a, b], (LP) > 2 > <> x(a)=xa, > u( ) U, > > · 2 > > x(b) E. > 2 Lastly,> the Mayer form is obtained also as special form from Bolza problem when the running cost :> is identically zero (L 0). By state augmentation, let us define: ⌘ y˙(t)=L (t, x(t),u(t)) ,y(a)=0. Therefore, the problem (BP) can be rewrite as follows:

Maximize l (x(b),b)+y(b), 8 >subject to, > > > x˙(t)=f (t, x(t),u(t)) ,a.et[a, b], > 2 > (MP) > y˙(t)=L (t, x(t),u(t)) ,a.et[a, b] > <> 2 x(a)=xa,y(a)=0, > > > u( ) U, > · 2 > > x(b) E. > 2 > For more studies on the:> transformation of optimal control problems from one to another one, we refer readers to [23, 73]. In fact, all the problems we have discussed here are of fixed time optimal control problems as the time interval [a, b] is fixed. But, by passing of time and development of the theory, several variants of optimal control problems appear in literature. 14 2. MATHEMATICAL BACKGROUND

3.2. Contraints problems. Constraints optimal control problems are frequently encountered in modeling real-life problems. For example, in epidemiology, there may be insucient resources (in number of vaccines or heath care workers) to vaccinate all the susceptible individuals within a relatively short period of time compared to the course of an outbreak. In such situations and for a better assignment of resources, it is suitable to consider limitations on the number of vaccines available and on the capacity of administration of the vaccines by the health care services and humanitarian teams working in the a↵ected areas (see for instance [11], where such problems have been studied in details). Thus, it is important to consider restrictions or limitations on resources in order to deal with these physical and biological constraints. In practice, there exists three main constraints to fixed time optimal control problems: (1) End-points constrained problems: these constraints are usually imposed at the initial point and/or terminal point of a fixed interval [a, b]; With respect to general Bolza problem (BP), the relations x(a)=x , and x(b) E, a 2 constitute an example of an end-point constrained problem. (2) Control constrained problems: these constraints are usually imposed on the control vari- ables; The relation u( ) U, · 2 given in Bolza problem (BP), exemplifies such kind of constraints. (3) Pathways constrained problems: these constraints are usually imposed to both functions (control function as well as the associated state dynamics) with the aim to restrict the range of values taken by these functions. Note that the restriction can be to the whole interval [a, b] or to its nonempty sub-interval. In most cases, such constraints take the form of scalar functional inequality constraints and thus explicit form can be derived easily. We have for example state constraints: expressed as equality constraints relation • h (t, x(t)) = 0 for all t [a, b], 2 inequality constraints relation

h (t, x(t)) 0 for all t [a, b], > 2 or as set constraints

x(t) X for all t [a, b], 2 2 where h :[a, b] Rn Rp,p 0 and X Rn; ⇥ ! > ✓ mixed constraints: expressed as equality relation • g (t, x(t),u(t)) = 0 a.e. t [a, b], 2 inequality relation

g (t, x(t),u(t)) 0 a.e. t [a, b], > 2 or both equality and inequality relation or in a general form as

(x(t),u(t)) Ca.e.t [a, b], 2 2 where g :[a, b] Rn Rm Rp,p 0 and C Rn Rm. ⇥ ⇥ ! > ✓ ⇥ We would like to emphasize that, equality and inequality constraints relation can be rewritten as set constraints and can be generalize to include moving set [17, 30]. 3. OPTIMAL CONTROL 15

3.3. The Maximun Principle. The maximum Principle (MP) gives a necessary optimality condi- tion that every solution to an optimization problem must verify. In optimal control theory, the so-called Pontryagin Maximum Principle (PMP) is mostly considered in the derivation of necessary optimality condition. The idea behind this derivation is to obtain the smallest set of candidates solution to the original optimal control problem. For illustration, let us consider the following problem

b Maximize a L (t, x(t),u(t)) dt, 8 R >subject to, > > > x˙(t)=f (t, x(t),u(t)) ,a.et[a, b], > 2 > (OCP) > u( ) U, > <> · 2 h (t, x(t)) 6 0 t [a, b], > 8 2 > > x(b) E, > 2 > > x(a)=xa. > > We assume that in what:> follows all the data are smooth, that is functions f,L are continuously di↵erentiable, U is supposed to be a closed set and E convex and closed. Next, we define the Pseudo Hamiltonian (or the unmaximized Hamiltonian or Pontryagin) function

H(t, x, p, u)= L(t, x, u)+ p, f(t, x, u) . 0 h i (1) The Maximum Principle without state constraints. Here, we omit in the problem (OCP), the state constraints h (t, x(t)) 0 for all t 6 2 [a, b]. The next theorem gives the Maximum Principle under theses considerations ((this is an adaption of Theorem 2.5.1 in [17]))

Theorem 3.1. (PMP for (OCP) without state constraints): Let (x⇤,u⇤) be an optimal pair for the problem (OCP). Then there exist an arc p 1,1 n 2 W ([a, b]; R ) and a scalar 0 0, 1 satisfying 2 { } i) the Nontriviality condition [NT]:

(p, ) = (0; 0) 0 6 2i) the Euler Adjoint Equation [EA]:

p˙(t)= L (t, x⇤(t),u⇤(t)) + p(t),f(t, x⇤(t),u⇤(t)) a.e., 0rx rxh 3i) the global Maximality condition [W]:

u U, H (t, x⇤(t),p(t),u) H (t, x⇤(t),p(t),u⇤(t)) a.e., 8 2 6 4i) and the transversality condition [T]:

p(t)=n 2

for some n N (x⇤(b)), where N (x⇤(b)) is the normal cone to E. 2 2 E E We would like to mention that, since E Rn is closed and convex, the normal cone to E at ⇢ x⇤ E is defined by 2 n NE(x⇤)= ✏ R : ✏,x x⇤ 0 x E . { 2 h i 6 8 2 } 16 2. MATHEMATICAL BACKGROUND

(2) The Maximum Principle with state Constraints Here the novelty is an introduction of measures as multipliers. Thus, the adjoint multiplier p is replaced by a function of bounded variation defined in the following

p(t)+ a,t (s)µ(ds) t [a, b), (7) q(t)= 2 8 R <>p(t)+ a,b (s)µ(ds) t = b. Where :[a, b] Rn is a multiplier> R associated with the state constraint. Hence the Maximum ! : Principle for the state constraint optimal control problem (this is an adaption of Theorem 2.5.1 in [17]) is Theorem 3.2. (PMP for (OCP) with state constraints): Let (x⇤,u⇤) be an optimal pair for the problem (OCP). Then there exist an arc p 1,1 n 2 W ([a, b]; R ),ascalar0 0, 1 , µ C ([a, b]), and a measurable function (t):[a, b] 2 { } 2 ! Rn such that the following conditions are satisfied:L i) the Nontriviality condition [NT]: (p, µ, ) =(0, 0, 0) 0 6 2i) the Euler Adjoint Equation [EA]:

p˙(t)= L (t, x⇤(t),u⇤(t)) + q(t),f(t, x⇤(t),u⇤(t)) a.e., 0rx rxh 3i) the global Weierstrass condition [W]:

u U, H (t, x⇤(t),q(t),u) H (t, x⇤(t),q(t),u⇤(t)) a.e., 8 2 6 4i) and the transversality condition [T]: q(b)=n 2 for some n N (x⇤(b)). 2 2 E 5i supp µ I(x⇤) { } ⇢ where I(x⇤ := t : h(t, x⇤(t)) = 0 ) and q is as in (7). { }

(3) A specific Maximum Principle for problems with mixed inequality constraints Let us consider the following mixed optimal control problem with terminal condition: T Maximize 0 F (t, x(t),u(t)) dt + S (T,x(T )) , 8 R >subject to, > > > x˙(t)=f (t, x(t),u(t)) , > > (ACP) > g (t, x(t),u(t)) 0 t [0,T], > > <> 8 2 a (T,x(T )) > 0, > > b (T,x(T )) = 0, > > > x(a)=xa, > n >m n n m n n where F : R >R R R, S : R R R, f : R R R R , a : R R R and ⇥ : ⇥ ! ⇥ ! ⇥ ⇥ ! ⇥ ! b : Rn R R are assumed to be piecewise continuously di↵erentiable in all their arguments. ⇥ ! Here T denotes the terminal time. Following [115], the following version of Pontryagin’s maximum principle yields:

Theorem 3.3 (PMP for problem (ACP)). Let (x⇤,u⇤) be an optimal pair for the problem (ACP)). Then there exists continuous and piecewise continuously di↵erentiable functions , piecewise continuous functions µ and constants ↵, such that the following conditions hold 3. OPTIMAL CONTROL 17

the adjoint equation: • ˙ = L (t, x⇤,u⇤, ,µ) x the transversality condition: • (T )=Sx (T,x⇤(T )) + ↵ax (T,x⇤(T )) + bx (T,x⇤(T )) ,

↵ > 0 and ↵ax (T,x⇤(T )) = 0 the Hamiltonian maximizing condition • H (t, x⇤(t),u⇤(t), (t)) > H (t, x⇤(t),u(t), (t))

at each t [0,T] for all u satisfying g (t, u, x⇤(t)) 0, 2 > the terminal constraints • a (T,x⇤(T )) > 0 and b (T,x⇤(T )) = 0, the Lagrange multipliers µ(t) are such that • @L @H @g := + µ @u |u=u⇤(t) @u @u ku=u⇤(t) the complementary slackness conditions⇣ ⌘ • µ(t) > 0,µ(t)g(t, x⇤,u⇤)=0, where the Hamiltonian function H is defined as H (t, x, u, )=F (t, x, u)+f(t, x, u), and the Lagrangian function L as L (t, x, u, ,µ)=H(t, x, u, )+µg(t, x, u).

CHAPTER 3

Ebola model and optimal control with vaccination constraints

Ebola is a highly lethal virus with at least 18 confirmed outbreaks in Africa between 1976 and 2014. By 27th March, 2014 it started a new outbreak of Ebola in West Africa. Historically, up to 2012, about 2,400 cases and 1,600 deaths were registered due to Ebola virus(es). But in the outbreak of 2014 in West Africa we have about 28,602 confirmed cases and 11,301 deaths by report of January 20, 2016. In fact, the burden of the 2014 Ebola disease is considerable with a mortality rate varying from 50% to 90%. A new study provided strong evidence that individual genetic di↵erences play a major role in whether people die from the disease [50]. In the 2014 outbreak the mean incubation period is 11.4 days, and does not vary by country [41]. The mean time from the onset of symptoms to hospitalization, a measure of the period of infectiousness in the community, is 5.0 4.7 days, and it is ± not shorter for health care workers than for other case patients. The mean time to death after admission to the hospital is 4.2 6.4 days, and the mean time to discharge is 11.8 6.1days[41]. Note that ± ± knowledge about incubation period for deadliest disease like Ebola is crucial for public health authorities or epidemiologist. Basically, suspected people must be identified to follow some rules of quarantine type during a period nearly as the incubation period before reentering in contact with the rest of susceptible population. In addition, the subject of transmission dynamics and control of Ebola become more more attractive to many researchers from di↵erent fields. There exists approximatively, 169 papers in the literature of Ebola modelling published in international journal, with some recent papers in the domain of applied and engineering mathematics [18, 86, 102, 106]. The results presented in this chapter have been published [11].

1. Introduction Ebola is a lethal virus for humans that is currently under strong research due to the recent outbreak in West Africa and its socioeconomic impact (see, e.g., [49, 51, 58, 70, 71, 95, 99, 107, 113] and references therein). World Health Organization (WHO) has declared Ebola virus disease epidemic as a public health emergency of international concern with severe global economic burden. At fatal Ebola infection stage, patients usually die before the antibody response. Mainly after the 2014 Ebola outbreak in West Africa, some attempts to obtain a vaccine for Ebola disease have been realized. According to the WHO, results in July 2015 from an interim analysis of the Guinea Phase III ecacy vaccine trial show that VSV-EBOV (Merck, Sharp & Dohme) is highly e↵ective against Ebola [8]. Since 2014, di↵erent mathematical models to analyze the spread of the 2014 Ebola outbreak have been presented (see, e.g., [9, 10, 54, 105, 133] and references therein). In these models the popula- tions under study are divided into compartments, and the rates of transfer between compartments are expressed mathematically as derivatives with respect to time of the size of the compartments. There exist di↵erent models for the spreading of Ebola, beginning with the simplest SIR and SEIR models [8, 27, 28] and later more complex but also more realistic models have been considered [10, 70, 132]. In [71], a stochastic discrete-time Susceptible-Exposed-Infectious-Recovered (SEIR) model for infectious diseases is developed with the aim of estimating parameters from daily incidence and mortality time series for an outbreak of Ebola in the Democratic Republic of Congo in 1995. In [70], the authors use data from two epidemics (in Democratic Republic of Congo in 1995 and in Uganda in 2000) and built

19 20 3. EBOLA MODEL AND OPTIMAL CONTROL WITH VACCINATION CONSTRAINTS a SEIHFR (Susceptible-Exposed-Infectious-Hospitalized-F(dead but not yet buried)-Removed) mathe- matical model for the spread of Ebola haemorrhagic fever epidemics taking into account transmission in di↵erent epidemiological settings (in the community, in the hospital, during burial ceremonies). In [95], the authors propose a SIRD (Susceptible-Infectious-Recovered-Dead) mathematical model using classi- cal and beta derivatives. In this model, the class of susceptible individuals does not consider new born or immigration. The study shows that, for small portion of infected individuals, the whole country could die out in a very short period of time in case there is no good prevention. In [9], a fractional order SEIR Ebola epidemic model is proposed and the authors show that the model gives a good approximation to real data published by WHO, starting from March 27th, 2014. Optimal control is a mathematical theory that emerged after the Second World War with the for- mulation of the celebrated Pontryagin maximum principle, responding to practical needs of engineering, particularly in the field of aeronautics and flight dynamics [100]. In the last decade, optimal control has been largely applied to biomedicine, namely to models of cancer chemotherapy (see, e.g., [69]), and recently to epidemiological models [40, 84, 116]. In [133], the authors present a comparison between SIR and SEIR mathematical models used in the description of the Ebola virus propagation. They applied optimal control techniques in order to under- stand how the spread of the virus may be controlled, e.g., through education campaigns, immunization or isolation. In [5], the authors introduce a deterministic SEIR type model with additional hospital- ization, quarantine and vaccination components in order to understand the disease dynamics. Optimal control strategies, both in the case of hospitalization (with and without quarantine) and vaccination, are used to predict the possible future outcome in terms of resource utilization for disease control and the e↵ectiveness of vaccination on sick populations. Both in [5] and [133], the authors study optimal control problems with L2 cost functionals without any state or control constraints. Here, we modify the model analyzed in [10] in order to consider optimal control problems with vaccination constraints. More precisely, we introduce an extra variable for the number of vaccines used, and we compare the hypothetical results if the vaccine were available at the beginning of the outbreak with the results of the model without vaccines. Firstly, we consider an optimal control problem with an end-point state constraint, that is, the total number of available vaccines, in a fixed period of time, is limited. Secondly, we analyze an optimal control problem with a mixed state constraint, in which there is a limited supply of vaccines at each instant of time for a fixed interval of time. Both optimal control problems have been analytically solved. Moreover, we have performed a number of numerical simulations in three di↵erent scenarios: unlimited supply of vaccines; limited total number of vaccines to be used; and limited supply of vaccines at each instant of time. From the results obtained in the first two cases, when there is no limit in the supply of vaccines or when the total number of vaccines used is limited, the optimal vaccina- tion strategy implies a vaccination of 100% of the susceptible population in a very short period of time (smaller than one day). In practice, this is a very dicult task because limitations in the number of vaccines and also in the number of humanitarian and medical teams in the a↵ected regions are common. In this direction, the third analyzed case is extremely important since we consider a limited supply of vaccines at each instant of time. The chapter is organized as follows. In Section 2, we recall a mathematical model for Ebola virus. In Section 3, the introduction of e↵ective vaccination for Ebola virus is modeled. An optimal control problem with an end-point state constraint is formulated and solved analytically in Section 4, which models the case where the total number of available vaccines in a fixed period of time is limited. In Section 5, the limited supply of vaccines at each instant of time for a fixed interval of time is mathematically translated into an optimal control problem with a mixed state control constraint. A closed form of the unique optimal control is given. In Section 6, we solve numerically the optimal control problems proposed in Sections 4 and 5. Finally, we end with Section 7 of discussion of the results. 2. INITIAL MATHEMATICAL MODEL FOR EBOLA 21

2. Initial mathematical model for Ebola The total population N under study is subdivided into eight mutually exclusive groups: susceptible (S), exposed (E), infected (I), hospitalized (H), asymptomatic but still infectious (R), dead but not buried (D), buried (B), and completely recovered (C). This model is adapted from [57] and analyzed in [10], where the birth and death rate are assumed to be equal and are denoted by µ, and the contact rate of susceptible individuals with infective, dead, hospitalized and asymptomatic individuals are denoted by i, d, h and r, respectively. Exposed individuals become infectious at a rate . The per capita rate of progression of individuals from the infectious class to the asymptomatic and hospitalized classes are denoted by 1 and ⌧, respectively. Individuals in the dead class progress to the buried class at a rate 1. Hospitalized individuals progress to the buried class and to the asymptomatic class at rates 2 and 2, respectively. Asymptomatic individuals become completely recovered at a rate 3. Infectious individuals progress to the dead class at a fatality rate ✏. Dead and buried bodies are incinerated at a rate ⇠. We assume that the total population, N = S + E + I + R + H + D + B + C, is constant, that is, the birth and death rates, both denoted by µ, are equal to the incineration rate ⇠.Themodel is mathematically described by the following system of eight nonlinear ordinary di↵erential equations: dS (t)=µN i S(t)I(t) h S(t)H(t) d S(t)D(t) dt N N N 8 > r S(t)R(t) µS(t), > N > >dE i h d > (t)= S(t)I(t)+ S(t)H(t)+ S(t)D(t) > dt N N N > > > + r S(t)R(t) E(t) µE(t), > N > >dI > (t)=E(t) ( + ✏ + ⌧ + µ)I(t), > 1 (8) > dt >dR > (t)= I(t)+ H(t) ( + µ)R(t), < dt 1 2 3 dD > (t)=✏I(t) ( + ⇠)D(t), > 1 > dt >dH > (t)=⌧I(t) ( + + µ)H(t), > 2 2 > dt >dB > (t)= D(t)+ H(t) ⇠B(t), > 1 2 > dt >dC > > (t)=3R(t) µC(t). > dt > In Fig. 1, we give a flowchart:> presentation of model (8). In this flowchart, we identify the compartmental classes as well as the parameters appearing in the model. Moreover, the values of the parameters are given in Table 1. The basic reproduction number (that is, the number of cases one case generates on average over the course of its infectious period, in an otherwise uninfected population) of model (8) can be computed using the associated next-generation matrix method [36]. It is obtained as the spectral radius of the following matrix, known as the next-generation-matrix:

A11 A12 A13 A14 A15 00000 0 1 1 00000 FV = B C , B 00000C B C B 00000C B C B 00000C B C where @ A a3ia1a4 + a3r (a41 + ⌧2)+d✏a1a4 + a3h⌧a1 A11 = , a1a2a3a4a5 22 3. EBOLA MODEL AND OPTIMAL CONTROL WITH VACCINATION CONSTRAINTS

µN

i d 1 3 S E I R C h r µ µ µ µ ✏ ⌧ µ 2 ⇠ D H µ

1

⇠ 2 B

Figure 1. Flowchart presentation of the compartmental model (8) for Ebola.

r r(a41 + ⌧2) d✏ h⌧ A12 = + + + , a1 a1a2a3 a2a3 a2a4 r d r2 h A13 = ,A14 = ,A15 = + a1 a3 a1a4 a4 with

a1 = 2 + 2 + µ, a2 = 3 + µ, a3 = 1 + ⇠,a4 = + µ, and a5 = 1 + ✏ + ⌧ + µ.

Therefore, the basic reproduction number R0 is given by

2 R0 = a3 i(3a1 + µ(2 + 2)+µ )+r(1a1 + 2⌧)+h⌧a2 a1a2a3a4a5 2 ⇥ +d✏ (3a1 + µ(2 + 2)) + µ .

As it is well-known, if the basic reproduction number R0 < 1, then the infection will stop in the⇤ long run; but if R0 > 1, then the infection will spread in population. In this section, we have recalled a model for describing the Ebola virus transmission. Now we want to address the question about how to introduce vaccination as a prevention measure. This is analyzed in the next section. 3. MATHEMATICAL MODEL FOR EBOLA WITH VACCINATION 23

Symbol Description Value per capita rate at which exposed individuals become infectious 1/11.4 µ death rate 14/1000

i contact rate of infective and susceptible individuals 0.14

d contact rate of infective and dead individuals 0.40

h contact rate of infective and hospitalized individuals 0.29

r contact rate of infective and asymptomatic individuals 0.185

1 per capita rate of progression of individuals from the infectious class to the asymptomatic class 1/10 ✏ fatality rate 1/9.6

1 per capita rate of progression of individuals from the dead class to the buried class 1/2

2 per capita rate of progression of individuals from the hospitalized class to the buried class 1/4.6

2 per capita rate of progression of individuals from the hospitalized class to the asymptomatic class 1/5 ⌧ per capita rate of progression of individuals from the infectious class to the hospitalized class 1/5

3 per capita rate of progression of individuals from the asymptomatic class to the completely recovered class 1/30 ⇠ incineration rate 14/1000

Table 1. Parameter values for model (8), corresponding to a basic reproduction num- ber R0 =2.287. The values of the parameters come from [22, 39, 62, 70, 94, 107, 112].

3. Mathematical model for Ebola with vaccination We now introduce vaccination of the susceptible population with the aim of controlling the spread of the disease. We assume that the vaccine is e↵ective so that all vaccinated susceptible individuals become completely recovered (see, e.g., [16, 87] for vaccination in a SEIR model that corresponds to a system of four nonlinear ordinary di↵erential equations). Let us introduce in model (8) a control function u(t), which represents the percentage of susceptible individuals being vaccinated at each instant of time t with t [0,t ]. Unfortunately, in many situations, the number of available vaccines does not fulfill the 2 f necessities in order to eradicate the disease. In this paper, we consider limitations on the total number of vaccines during a fixed interval of time [0,tf ] and on the number of available vaccines at each instant of time t with t [0,t ]. In order to translate this real situation mathematically, we introduce an extra 2 f variable W that denotes the number of vaccines used:

dW (t)=u(t)S(t), subject to the initial condition W (0) = 0. dt 24 3. EBOLA MODEL AND OPTIMAL CONTROL WITH VACCINATION CONSTRAINTS

Hence, the model with vaccination is given by the following system of nine nonlinear ordinary di↵erential equations: dS (t)=µN i S(t)I(t) h S(t)H(t) d S(t)D(t) dt N N N 8 r > S(t)R(t) µS(t) S(t)u(t), > N > > >dE i h d > (t)= S(t)I(t)+ S(t)H(t)+ S(t)D(t) > dt N N N > > r > + S(t)R(t) E(t) µE(t), > N > > >dI > (t)=E(t) (1 + ✏ + ⌧ + µ)I(t), > dt > >dR (9) > (t)=1I(t)+2H(t) (3 + µ)R(t), > dt >

dt 1 > > >dH > (t)=⌧I(t) (2 + 2 + µ)H(t), > dt > >dB > (t)=1D(t)+2H(t) ⇠B(t), > dt > > >dC > (t)=3R(t) µC(t)+S(t)u(t), > dt > >dW > (t)=S(t)u(t). > dt > > In model (9), the:> vaccination parameter is fixed. In the next section, we address the question of how to choose this parameter in an optimal way along time.

4. Optimal control with an end-point state constraint We start by considering the case where the total number of available vaccines, in a fixed period of time, is limited. We formulate and solve analytically such optimal control problem with end-point state constraint, which will be then solved numerically in Section 6. We consider the model with vaccination (9) and formulate the optimal control problem with the aim to determine the vaccination strategy u over a fixed interval of time [0,tf ] that minimizes the cost functional

tf 2 (10) J(u)= w1I(t)+w2u (t) dt, Z0 ⇥ ⇤ where the constants w1 and w2 represent the weights associated with the number of infected individuals and on the cost associated with the vaccination program, respectively. We assume that the control function u takes values between 0 and 1. When u(t) = 0, no susceptible individual is vaccinated at time t;ifu(t) = 1, then all susceptible individuals are vaccinated at t. Let # denote the total amount of available vaccines in a fixed period of time [0,tf ]. This constraint is represented by

(11) W (t ) #. f  Let 9 x(t)=(x1(t),...,x9(t)) = (S(t),E(t),I(t),R(t),D(t),H(t),B(t),C(t),W(t)) R . 2 4. OPTIMAL CONTROL WITH AN END-POINT STATE CONSTRAINT 25

The optimal control problem consists to find the optimal trajectoryx ˜ associated with the optimal control u˜, satisfying the control system (9), the initial conditions

(12) x(0) = (18000, 0, 15, 0, 0, 0, 0, 0, 0)

(see [10]), the constraint (11), and where the controlu ˜ ⌦ minimizes the objective functional (10) with 2

(13) ⌦ = u( ) L1(0,t ) 0 u(t) 1 . · 2 f |   ⇢ The existence of an optimal controlu ˜ and associated optimal trajectoryx ˜ comes from the convexity of the integrand of the cost function (10) with respect to the control u and the Lipschitz property of the state system with respect to state variables x (see, e.g., [23, 42] for existence results of optimal solutions). According to the Pontryagin Maximum Principle [100], ifu ˜ ⌦ is optimal for the problem 2 9 (9), (10) with initial conditions (12) and fixed final time tf ,thenthereexists AC([0,tf ]; R ), 2 (t)=(1(t),...,9(t)), called the adjoint vector, such that @ @ x˙ = H1 and ˙ = H1 , @ @x where the Hamiltonian is defined by H1 (x, u, )=w x + w u2 + (f(x)+Ax + Bxu) H1 1 3 2 with

f =(f1 f2 0 0 0 0 0 0 0) ,

f (x)=µN i SI h SH d SD r SR, 1 N N N N f (x)= i SI + h SH + d SD + r SR, 2 N N N N

µ 00 0 0 0 00 0 µ 00 0 0 00 0 1 0 ⇤ 00 000 B C B 001 (3 + µ)0 2 00C B C A = B 00✏ 0 (1 + ⇠)0 00C B C B 00⌧ 00(2 + 2 + µ)0 0C B C B 000 0 1 2 ⇠ 0 C B C B 000 000µC B 3 C B C B 000 0 0 0 00C @ A with ⇤ = ( + ✏ + ⌧ + µ), and 1 B =(bZ) , where b =( 1 0 0 0 0 0 0 1 1)T and Z =0with0the8 9 null matrix. The minimization condition ⇥

(14) 1 x˜(t), u˜(t), ˜(t) =min 1 x˜(t),u,˜(t) H u ⌦ H ⇣ ⌘ 2 ⇣ ⌘ holds almost everywhere on [0,tf ]. Moreover, transversality conditions ˜i(tf ) = 0, i =1,...,8, hold. Solving the minimality condition (14) on the interior of the set of admissible controls ⌦ gives

˜ ˜ ˜ 1 1(t)+8(t)+9(t) x˜1(t) u˜(t)= , 2 ⇣ !2 ⌘ 26 3. EBOLA MODEL AND OPTIMAL CONTROL WITH VACCINATION CONSTRAINTS where the adjoint functions satisfy

˙ i h d r ˜ = ˜ x˜ x˜ x˜ x˜ µ u˜ 1 1 N 3 N 6 N 5 N 4 8 ✓ ◆ i h d r > ˜2 x˜3 + x˜6 + x˜5 + x˜4 ˜8u˜ ˜9u,˜ > N N N N > ✓ ◆ >˜˙ ˜ ˜ >2 = 2 ( µ) 3 , > > >˜˙ ˜ i ˜ i ˜ ˜ ˜ ˜ >3 = !1 + 1 x˜1 2 x˜1 3 ( 1 ✏ ⌧ µ) 41 5✏ 6⌧, > N N > >˜˙ ˜ r ˜ r ˜ ˜ >4 = 1 x˜1 2 x˜1 4 ( 3 µ) 83, (15) > N N > > ˙ d d <>˜ = ˜ x˜ ˜ x˜ ˜ ( ⇠) ˜ , 5 1 N 1 2 N 1 5 1 7 1 > ˙ h h >˜ = ˜ x˜ ˜ x˜ ˜ ˜ ( µ) ˜ , > 6 1 1 2 1 4 2 6 2 2 7 2 > N N > ˙ >˜7 = ˜7⇠, > > >˜˙ ˜ >8 = 8µ, > > >˜˙ >9 =0. > > ˜ Since W has:> initial and terminal conditions, the adjoint function 9, associated with the state variable W , has no transversality condition. From (15), ˜ (t) k, where the constant k must be such that the 9 ⌘ end point conditions W (0) = 0 and W (tf )=# are satisfied. As the optimal controlu ˜ can take values on the boundary of the control set [0, 1], the optimal controlu ˜ must satisfy ˜ ˜ ˜ 1 1(t) 8(t) 9(t) x˜1(t) (16) u˜(t)=min 1, max 0, . 8 8 2 ⇣ !2 ⌘ 99 < < == The optimal controlu ˜ given by (16): is unique: due to the boundedness of the state;; and adjoint functions and the Lipschitz property of systems (9) and (15). We would like to note that if we consider the optimal control problem without any restriction on the number of available vaccines, that is, to find the optimal solution (˜x, u˜), withu ˜ ⌦,which 2 minimizes the cost functional (10) subject to the control system (9), initial conditions (12), and free final conditions (x1(tf ),...,x9(tf )), then the adjoint functions (1,...,9) must satisfy transversality conditions i(tf ) = 0, i =1,...,9, and, since ˜9 = 0, the optimal control is given by ˜ ˜ 1 1(t) 8(t) x˜1(t) u˜(t)=min 1, max 0, . 8 8 2 ⇣ !2 ⌘ 99 < < == In a concrete situation, the number: of available: vaccines is always; limited.; Therefore, it is also important to study the optimal control problem with such kind of constraints. This is done in Section 5. Both problems, with and without constraints, are numerically solved in Section 6. 5. Optimal control with a mixed state control constraint A particularly challenging situation in vaccination programs happens when there is a limited supply of vaccines at each instant of time for a fixed interval of time [0,tf ]. In order to study this health public problem, from the optimal point of view, we formulate an optimal control problem with a mixed state control constraint (see, e.g., [16]). The cost functional (10) remains (10), the one considered in previous section, as well as the set of admissible controls ⌦ (13). The end point state constraint (11) is replaced by the following mixed state control constraint: S(t)u(t) # , # 0 , for almost all t [0,t ],  2 f 5. OPTIMAL CONTROL WITH A MIXED STATE CONTROL CONSTRAINT 27 which should be satisfied at almost every instant of time during the whole vaccination program. Anal- ogously to [16], we observe that in our optimal control problem the di↵erential equation dW (t)=S(t)u(t) dt does not appear neither in the cost and in any other di↵erential equation, nor in the mixed state control constraint. Thus, in this section, the control system does not include the last equation and x is used to denote 8 x(t)=(x1(t),...,x8(t)) = (S(t),E(t),I(t),R(t),D(t),H(t),B(t),C(t)) R . 2 Let us consider the initial conditions (12). The control system can be rewritten in the following way: dx(t) = f(x(t)) + Ax(t)+Bx(t)u(t), dt with µ 00 0 0 0 00 0 µ 00 0 0 00 0 1 0 ⇤ 00 000 B C B 001 (3 + µ)0 2 00C A = B C , B 00✏ 0 (1 + ⇠)0 00C B C B 00⌧ 00(2 + 2 + µ)0 0C B C B 000 0 ⇠ 0 C B 1 2 C B 000 000µC B 3 C @ A ⇤ = ( + ✏ + ⌧ + µ), 1 B =(bZ) , where b =( 1 0 0 0 0 0 0 1)T and Z =0with0the7 8 null matrix, and ⇥ f =(f1 f2 0 0 0 0 0 0) , with f (x)=µN i SI h SH d SD r SR 1 N N N N and f (x)= i SI + h SH + d SD + r SR. 2 N N N N It follows from Theorem 23.11 in [29] that problem (9)–(13) has a solution (see also [16]). Let (˜x, u˜)de- note such solution. To determine it, we apply the Pontryagin Maximum Principle (see, e.g., Theorem 7.1 8 1 in [30]): there exists multipliers 0 0, AC([0,tf ]; R ), and L ([0,tf ]; R), such that  2 2 min (t) : t [0,tf ] > 0 (nontriviality condition); • d(t){| |@ 2 } = H2 (˜x(t), u˜(t), , (t), (t)) (adjoint system); • dt @x 0 (t)Bx˜(t)+ (t)˜x (t)+ w u˜2(t) (˜u(t)) a.e. and • 1 0 2 2 N[0,1] (˜x(t), u˜(t), , (t), (t)) (˜x(t),v, , (t), (t)), v [0, 1] :x ˜ (t)v # H2 0  H2 0 8 2 1  (minimality condition); (t)(˜x (t)˜u(t) #) = 0 and (t) 0 a.e.; • 1  (t )=(0,...,0) (transversality conditions); • f where the Hamiltonian for problem (9)–(13) is defined by H2 (x, u, , , )= w x + w u2 + (f(x)+Ax + Bxu)+ (Su #), H2 0 0 1 3 2 and (˜u(t)) stands for the normal cone from convex analysis to [0, 1] at the optimal controlu ˜(t) N[0,1] (see, e.g., [29]). The optimal solution (˜x, u˜) is normal (see [16] for details), so we can choose 0 = 1. 28 3. EBOLA MODEL AND OPTIMAL CONTROL WITH VACCINATION CONSTRAINTS

Analogously to previous section, we obtain a closed form of the unique optimal controlu ˜: ˜ ˜ 1 1(t) 8(t) (t) x˜1(t) u˜(t)=min 1, max 0, . 8 8 2 ⇣ !2 ⌘ 99 < < == The theoretical results obtained: in Sections: 4 and 5 are illustrated numerically;; in the next section. 6. Numerical simulations We start the numerical simulations by considering an intervention of 90 days, initial conditions given in (12), and the evolution of cumulative confirmed cases based on the data from the World Health Organization (WHO), following all the reports of the disease in the three main a↵ected countries of Western Africa of the 2014 Ebola outbreak, namely, Liberia, Guinea and Sierra Leone. The model (8) with the parameter values from Table 1 fits the real data from WHO, see Fig. 2a. We would like to emphasize that we are just considering the initial period of spreading of the disease, in which the vaccination should be introduced. Looking to a longer period of time (as considered in [10]), then the model fits quite well the real data: in [10, Figure 2] the `2 norm of the di↵erence between the real data and the prediction is 3181, which gives an error of less than 7.3 cases per day, as compared with about 15,000 cases at the end of the outbreak.

900 16

800 14

700 12 600

500 10

400 8

300 6 Cumulative confirmed cases Cumulative confirmed cases 200

4 100

0 2 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 time (days) time (days)

(a) Real data and solution of (8) (b) Unlimited supply of vaccines

Figure 2. (a) Cumulative confirmed cases: in dashed circle line the real data from WHO and in continuous line the values of I(t)+R(t)+D(t)+H(t)+B(t)+C(t) µ(N S(t) E(t)) from (8) with the parameter values from Table 1. (b) Cumulative confirmed cases given in (9), when available an unlimited supply of vaccines, also with the parameter values from Table 1.

Assuming that, in the near future, an e↵ective vaccine against the Ebola virus will be available, as expected by WHO by the end of 2015, we study three di↵erent scenarios, which illustrate limitations on the number of vaccines available and on the capacity of administration of the vaccines by the health care services and humanitarian teams working in the a↵ected countries. The three vaccination scenarios are the following: unlimited supply of vaccines; limited total number of vaccines to be used; and limited supply of vaccines at each instant of time.

6.1. Unlimited supply of vaccines. Assume w1 = w2 = 1. If we administer from an unlimited supply of vaccines, then the number of total individuals who have an active infection I(t)+R(t)+D(t)+ H(t)+B(t) µ(N S(t) E(t) C(t)) during the 90 days is a decreasing function in time, and is equal to 3.56 individuals at the final time (see Fig. 2b). 6. NUMERICAL SIMULATIONS 29

If we compare the case where there is no vaccination with the opposite case of unlimited supply of vaccines, we observe that at the end of 90 days the class of completely recovered individuals has approx- imately 86.5 individuals in the case of no vaccination and 13468 in the case of unlimited vaccination, which represents 74.82 per cent of the total population (see Fig. 3–4). If vaccines are available, then the number of individuals that develop active disease is less than one at the end of 7.5 days and less that 0.1 at the end of 32.4 days. In the case of no vaccination, the class of active infected individuals has 61.7 individuals at the end of 90 days (see Fig. 3). If no vaccination is provided, then the number of deaths, hospitalizations and burials increases from 1.2 to 262.6, when compared to the case of unlimited supply of vaccines (see Fig. 4). The optimal vaccination policy suggested by the solution of the optimal

4 x 10 2 400

1.5 300

S 1 E 200

0.5 100

0 0 0 50 100 0 50 100 time (days) time (days)

80 150

60 100 I 40 R 50 20

0 0 0 50 100 0 50 100 time (days) time (days)

Figure 3. Individuals S(t), E(t), I(t) and R(t). In dashed line, the case of vaccination without limit on the supply of vaccines; in continuous line, the case with no vaccination with the parameter values from Table 1. problem, implies a vaccination of 100 per cent of the susceptible population for approximately 1.62 days followed by a fast reduction of the fraction of susceptible population that is vaccinated. This is based on the fact that the vaccine is e↵ective and once all the susceptible population is vaccinated in a short period of time, then the number of susceptible individuals immediately decreases, since they are trans- ferred to the class of completely recovered individuals, as well as the need of vaccination (see Fig. 5a). The previous results show the importance of an e↵ective vaccine for Ebola virus and the very good results that can be attained if the number of available vaccines satisfies the needs of the population.

6.2. Limited total number of vaccines. In Fig. 5b, we observe that at the end of 90 days, 33786 vaccines were used, if the supply of vaccines is unlimited. In this section, we consider the case where the total number of vaccines used in the 90 days period is limited. We consider the case where the total number of vaccines available is lower or equal than the initial number of susceptible individuals (W (90) 10000, W (90) 11000, W (90) 13000, W (90) 15000, W (90) 16000, W (90) 18000)       and the case where the total number of vaccines available is bigger than the initial number of susceptible individuals (W (90) 20000). We first consider w = w = 1. The cumulative confirmed cases (see  1 2 Fig. 6) increases in time in the case W (90) 10000 and decreases in the case W (90) 20000, t [0, 90].   2 30 3. EBOLA MODEL AND OPTIMAL CONTROL WITH VACCINATION CONSTRAINTS

15 30

10 20 D H 5 10

0 0 0 50 100 0 50 100 time (days) time (days) 4 x 10 300 2

1.5 200 B C 1 100 0.5

0 0 0 50 100 0 50 100 time (days) time (days)

Figure 4. Individuals D(t), H(t), B(t) and C(t), with the parameter values from Table 1. In dashed line, the case of vaccination without limit on the supply of vaccines; in continuous line, the case of no vaccination.

4 x 10 3.5 1

0.9 3

0.8 2.5 0.7

0.6 2 u 0.5 W 1.5 0.4

0.3 1

0.2 0.5 0.1

0 0 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 time (days) time (days)

(a) Optimal control u(t) (b) Number of vaccines W (t)

Figure 5. Optimal control and number of vaccines with the parameter values from Table 1, when an unlimited supply of vaccines is available.

At the end of the 90 days period, the total number of individuals who got active infection is approximately 76 and 9.5 individuals in the case W (90) 10000 and W (90) 20000, respectively. In the case   W (90) 10000, the optimal control takes the maximum value for less than one day (approximately  0.72 days) with a cost equal to 322.74, and in the case W (90) 20000, the optimal control takes the  maximum value for approximately 2.2 days with a cost equal to 72.35. The cost associated to the case W (90) 20000 is lower than the one in the case W (90) 10000, although more individuals are   vaccinated, since the number of individuals in the class I is lower. Namely, in the case W (90) 10000,  the number of individuals with active infection at the end of 90 days is equal to I(90) = 8.4 and in 6. NUMERICAL SIMULATIONS 31

80 ≤ W(90)≤10000 W(90) 10000 1 W(90)≤20000 70 W(90)≤20000

60 0.8

50 0.6

40 u

0.4 30

Cumulative confirmed cases 20 0.2

10 0

0 0 20 40 60 80 100 0 10 20 30 40 50 60 70 80 90 time (days) time (days)

(a) Cumulative confirmed cases (b) Optimal control

Figure 6. (a) Cumulative confirmed cases. (b) Optimal control for the case of limited total number of vaccines. Dashed line for W (90) 10000 and continuous line for  W (90) 20000. 

4 x 10 2 W(90)≤10000 60 W(90)≤10000 W(90)≤20000 W(90)≤20000 1.5 40

S 1 E 20 0.5

0 0 0 50 100 0 50 100 time (days) time (days)

15 30 W(90)≤10000 ≤ W(90) 10000 W(90)≤20000 W(90)≤20000 10 20 I R

5 10

0 0 0 50 100 0 50 100 time (days) time (days)

Figure 7. Individuals S(t), E(t), I(t) and R(t). The dashed line represents the case where W (90) 10000 and the continuous line represents the case where W (90) 20000.   the case W (90) 20000 the respective number is equal to I(90) = 0.67. This means that in the case  W (90) 10000, in a epidemiological scenario corresponding to a basic reproduction number greater  than one, 10000 vaccines will not be enough to eradicate the disease. Additionally, if we consider the maximum value for the total number of vaccines used during the period of 90 days to be equal to 11000, 32 3. EBOLA MODEL AND OPTIMAL CONTROL WITH VACCINATION CONSTRAINTS

2 W(90)≤ 10000 4 W(90)≤10000 W(90)≤20000 W(90)≤20000 1.5 3

D 1 H 2

0.5 1

0 0 0 50 100 0 50 100 time (days) time (days) 4 x 10 60 2 ≤ W(90)≤10000 W(90) 10000 W(90)≤20000 W(90)≤20000 1.5 40 B C 1 20 0.5

0 0 0 50 100 0 50 100 time (days) time (days)

Figure 8. Individuals D(t), H(t), B(t) and C(t). The dashed line represents the case where W (90) 10000 and the continuous line represents the case where W (90) 20000.  

13000, 15000, 16000 and 18000, then we observe that the optimal control u remains more time at the maximum value 1 when the supply of vaccines is bigger, which means that when the total number of available vaccines is increased there will be resources to vaccinate all susceptible individuals for a longer period of time, which implies a bigger reduction of the number of individuals who get infected by the virus (see Fig. 9a and 9b for the optimal control strategy and respective zoom in the period of vaccination).

4 x 10 2.5 W(90)≤10000 1 W(90)≤10000 1 W(90)≤11000 W(90)≤11000 0.9 W(90)≤13000 W(90)≤13000 2 W(90)≤15000 0.8 W(90)≤15000 0.8 W(90)≤16000 W(90)≤16000 W(90)≤18000 0.7 W(90)≤18000 W(90)≤20000 W(90)≤20000 0.6 0.6 1.5 u u

0.5 W

0.4 0.4 1 W(90)≤10000 W(90)≤11000 0.3 W(90)≤13000 0.2 0.2 W(90)≤15000 0.5 W(90)≤16000 0.1 W(90)≤18000 0 ≤ 0 W(90) 20000 0 0 10 20 30 40 50 60 70 80 90 2 4 6 8 10 12 14 16 18 20 0 10 20 30 40 50 60 70 80 90 time (days) time (days) time (days)

(a) Optimal control (b) Optimal control (c) Number of vaccines

Figure 9. Optimal control u(t) and number of vaccines W (t) for W (90) 10000,  W (90) 11000, W (90) 13000, W (90) 15000, W (90) 16000, W (90) 18000 and      W (90) 20000.  6. NUMERICAL SIMULATIONS 33

Consider now the case where the weight constant associated with the cost of implementation of the vaccination strategy, designated by the optimal control u, is bigger than one, for example, consider w = 1 and w = 50, and w = 1 and w = 500. To simplify, consider in both cases W (90) 10000 and 1 2 1 2  W (90) 20000. When we increase the weight constant w , the maximum value attained by the optimal  2 control becomes lower than one (see Fig. 10). In the case W (90) 10000 for w = 50, the optimal  2 control starts with the value u(0) = 0.54 and is a decreasing function with a cost function 344.3. At the end of approximately 3.7 days, the control remains equal to zero. For w2 = 500, the optimal control starts with the value u(0) = 0.16 and is also a decreasing function, with a cost 399.62. At the end of 13.5 days, it remains equal to zero. The behavior of the optimal state variables S, E, I, R, D, H, B and C are similar.

12000 w =50 2 w =500 0.6 2 10000 w =50 0.5 2 w =500 2 8000 0.4 u

0.3 W 6000

0.2 4000

0.1

2000 0

−0.1 0 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 time (days) time (days)

(a) Control (b) Number of vaccines

Figure 10. Optimal control u(t) and number of vaccines W (t) for W (90) 10000. In  dashed line the case w2 = 50 and in continuous line the case w2 = 500.

6.3. Limited supply of vaccines at each instant of time. ¿From previous results, we observe that when there is no limit on the supply of vaccines, or when the total number of used vaccines is limited, the optimal vaccination strategy implies a vaccination of 100 per cent of the susceptible population in a very short period of time, sometimes smaller than one day. But we know that in practice this is a very dicult task, since there are limitations in the number of vaccines available and also in the number of health care workers or humanitarian teams in the regions a↵ected by Ebola virus with capacity to vaccinate such a big number of individuals almost simultaneously. From this point of view, it is important to study the case where there is a limited supply of vaccines at each instant of time. In this section, we consider w1 = w2 = 1, a shorter interval of time [0,tf ], with tf = 10, 15, 16, and we assume that at each instant of time there exist only 1000, 1200 and 900 available vaccines, respectively. From our point of view, these numbers of available vaccines at each instant of time and the number of days considered, correspond to possible real scenarios, which are possible to implement in a concrete endemic region and at the same time characterize lack of human and material resources to vaccinate the susceptible population in a short period of time. From the numerical simulations, for such mixed constraints, the number of cumulative confirmed cases increases with time (see Figure 11a). The cost associated with the vaccination campaign, associated with the solution of the optimal control problem with the mixed constraint S(t)u(t) 1000, t [0, 10], is equal to 45.8879. Such solution is the less costly  2 of the three considered, followed by the constraint S(t)u(t) 1200 for t [0, 15] with a cost of 55.079.  2 The most expensive vaccination strategy is the one associated with the mixed constraint S(t)u(t) 900,  t [0, 16], with a cost of 59.109. The strategy associated with the constraint S(t)u(t) 1000 is the one 2  where a lowest number of susceptible individuals completely recover through vaccination, with 7540.9 34 3. EBOLA MODEL AND OPTIMAL CONTROL WITH VACCINATION CONSTRAINTS individuals in the class C at the end of 10 days. If we consider that at each instant of time there are 1200 vaccines available during a period of 15 days, then 12438 completely recover. This is the strategy with more individuals in the class C. If we consider 16 days, but only 900 vaccines available for each instant of time, then only 10839 individuals completely recover (see Fig. 11b). For all three mixed constraint situations, the number of individuals in the classes E, I, R, D, H and B does not change significantly (therefore, the figures with these classes are omitted). As the number of available vaccines represent a small percentage of the susceptible population, in the three cases the optimal vaccination strategies for the constraints S(t)u(t) 1200 and S(t)u(t) 900 suggest that the percentage of the   susceptible population that is vaccinated is always inferior than 18 percent. In the case of the constraint S(t)u(t) 1000, this percentage is always inferior to 8 percent (see Fig. 11c). 

14000 21 0.2 ≤ S*u ≤ 1000 S*u 1000 S*u ≤ 1000 ≤ S*u ≤ 1200 0.18 S*u 1200 12000 S*u ≤ 1200 ≤ 20 S*u ≤ 900 S*u 900 ≤ 0.16 S*u 900 10000 0.14 19 8000 0.12 C

18 u 0.1 6000 0.08 17 4000 0.06 Cumulative confirmed cases

0.04 16 2000 0.02

15 0 0 0 5 10 15 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 time (days) time (days) time (days)

(a) Total of active infected (b) Completely recovered (c) Optimal control

Figure 11. (a) Cumulative confirmed cases, (b) completely recovered, (c) optimal control. In (a), (b) and (c) the following mixed constraints are considered: S(t)u(t)  1000 for all t [0, 10], S(t)u(t) 1200 for all t [0, 15], and S(t)u(t) 900 for all 2  2  t [0, 16]. 2

7. Discussion We assume that, in a near future, an e↵ective vaccine against the Ebola virus will be available. Under this assumption, three di↵erent scenarios have been studied: unlimited supply of vaccines; limited total number of vaccines to be used; and limited supply of vaccines at each instant of time. We have solved the optimal control problems analytically and we have performed a number of numerical simulations in the three aforementioned vaccination scenarios. Some authors have already considered the optimal control problem with vaccination for Ebola disease, but always with unlimited supply of vaccines [5, 133]. It turns out that the solution to this mathematical problem is obvious: the solution consists to vaccinate all susceptible individuals in the beginning of the outbreak. This is a very particular case of our work, investigated in Section 6.1 (see Figure 5). If vaccines are available without any restriction, then one could completely eradicate Ebola in a very short period of time. These results show the importance of an e↵ective vaccine for Ebola virus and the very good results that can be attained if the number of available vaccines satisfy the needs of the population. Unfortunately, such situation is not realistic: in case an e↵ective vaccine for Ebola virus will appear, there always will be restrictions on the number of available vaccines as well as constraints on how to inoculate them in a proper way and in a short period of time; economic problems might also exist. In our work, for first time in the literature of Ebola, an optimal control problem with state and control constraints has been considered. Mathematically, it represents a health public problem of limited total number of vaccines. The results obtained in Section 6.2 provide useful information on the number of vaccines to be bought, in order to reduce the number of new infections with minimum cost. For 7. DISCUSSION 35 example, the results between 10000 and 20000 vaccines (in 90 days) are completely di↵erent. With 10000 vaccines, the number of cumulative infected cases continues to increase, while with 20000 vaccines it is already possible to decrease the new infections. The optimal solution, in this case, is similar to the case of unlimited supply of vaccines, that is, it implies a vaccination of 100 per cent of the susceptible population in a very short period of time. In practice, this is an unrealistic task, due to the necessary number of vaccines and humanitarian teams in the regions a↵ected by Ebola. Therefore, we conclude that it is important to study the case where there is a limited supply of vaccines at each instant of time. This was investigated in Section 6.3. This situation is much richer and the optimal control solution is not obvious. For a given number of available vaccines at each instant of time, we have a di↵erent solution, which is the optimal rate of susceptible individuals that should be vaccinated. In this case, the optimal control implies the vaccination of a small subset of the susceptible population. It remains the ethical problem of how to choose the individuals to be vaccinated.

CHAPTER 4

Mathematical modeling of COVID-19 dynamics with a case study of Wuhan

Mathematical models of infectious disease transmission dynamics are now ubiquitous. Such models play an important role in helping to quantify possible infectious disease control and mitigation strategies [38, 92, 104]. There exist a number of models for infectious diseases; as for compartmental models, starting from the very classical SIR model to more complex proposals [20]. This chapter is devoted to the study of the ongoing pandemic (COVID-19) with focus on the reality of Wuhan. Our model is a deterministic one, well suited to Wuhan reality. We start by Section 2, in which we present the ODE model for the the transmission dynamics COVID-19 disease. A qualitative analysis of the model is investigated in Section 3: in Section 3.1, we compute the basic reproduction number R0 of the COVID-19 system model; in Section 3.2, we study the local stability of the disease free equilibrium in terms of R0. The sensitivity of the basic reproduction number R0 with respect to the parameters of the system model is given in Section 4. The usefulness of our model is then illustrated in Section 5 of numerical simulations, where we use real data from Wuhan. We close this part with Section 6 of discussion, and future research.

1. Introduction Coronavirus disease 2019 (COVID-19) is an infectious disease caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). The disease was first identified December 2019 in Wuhan, the capital of Hubei, China, and has since spread globally, resulting in the ongoing 2020 pandemic outbreak [96]. The COVID-19 pandemic is considered as the biggest global threat worldwide because of thousands of confirmed infections, accompanied by thousands deaths over the world. Notice, by March 26, 2020, report 503,274 confirmed cumulative cases with 22,342 deaths. At the time of this revision, the numbers have increased to 1,353,361 confirmed cumulative cases with 79,235 deaths, according to the report dated by April 8, 2020, by the Word Health Organization. The global problem of the outbreak has attracted the interest of researchers of di↵erent areas, giving rise to a number of proposals to analyze and predict the evolution of the pandemic [24, 80]. Our main contribution is related with considering the class of super-spreaders, which is now appearing in medical journals (see, e.g., [124, 130]). This new class, as added to any compartmental model, implies a number of analysis about disease free equilibrium points, which is also considered in this work.

2. The Proposed COVID-19 Compartment Model Based on a 2016 model [64], and taking into account the existence of super-spreaders in the family of corona virus [6], we propose a new epidemiological compartment model that takes into account the super-spreading phenomenon of some individuals. Moreover, we consider a fatality compartment, related to death due to the virus infection. In doing so, the constant total population size N is subdivided into eight epidemiological classes: susceptible class (S), exposed class (E), symptomatic and infectious class (I), super-spreaders class (P ), infectious but asymptomatic class (A), hospitalized (H), recovery class

37 38 4. MATHEMATICAL MODELING OF COVID-19 DYNAMICS WITH A CASE STUDY OF WUHAN

(R), and fatality class (F ). The model takes the following form:

dS I H P = S l S 0 S, dt N N N 8 dE I H P > 0 > = S + l S + S E, > dt N N N > >dI > = ⇢1E (a + i)I iI, > dt > > >dP > = ⇢2E (a + i)P pP, > dt (17) > >dA <> = (1 ⇢1 ⇢2)E, dt > >dH > = a(I + P ) rH hH, > dt > >dR > = i(I + P )+rH, > dt > > >dF > = iI(t)+pP (t)+hH(t), > dt > > with quantifying the human-to-human: transmission coecient per unit time (days) per person, 0 quantifies a high transmission coecient due to super-spreaders, and l quantifies the relative transmis- sibility of hospitalized patients. Here  is the rate at which an individual leaves the exposed class by becoming infectious (symptomatic, super-spreaders or asymptomatic); ⇢1 is the proportion of progres- sion from exposed class E to symptomatic infectious class I; ⇢2 is a relative very low rate at which exposed individuals become super-spreaders while 1 ⇢ ⇢ is the progression from exposed to asymp- 1 2 tomatic class; a is the average rate at which symptomatic and super-spreaders individuals become hospitalized; i is the recovery rate without being hospitalized; r is the recovery rate of hospitalized patients; and i, p, and h are the disease induced death rates due to infected, super-spreaders, and hospitalized individuals, respectively. At each instant of time,

dF (t) (18) D(t):= I(t)+ P (t)+ H(t)= i p h dt gives the number of death due to the disease. The transmissibility from asymptomatic individuals has been modeled in this way since it was not apparent their behavior. Indeed, at present, this question is a controversial issue for epidemiologists. A flowchart of model (17) is presented in Figure 1.

3. Qualitative Analysis of the Model One of the most significant thresholds when studying infectious disease models, which quantifies disease invasion or extinction in a population, is the basic reproduction number [125]. In this section we obtain the basic reproduction number for our model (17) and study the locally asymptotically stability of its disease free equilibrium (see Theorem 3.1).

3.1. The Basic Reproduction Number. The basic reproduction number, as a measure for disease spread in a population, plays an important role in the course and control of an ongoing outbreak. It can be understood as the average number of cases one infected individual generates, over the course of its infectious period, in an otherwise uninfected population. Using the next generation matrix approach outlined in [126] to our model (17), the basic reproduction number can be computed by considering the below generation matrices F and V , that is, the Jacobian matrices associated to the rate of appearance 3. QUALITATIVE ANALYSIS OF THE MODEL 39

F

h

i

p 0 , , l ⇢1 a S E IH

⇢2 a

(1 ⇢ ⇢ ) 1 2 r

i

i A PR

Figure 1. Flowchart of model (17). of new infections and the net rate out of the corresponding compartments, respectively,

0 0 l  000 00 0 0 ⇢1 $i 00 J = 2 3 and J = 2 3 , F 00 0 0 V ⇢2 0 $p 0 6 7 6 7 6 00 0 07 6 0 a a $h 7 4 5 4 5 where

(19) $i = a + i + i, $p = a + i + p and $h = r + h.

1 The basic reproduction number R is obtained as the spectral radius of F V ,precisely, 0 · 0 ⇢1(al + $h) (al + $h)⇢2 (20) R0 = + . $i$h $p$h For the parameters used in our simulations (see Table 1), one computes this basic reproduction number to obtain R0 =4.3751. This confirm the severity of this outbreak.

3.2. Local Stability in Terms of the Basic Reproduction Number. Noting that the two last equations and the fifth of system (17) are uncoupled to the remaining equations of the system, we can easily obtain, by direct integration, the following analytical results:

A(t)=(1 ⇢ ⇢ ) t E(s)ds 1 2 0 8 t R t (21) >R(t)=i I(s)+P (s) ds + r H(s)ds > 0 0 <> R t ⇣ ⌘t R t F (t)=i 0 I(s)ds + p 0 P (s)ds + h 0 H(s)ds. > > Furthermore, since the total:> populationR size N is constant,R one hasR

(22) S(t)=N [E(t)+I(t)+P (t)+A(t)+H(t)+R(t)+F (t)] . Therefore, the local stability of model (17) can be studied through the remaining coupled system of states variables, namely, the variables E, I, P , and H in (17). The Jacobian matrix associated to these 40 4. MATHEMATICAL MODELING OF COVID-19 DYNAMICS WITH A CASE STUDY OF WUHAN variables of (17) is the following one:  0 l ⇢1 $i 00 (23) JM = 2 3 , ⇢2 0 $p 0 6 7 6 0 a a $h 7 4 5 where $i, $p, and $h are defined in (19). The eigenvalues of the matrix JM are the roots of the following characteristic polynomial: 4 3 2 Z()= + a1 + a2 + a1 + a4, where

a1 =  + $h + $i + $p,

a = ⇢ 0 ⇢ + $ + $ + $ $ + $ + $ $ + $ $ , 2 1 2 h i h i p h p i p

a = l⇢ l⇢ ⇢ $ 0 ⇢ $ ⇢ $ 0 ⇢ $ 3 a 1 a 2 1 h 2 h 1 p 2 i + $h$i + $h$p + $i$p + $h$i$p,

a = l⇢ $ l⇢ $ 0 ⇢ $ $ ⇢ $ $ + $ $ $ . 4 a 2 i a 1 p 2 i h 1 h p h i p Next, by using the Li´enard–Chipard test [45, 75], all the roots of Z() are negative or have negative real part if, and only if, the following conditions are satisfied:

1. ai > 0, i =1, 2, 3, 4; 2. a1a2 >a3.

In order to check these conditions of the Li´enard–Chipard test, we rewrite the coecients a1, a2, a3, and a4 of the characteristic polynomial in terms of the basic reproduction number given by (20):

a1 =  + $h + $i + $p,

⇢ 0 ⇢ 1 $ a =(1 R )($ + $ )+$ 1 + $ 2 + l⇢  + p 2 0 i p p $ i $ a 1 $ $ $ i p ✓ h h i ◆ 1 $ + l⇢  + i +( + $ )$ +($ + $ )$ , a 2 $ $ $ i h h i p ✓ h h p ◆

0 ⇢1$h ⇢2$h a3 = (1 R0)($h$p + $h$i + $i$p)+$p + $i $i $p

1 1 1 1 +$ l⇢ + + $ l⇢ + + $ $ $ , p a 1 $ $ i a 2 $ $ i h p ✓ h i ◆ ✓ h p ◆ a = $ $ $ (1 R ). 4 i h p 0 Moreover, we also compute, in terms of R0, the following expression: a a a =(1 R )( + $ )$ +(1 R )( + $ + $ )$ 1 2 3 0 i i 0 h p p

0 ⇢1 al⇢1 ⇢2 al⇢2 +( + $p + $i) + $p +( + $p + $i) + $i $p $i $p $p ⇣ ⌘ ⇣ ⌘

al⇢1 al⇢2 +( + $h + $i) +( + $h + $p) +( + $i)$h +($h + $i)$p. $h $h 4. SENSITIVITY ANALYSIS 41

From these previous expressions, it is clear that if R0 < 1, then the conditions of the Li´enard–Chipard test are satisfied and, as a consequence, the disease free equilibrium is stable. In the case when R0 > 1, we have that a4 < 0 and, by using Descartes’ rule of signs, we conclude that at least one of the eigenvalues is positive. Therefore, the system is unstable. In conclusion, we have just proved the following result: Theorem 3.1. The disease free equilibrium of system (17),thatis,(N,0, 0, 0, 0, 0, 0),islocally asymptotically stable if R0 < 1 and unstable if R0 > 1. Next we investigate the sensitiveness of the COVID-19 model (17), with respect to the variation of each one of its parameters, for the endemic threshold (20).

4. Sensitivity Analysis As we saw in Section 3, the basic reproduction number for the COVID-19 model (17), which we propose in Section 2, is given by (20). The sensitivity analysis for the endemic threshold (20) tells us how important each parameter is to disease transmission. This information is crucial not only for experimental design, but also to data assimilation and reduction of complex models [101]. Sensitivity analysis is commonly used to determine the robustness of model predictions to parameter values, since there are usually errors in collected data and presumed parameter values. It is used to discover param- eters that have a high impact on the threshold R0 and should be targeted by intervention strategies. More accurately, sensitivity indices’ allows us to measure the relative change in a variable when a pa- rameter changes. For that purpose, we use the normalized forward sensitivity index of a variable with respect to a given parameter, which is defined as the ratio of the relative change in the variable to the relative change in the parameter. If such variable is di↵erentiable with respect to the parameter, then the sensitivity index is defined as follows.

Definition 4.1 (See [26, 110]). The normalized forward sensitivity index of R0,whichisdi↵eren- tiable with respect to a given parameter ✓,isdefinedby

R0 @R0 ✓ ⌥✓ = . @✓ R0 The values of the sensitivity indices for the parameters values of Table 1, are presented in Table 2.

Name Description Value Units 1 Transmission coecient from infected individuals 2.55 day l Relative transmissibility of hospitalized patients 1.56 dimensionless 0 1 Transmission coecient due to super-spreaders 7.65 day 1  Rate at which exposed become infectious 0.25 day ⇢1 Rate at which exposed people become infected I 0.580 dimensionless ⇢2 Rate at which exposed people become super-spreaders 0.001 dimensionless 1 a Rate of being hospitalized 0.94 day 1 i Recovery rate without being hospitalized 0.27 day 1 r Recovery rate of hospitalized patients 0.5 day 1 i Disease induced death rate due to infected class 1/23 day 1 p Disease induced death rate due to super-spreaders 1/23 day 1 h Disease induced death rate due to hospitalized class 1/23 day

Table 1. Values of the model parameters corresponding to the situation of Wuhan, for which R0 =4.3751.

These values have been determined experimentally in such a way the mathematical model describes well the real data, giving rise to Figures 2 and 3. Other values for the parameters can be found, e.g., in [3]. 42 4. MATHEMATICAL MODELING OF COVID-19 DYNAMICS WITH A CASE STUDY OF WUHAN

Parameter Sensitivity index 0.999 l 0.729 0 0.00139  0.000

⇢1 0.997

⇢2 0.00265

a -0.0210

i -0.0215

r -0.671

i -0.0346

p -0.0000919

h -0.0583

Table 2. Sensitivity of R0 evaluated for the parameter values given in Table 1.

Note that the sensitivity index may depend on several parameters of the system, but also can be R0 constant, independent of any parameter. For example, ⌥✓ = +1 means that increasing (decreasing) ✓ by a given percentage increases (decreases) always R0 by that same percentage. The estimation of a sensitive parameter should be carefully done, since a small perturbation in such parameter leads to relevant quantitative changes. On the other hand, the estimation of a parameter with a rather small value for the sensitivity index does not require as much attention to estimate, because a small perturbation in that parameter leads to small changes. From Table 2, we conclude that the most sensitive parameters to the basic reproduction number R0 of the COVID-19 model (17) are , ⇢1 and i. In concrete, an increase of the value of will increase the basic reproduction number by 96.3% and this happens, in a similar way, for the parameter ⇢1.In contrast, an increase of the value of i will decrease R0 by 3.46%.

5. Numerical Simulations: The Case Study of Wuhan We perform numerical simulations to compare the results of our model with the real data obtained from several reports published by WHO [34, 35] and worldometer [96]. The starting point of our simulations is 4 January 2020 (day 0), when the Chinese authorities informed about the new virus [34], with already 6 confirmed cases in one day. From this period up to January 19, there is less information about the number of people contracting the disease. Only on January 20, we have the report [35], with 1460 new reported cases in that day and 26 the dead. Thus, the infection gained much more attention from 21 January 2020, with 1739 confirmed cases and 38 the dead, up to 4 March 2020, when the numbers in that day were as low as 11 and 7, respectively infected and dead, after a pick of 3892 confirmed cases on 27 January 2020 and a pick of 254 dead on 4 February 2020. Here we follow the data of the daily reports published by [96]. We show that our COVID-19 model describes well the real data of daily confirmed cases during the 2 months outbreak (66 days to be precise, from January 4 to March 9, 2020). The total population of Wuhan is about 11 million. During the COVID-19 outbreak, there was a restriction of movements of individuals due to quarantine in the city. As a consequence, there was a limitation on the spread of the disease. In agreement, in our model we consider, as the total population under study, N = 11000000/250. This denominator has been determined in the first days of the outbreak and later has been proved to be a correct value: according to the real data published by the WHO, it 6. DISCUSSION 43 is an appropriate value for the restriction of movements of individuals. As for the initial conditions, the following values have been fixed: S = N 6, E = 0, I = 1, P = 5, A = 0, H = 0, R = 0, and 0 0 0 0 0 0 0 F0 = 0.

4000

3500

3000

2500

2000

1500 Confirmed cases per day

1000

500

0 0 10 20 30 40 50 60 70 Time (in days)

Figure 2. Number of confirmed cases per day. The green line corresponds to the real data obtained from reports [34, 35, 96] while the black line (I + P + H) has been obtained by solving numerically the system of ordinary di↵erential equations (17), by using the Matlab code fde12 with ↵ = 1.

We would like to mention that there exist gaps in the reports of the WHO at the beginning of the outbreak. For completeness, we give here the list LC of the number of confirmed cases in Wuhan per day, corresponding to the green line of Figure 2, and the list LD of the number of dead individuals in Wuhan per day, corresponding to the red line of Figure 3:

LC =[6, 12, 19, 25, 31, 38, 44, 60, 80, 131, 131, 259, 467, 688, 776, 1776, 1460, 1739, 1984, 2101, 2590, 2827, 3233, 3892, 3697, 3151, 3387, 2653, 2984, 2473, 2022, 1820, 1998, 1506, 1278, 2051, 1772, 1891, 399, 894, 397, 650, 415, 518, 412, 439, 441, 435, 579, 206, 130, 120, 143, 146, 102, 46, 45, 20, 31, 26, 11, 18, 27, 29, 39, 39],

LD =[0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 4, 8, 15, 15, 25, 26, 26, 38, 43, 46, 45, 57, 64, 66, 73, 73, 86, 89, 97, 108, 97, 254, 121, 121, 142, 106, 106, 98, 115, 118, 109, 97, 150, 71, 52, 29, 44, 37, 35, 42, 31, 38, 31, 30, 28, 27, 23, 17, 22, 11, 7, 14, 10, 14, 13, 13].

Lists LC and LD have 66 numbers, where LC (0) represents the number of confirmed cases 04 January 2020 (day 0) and LC (65) the number of confirmed cases 09 March 2020 (day 65) and, analogously, LD(0) represents the number of dead on January 4 and LD(65) the number of dead on March 9, 2020.

6. Discussion Classical models consider SIR populations. Here we have taken into consideration the super- spreaders (P ), hospitalized (H), and fatality class (F ), so that its derivative (see formula (18)) gives the number of deaths (D). Our model is an ad hoc compartmental model of the COVID-19, taking into account its particularities, some of them still not well-known, giving a good approximation of the reality of the Wuhan outbreak (see Figure 2) and predicting a diminishing on the daily number of confirmed cases of the disease. This is in agreement with our computations of the basic reproduction number in 44 4. MATHEMATICAL MODELING OF COVID-19 DYNAMICS WITH A CASE STUDY OF WUHAN

Section 20 that, surprisingly, is obtained less than 1. Moreover, it is worth to mention that our model fits also enough well the real data of daily confirmed deaths, as shown in Figure 3.

300

250

200

150

100 Confirmed deads per day 50

0 0 10 20 30 40 50 60 70 Time (in days)

Figure 3. Number of confirmed deaths per day. The red line corresponds to the real data obtained from reports [34, 35, 96] while the black line has been obtained by solving numerically, using the Matlab code fde 12 with ↵ = 1, our system of ordinary di↵erential equations (17) to derive D(t) given in (18).

Our theoretical findings and numerical results adapt well to the real data and it reflects or reflected the reality in Wuhan, China. The number of hospitalized persons is relevant to give an estimate of the Intensive Care Units (ICU) needed. Some preliminary simulations indicate that this would be useful for the health authorities. Our model can also be used to study the reality of other countries, whose outbreaks are currently on the rise. We claim that some mathematical models like the one we have proposed here will contribute to reveal some important aspects of this pandemia. Of course, this investigation has some limitations, being the first on the relative recent spread of the new coronavirus and therefore the limited data accessible at the beginning of this study. In the future, we can develop further this prototype. Even with these shortcomings, the model can be useful due to the high relevance of the topic. Finally, we suggest new directions for further research: (1) the transmissibility from asymptomatic individuals; (2) compare, in the near future, our results with other models; (3) consider sub-populations related to age, gender, etc.; (4) introduce preventive measures in this COVID-19 epidemic and for future viruses; (5) integrate into the model some imprecise data by using fuzzy di↵erential equations; (6) include the viral load of the infectious into the model. These and other questions are under current investigation and will be addressed elsewhere. CHAPTER 5

Some models related to mosquitoes, water and environments

Water-related diseases are infectious diseases caused by a multiple variety types of agents (pathogens) including bacteria, viruses, protozoa, parasites, mosquitoes, insects, chemicals toxin and personal phys- ical factors. These micro-organisms agents are in general fundamental in the context of microbiology in order to understand the disease transmission process. Indeed, science of microbiology already suggested that diseases are spread by contact through a virus or bacterium. This represents a true means of spread of a disease through invisible micro-organism. From historical point of view, Antonie van Leeuwenhoek (1632–1723) is considered a the father of microbiology and was the first to use microscopes to show the existence of micro-organism of unicellular type and also multicellular type in pond water. Water-related diseases can be classified into groups reflecting specific routes of transmission of a disease [12, 13] and thus can give perspectives for control measures for each groups. It might be interesting to notice that diseases related to water are major cause of morbidity and mortality worldwide. Among the most influential and significant burden by group of classification are Water-borne diseases: transmission by contaminated fresh water, mainly through drinking or • bathing or in the preparation of food. Major and responsible for high morbidity and mortality are Cholera with an estimate of 120 thousand deaths every year; acute diarrhea with an estimate of 801 thousand deaths mainly killing children, and typhoid with more 128 thousand deaths every year. Mosquito-borne diseases: transmission by insects having aquatic immature stages and com- • monly by the female mosquitoes while obtaining blood meal necessary for the development of their eggs. Major and responsible for high morbidity and mortality are Malaria with about 405 thousand of deaths every year; And very recently A group of Severe Acute Respiratory Syndrome (SARS): transmission through droplets, either • fecal or respiratory droplets. Major responsible of high morbidity and mortality is Coronavirus disease 2019 (COVID-19) with more than 440,593. Moreover, waterborne and water-related vector borne diseases are sensitive to environmental con- ditions, and could be a↵ected by climate change issues. In fact, changes in climate may lead to higher water temperature, melting of polar ice and glaciers, and also intensification of water cycle [33, 114]. As consequence, there would be higher exposure to pathogens or chemical toxicants via the food chain, for instance, the result of irrigating plants with contaminated water and of bioaccumulation of toxic chem- icals by aquatic organisms, including seafood and fish or during recreation like swimming in polluted surface water[82]. The Chapter is organized as follows. In Section 1, we study and analyzed a mathematical model for the Zika outbreak occured in Brazil in 2016. This part is stuctured in the following: in Subsection 1.1, we introduce the model. Then, in Subsection 1.2, we prove that the model is biologically well-posed, in the sense that the solutions belong to a biologically feasible region (see Theorem 1.1). In Subsection 1.3, we give analytical expressions for the two disease free equilibria of our dynamical system. We compute the basic reproduction number R0 of the system and study the relevant equilibrium point of interaction between women and mosquitoes, showing its local asymptotically stability when R0 is less than one (see Theorem 1.2). The sensitivity of the basic reproduction number R0, with respect to the parameters of the system, is investigated

45 46 5. SOME MODELS RELATED TO MOSQUITOES, WATER AND ENVIRONMENTS in Subsection 1.4 in terms of the normalized forward sensitivity index. The possibility of occurrence of an endemic equilibrium is discussed in Subsection 1.5. We end with Subsections 1.6 of numerical simulations.

1. Mathematical modelling of Zika disease Zika virus infection on humans is mainly caused by the bite of an infected Aedes mosquito, either A. aegypti or A. albopictus. The infection on human usually causes rash, mild fever, conjunctivitis, and muscle pain. These symptoms are quite similar to dengue and chikungunya diseases, which can be transmitted by the same mosquitoes. Other modes of transmission of Zika disease have been observed, as sexual transmission, though less common [89]. Such modes of transmission are included in mathematical models found in the recent literature: see [125, Section 8] for a good state of the art. The name of the virus comes from the Zika forest in Uganda, where the virus was isolated for the first time in 1947. Up to very recent times, most of the Zika outbreaks have occurred in Africa with some sporadic outbreaks in Southeast Asia and also in the Pacific Islands. Since May 2015, Zika virus infections have been confirmed in Brazil and, since October 2015, other countries and territories of the Americas have reported the presence of the virus: see [131], where evolutionary trees, constructed using both newly sequenced and previously available Zika virus genomes, reveal how the recent outbreak arose in Brazil and spread across the Americas. The subject attracted a lot of attention and is now under strong current investigations. In [4], a deterministic model, based on a system of ordinary di↵erential equations, was proposed for the study of the transmission dynamics of the Zika virus. The model incorporates mother-to-child transmission as well as the development of microcephaly in newly born babies. The analysis shows that the disease-free equilibrium of the model is locally and globally asymptotically stable, whenever the associated reproduction number is less than one, and unstable otherwise. A sensitivity analysis was carried out showing that the mosquito biting, recruitment and death rates, are among the parameters with the strongest impact on the basic reproduction number. Then, some control strategies were proposed with the aim to reduce such values [4]. A two-patch model, where host-mobility is modeled using a Lagrangian approach, is used in [89], in order to understand the role of host-movement on the transmission dynamics of Zika virus in an idealized environment. Here we are concerned with the situation in Brazil and its consequences on brain anomalies, in particular microcephaly, which occur in fetuses of infected pregnant woman. This is a crucial question as far as the main problem related with Zika virus is precisely the number of neurological disorders and neonatal malformations [121]. Our study is based on the Zika virus situation reports for Brazil, as publicly available at the World Health Organization (WHO) web page. Based on a systematic review of the literature up to 30th May 2016, the WHO has concluded that Zika virus infection during pregnancy is a cause of congenital brain abnormalities, including microcephaly. Moreover, another important conclusion of the WHO is that the Zika virus is a trigger of Guillain-Barr´esyndrome [67]. Our analysis is focused on the number of confirmed cases of Zika in Brazil. For this specific case, an estimate of the population of the country is known, as well as the number of newborns. Moreover, from WHO data, it is possible to have an estimation of the number of newborn babies with neurological disorder. Our mathematical model allows to predict the number of cases of newborn babies with neurological disorder. Next, we will introduce the formulation of Zika model.

1.1. The Zika model. We consider women as the population under study. The total women population, given by N, is subdivided into four mutually exclusive compartments, according to disease status, namely: susceptible pregnant women (S); infected pregnant women (I); women who gave birth to babies without neurological disorder (W ); women who gave birth to babies with neurological disorder due to microcephaly (M). 1. MATHEMATICAL MODELLING OF ZIKA DISEASE 47

As for the mosquitoes population, since the Zika virus is transmitted by the same virus as Dengue disease, we shall use the same scheme as in [108]. There are four state variables related to the (female) mosquitoes, namely: Am(t), which corresponds to the aquatic phase, that includes the egg, larva and pupa stages; Sm(t), for the mosquitoes that might contract the disease (susceptible); Em(t), for the mosquitoes that are infected but are not able to transmit the Zika virus to humans (exposed); Im(t), for the mosquitoes capable of transmitting the Zika virus to humans (infected). The following assumptions are considered in our model: (A.1) there is no immigration of infected humans; (A.2) the total human populations N is constant; (A.3) the coecient of transmission of Zika virus is constant and does not varies with seasons; (A.4) after giving birth, pregnant women are no more pregnant and they leave the population under study at a rate µh equal to the rate of humans birth; (A.5) death is neglected, as the period of pregnancy is much smaller than the mean humans lifespan; (A.6) there is no resistant phase for the mosquito, due to its short lifetime. Note that the male mosquitoes are not considered in this study because they do not bite humans and consequently they do not influence the dynamics of the disease. The di↵erential system that describes the model is composed by compartments of pregnant women and women who gave birth: dS I = ⇤ (B m +(1 )⌧ + µ )S, dt mh N 1 h 8dI I > = B m S (⌧ + µ )I, > mh 2 h (24) > dt N >dW > =(1 )⌧ S +(1 )⌧ I µ W, < dt 1 2 h dM > = ⌧ I µ M, > 2 h > dt > where N = S +I +W +M is:> the total population (women). The parameter ⇤ denotes the new pregnant women per week, stands for the fraction of susceptible pregnant women that gets infected, B is the average daily biting (per day), mh represents the transmission probability from infected mosquitoes Im (per bite), ⌧1 is the rate at which susceptible pregnant women S give birth (in weeks), ⌧2 is the rate at which infected pregnant women I give birth (in weeks), µh is the natural death rate for pregnant women, denotes the fraction of infected pregnant women I that give birth babies with neurological disorder due to microcephaly. The above system (24) is coupled with the dynamics of the mosquitoes [109]:

dA A m = µ 1 m (S + E + I ) (µ + ⌘ ) A , dt b K m m m A A m 8dS ✓ ◆ I > m = ⌘ A B + µ S , > A m hm m m (25) > dt N >dE I⇣ ⌘ > m = B S (⌘ + µ )E , < dt hm N m m m m >dIm ⇣ ⌘ > = ⌘mEm µmIm, > dt > > where parameter hm:represents the transmission probability from infected humans Ih (per bite), µb stands for the number of eggs at each deposit per capita (per day), µA is the natural mortality rate of larvae (per day), ⌘A is the maturation rate from larvae to adult (per day), 1/⌘m represents the extrinsic incubation period (in days), 1/µm denotes the average lifespan of adult mosquitoes (in days), and K is the maximal capacity of larvae. See Table 1 for the description of the state variables and parameters of the Zika model (24)–(25). In Figure 1, we describe the behavior of the movement of individuals among these compartments. 48 5. SOME MODELS RELATED TO MOSQUITOES, WATER AND ENVIRONMENTS

Variable/Symbol Description S(t) susceptible pregnant women I(t) infected pregnant women W (t) women who gave birth to babies without neurological disorder M(t) women who gave birth to babies with neurological disorder due to microcephaly

Am(t) mosquitoes in the aquatic phase

Sm(t) susceptible mosquitoes

Em(t) exposed mosquitoes

Im(t) infected mosquitoes ⇤ new pregnant women (per week) fraction of S that gets infected B average daily biting (per day)

mh transmission probability from Im (per bite)

⌧1 rate at which S give birth (in weeks)

⌧2 rate at which I give birth (in weeks)

µh natural death rate fraction of I that gives birth to babies with neurological disorder

hm transmission probability from Ih (per bite)

µb number of eggs at each deposit per capita (per day)

µA natural mortality rate of larvae (per day)

⌘A maturation rate from larvae to adult (per day)

1/⌘m extrinsic incubation period (in days)

1/µm average lifespan of adult mosquitoes (in days) K maximal capacity of larvae

Table 1. Variables and parameters of the Zika model (24)–(25).

µ W h (1 )⌧ 1 ⇤ (1 )⌧2 B S mh I

µh µh ⌧2 µ M h

⌘m ⌘A µA Im Em Sm Am Bhm µm µm µm

µ (1 A /K) b m µ (1 A /K) b m µ (1 A /K) b m Figure 1. Flowchart presentation of the compartmental model (24)–(25) for Zika.

We consider system (24)–(25) with given initial conditions

S(0) = S0,I(0) = I0,W(0) = W0,M(0) = M0,Am(0) = Am0,

Sm(0) = Sm0,Em(0) = Em0,Im(0) = Im0, 1. MATHEMATICAL MODELLING OF ZIKA DISEASE 49

8 satisfying (S0,I0,W0,M0,Am0,Sm0,Em0,Im0) R . In what follows, we assume mh = hm. 2 + 1.2. Positivity and boundedness of solutions. Since the systems of equations (24) and (25) represent, respectively, human and mosquitoes populations, and all parameters in the model are non- negative, we prove that, given nonnegative initial values, the solutions of the system are nonnegative. More precisely, let us consider the biologically feasible region

8 ⇤ ⌦ = (S, I, W, M, Am,Sm,Em,Im) R+ : S + I + W + M , ( 2  µh (26)

Am kNh,Sm + Em + Im mNh .   ) The following result holds.

Theorem 1.1. The region ⌦ defined by (26) is positively invariant for model (24)–(25) with initial 8 conditions in R+.

Proof. Our proof is inspired by [109]. System (24)–(25) can be rewritten in the following way: dX = M(X)X + F ,whereX =(S, I, W, M, A ,S ,E ,I ), M(X)= M M with dt m m m m 1 2 B Im (1 )⌧ µ 000 mh N 1 h B Im ⌧ µ 00 0 mh N 2 h 1 (1 )⌧1 (1 )⌧2 µh 0 B C B 0 ⌧2 0 µhC M1 = B C , B 0000C B C B 0000C B C B 0000C B C B 0000C B C @ A 0000 0000 0 1 0000 B 0000C M = B C , 2 B Sm+Em+Im C B µb K µA ⌘A µb µb µb C B I C B ⌘A Bhm µm 00C B N C B 0 B I ⌘ µ 0 C B hm N m m C B 00⌘ µ C B m mC @ A and F =(⇤, 0, 0, 0, 0, 0, 0, 0)T . Matrix M(X)isMetzler,i.e.,theo↵ diagonal elements of A are nonneg- dX ative. Using the fact that F 0, system = M(X)X + F is positively invariant in R8 [1], which dt + 8 means that any trajectory with initial conditions in R+ remains in ⌦ for all t>0. ⇤

1.3. Existence and local stability of the disease-free equilibria. System (24)–(25) admits two disease free equilibrium points (DFE), obtained by setting the right-hand sides of the equations in the model to zero: the DFE E1, given by

E1 =(S⇤,I⇤,W⇤,M⇤,Am⇤ ,Sm⇤ ,Em⇤ ,Im⇤ ) ⇤ ⌧ ⇤ (1 ) = , 0, 1 , 0, 0, 0, 0, 0 , ⌧ (1 )+µ µ (⌧ (1 )+µ ) ✓ 1 h h 1 h ◆ 50 5. SOME MODELS RELATED TO MOSQUITOES, WATER AND ENVIRONMENTS which corresponds to the DFE in the absence of mosquitoes, and the DFE in the presence of mosquitoes, E2, given by

E2 =(S⇤,I⇤,W⇤,M⇤,Am⇤ ,Sm⇤ ,Em⇤ ,Im⇤ ) ⇤ ⌧ ⇤ (1 ) K% K% = , 0, 1 , 0, , , 0, 0 , ⌧ (1 )+µ µ (⌧ (1 )+µ ) µ ⌘ µ µ ✓ 1 h h 1 h b A b m ◆ where

(27) % = ⌘ (µ µ )+µ µ . A m b A m

In what follows, we consider only the DFE E2, because this equilibrium point considers interaction between humans and mosquitoes, being therefore more interesting from the biological point of view. The local stability of E2 can be established using the next-generation operator method on (24)–(25). Following the approach of [126], we compute the basic reproduction number R0 of system (24)–(25) writing the right-hand side of (24)–(25) as with F V 0 ⇤ +(B Im +(1 )⌧ + µ )S mh N 1 h B Im S (⌧ + µ )I 0 mh N 1 0 2 h 1 0 (1 )⌧ S +(1 )⌧ I + µ W 1 2 h B 0 C B µ M ⌧ I C = B C , = B h 2 C . B C B Am C F B 0 C V B µb(1 K )(Sm + Em + Im)+(µA + ⌘A)AmC B C B I C B 0 C B ⌘AAm +(Bhm + µm)Sm C B C B N C BB I S C B (⌘ + µ )E C B hm N m C B m m m C B 0 C B ⌘ E + µ I C B C B m m m m C @ A @ A Then we consider the Jacobian matrices associated with and : F V 00000000 0 ✏Im (I+W +M) ✏Im S ✏Im S ✏Im S 000✏S1 N N N N B 00000000C B C B C B 00000000C B C J = B C , F B 00000000C B C B C B 00000000C B C B C B ✏ ISm ✏ Sm(S+W +M) ✏ISm ✏ ISm 0 ✏ I 0 ✏ S C B N N N N mC B C B 00000000C B C @ A where ✏ = Bmh/N , and J = J 1 J 2 with V V V ✏I S ✏I S ✏I S A m m m N N N 0 0 ⌧2 + µh 001 B⌧ ( 1) ( 1) ⌧ µ 0 C B 1 2 h C B C B 0 ⌧2 0 µh C B C J 1 = B C , V B 0000C B C B C B ✏ISm ✏Sm(S+W +M) ✏ISm ✏ISm C B N N N N C B C B 0000C B C B C B 0000C B C @ A 1. MATHEMATICAL MODELLING OF ZIKA DISEASE 51

000✏S 0 00001 B 0000C B C B C B 0000C J = B C 2 B µ (S +E +I )+(µ +⌘ )K µ (A K) µ (A K) µ (A K) C V B b m m m A A b m b m b m C B K K K K C B C B ⌘ ✏I + µ 00C B A N m C B C B 00⌘ + µ 0 C B M m C B C B 00⌘ µ C B M m C and @ A B I (N S) A = ⌧ (1 )+µ + mh m 1 h N 2 1 . The basic reproduction number R0 is obtained as the spectral radius of the matrix J (J ) at F ⇥ V the disease-free equilibrium E2, and is given by µ ⇤ (µ + ⌧ )(⌧ (1 )+µ )(⌘ + µ ) K⌘ (⌘ (µ µ )+µ µ ) µ B (28) R = b h 2 1 h m m m A m b A m mh h 0 µ ⇤ (µ + ⌧ )(⌧ (1 )+µ )(⌘ + µ ) µ p b h 2 1 h m m m or B2 2 Kµ2 ⌘ (µ ⌘ µ (µ + ⌘ )) (29) R2 = mh h m b A m A A . 0 µ2 µ ⇤ (µ + ⌧ )(⌘ + µ )(⌧ (1 )+µ ) m b h 2 m m 1 h The disease-free equilibrium E2 is locally asymptotically stable if all the roots of the characteristic equation of the linearized system associated to (24)–(25) at the DFE E2 have negative real parts. The characteristic equation associated with E2 is given by

(30) p1()p2()p3()p4()=0 with p ()= + µ + ⌧ (1 ),p()=( + µ )2 , 1 h 1 2 h p ()= 2µ + µ ⌘ µ2 , + µ (µ (µ + ⌘ ) µ ⌘ ),p()=a 3 + a 2 + a + a , 3 m b A m m m A A b A 4 3 2 1 0 where 2 2 2 2 µmµb ⇤ (µh + ⌧2)(⌘m + µm)(⌧1(1 )+µh) B mh Kµh ⌘m (µb⌘A µm(µA + ⌘A)) a0 = 2 , ⇤ µmµb (⌧1(1 )+µh) 2 µm +(2⌧2 +2µh + ⌘m) µm + ⌘m(⌧2 + µh) µh + ⌘m + ⌧2 +2µm 1 a1 = ,a2 = ,a3 = . µm µm µm By the Routh–Hurwitz criterion, all the roots of the characteristic equation (30) have negative parts whenever R0 < 1. We have just proved the following result.

Theorem 1.2. The disease free equilibrium in the presence of non-infected mosquitoes, E2,islocally asymptotically stable if R0 < 1 and unstable if R0 > 1. 1.4. Sensitivity of the basic reproduction number. The sensitivity of the basic reproduction number R0 is an important issue because it determines the model robustness to parameter values. The sensitivity of R0 with respect to model parameters is here measured by the so called sensitivity index. Definition 1.3 (See [25, 66]). The normalized forward sensitivity index of a variable that @ p depends di↵erentiability on a parameter p is defined by ⌥ := . p @p ⇥ | | Remark 2. If ⌥p = +1, then an increase (decrease) of p by x% increases (decreases) by x%; if ⌥ = 1, then an increase (decrease) of p by x% decreases (increases) by x%. p 52 5. SOME MODELS RELATED TO MOSQUITOES, WATER AND ENVIRONMENTS

¿From (28) and Definition 1.3, it is easy to derive the normalized forward sensitivity index of R0 with respect to the average daily biting B and to the transmission probability mh from infected mosquitoes Im per bite.

Proposition 1. The normalized forward sensitivity index of R0 with respect to B and mh is one: ⌥R0 =1and ⌥R0 =1. B mh

Proof. It is a direct consequence of (28) and Definition 1.3. ⇤

The sensitivity index of R0 with respect to , ⌘m, µh, ⌘A, µm, ⌧1, ⌧2, µb, µA, K and ⇤ is given, respectively, by

µh + ⌧1 µm (⌧1(µh +2⌧2))(1 )+⌧2µh ⌥R0 = , ⌥R0 = , ⌥R0 = , 2(⌧ (1 )+µ ) ⌘m 2(⌘ + µ ) µh 2(µ + ⌧ )(⌧ (1 )+µ ) 1 h m m h 2 1 h (µm µb) ⌘A µm ((⌘A + µA)( ⌘m 2µm)+3⌘Aµb)+2⌘A⌘mµb ⌥R0 = , ⌥R0 = , ⌘A 2(µ (⌘ + µ ) ⌘ µ ) µm 2(⌘ + µ )(µ (⌘ + µ ) ⌘ µ ) m A A A b M M M A A A b ⌧1 ( 1+) ⌧2 µM (µA + ⌘A) ⌥R0 = , ⌥R0 = , ⌥R0 = , ⌧1 2(⌧ (1 )+µ ) ⌧2 2(µ + ⌧ ) µb 2(µ (⌘ + µ ) ⌘ µ ) 1 h h 2 M A A A b µAµm 1 1 ⌥R0 = , ⌥R0 = , ⌥R0 = . µA 2(µ (⌘ + µ ) ⌘ µ ) K 2 ⇤ 2 m A A A b In Section 1.6, we compute the previous sensitivity indexes for data from Brazil. 2 To analyze the sensitivity of R0 with respect to all the parameters involved, we compute appropriate derivatives: @R2 µ + ⌧ @R2 2 @R2 2 0 = h 1 R2, 0 = R2, 0 = R2, @ (µ +(1 )⌧ ) 0 @B B 0 @ 0 h 1 mh mh @R2 µ @R2 µ (⌘ + µ ) 0 = m R2, 0 = m a a R2, @⌘ ⌘ (µ + ⌘ ) 0 @µ µ (⌘ µ µ (⌘ + µ )) 0 m m m m b b a b m a a @R2 µ µ @R2 µ (31) 0 = b m R2, 0 = m R2, @⌘ ⌘ µ µ (⌘ + µ ) 0 @µ µ (⌘ + µ ) ⌘ µ 0 A a b m a a A m a a a b @R2 ⌘ + µ 1 2 0 = a a R2, @µ µ (⌘ + µ ) ⌘ µ ⌘ + µ µ 0 m ✓ m a a a b m m m ◆ 2 2 @R0 1 2 @R0 1 2 = R0, = R0. @⇤ ⇤ @⌧2 µh + ⌧2 In Section 1.6 we compute the values of these expressions according with the numerical values given 2 in Table 2, up to the R0 factor, which appears in all the right-hand expressions of (31), in order to determine which are the most and less sensitive parameters.

1.5. Existence and stability analysis of the endemic equilibrium point. The system (24)– (25) has one endemic equilibrium (EE) with biologic meaning whenever R0 > 1. This EE is given by + E = S+⇤ ,I+⇤ ,W+⇤ ,M+⇤ ,Am⇤ +,Sm⇤ +,Em⇤ +,Im⇤ + with

⇣S ⇣I ⇣W ⇣M S+⇤ = ,I+⇤ = ,W+⇤ = ,M+⇤ = , d⇤ d⇤(µh + ⌧2) d⇤ µh (µh + ⌧2) d⇤ µh (µh + ⌧2)

K% ⇣Sm ⇣Em ⌘m Am⇤ + = ,Sm⇤ + = ,Em⇤ + = ,Im⇤ + = Em⇤ + , µb⌘A dm⇤ (⌘m + µm)dm⇤ µm 1. MATHEMATICAL MODELLING OF ZIKA DISEASE 53 where % is defined in (27) and

d⇤ =[BK ⌘ µ % ⇤ µ µ (⌘ ⌧ (1 )+µ ⌧ (1 )+µ (⌘ + µ ))] µ B, mh m h m b m 1 m 1 h m m h mh dm⇤ = µb (Bmhµh + µmµh + µm⌧2) ⌘mBmhµh > 0, ⇣ = µ (⌘ + µ )(B µ + µ (µ + ⌧ )) µ ⇤2, S b m m mh h m h 2 m ⇣ = ⇤ B2K2 ⌘ µ2 % + ⇤ µ2 µ (µ + ⌧ )((1 ) ⌧ + µ )(⌘ + µ ) , I mh m h m b h 2 1 h m m 2 2 2 2 ⇣W = ⇤ K ⌘mµ ⌧2 (1 ) %B ⇤ mhµmµbµh⌧1 (1 )(µh + ⌧2)(⌘m+ µm) B mh h (⇤2µ2 µ (µ + ⌧ )(µ ⌧ ( 1+ )+ ⌧ ⌧ (1 )+µ ⌧ (1 )) (⌘ + µ )), m b h 2 h 2 1 2 h 1 m m ⇣ = ⇤ µ2 %K2 B2⌘ + ⇤µ2 µ (µ + ⌧ )(⌧ (1 )+µ )(⌘ + µ ) ⌧ , M h mh m m b h 2 1 h m m 2 ⇣ = (µ + ⌧ )(BK ⌘ µ % ⇤µ µ ((1 ) ⌧ + µ )(⌘ + µ )) , Sm h 2 mh m h m b 1 h m m ⇣ = µ2 %K2 B2⌘ (µ + ⌧ ) ⇤µ2 µ (⌘ + µ )((1 ) ⌧ + µ ) . Em h mh m h 2 m b m m 1 h

From (29), µb⌘A >µm(µA + ⌘A), < 1, and d⇤ < 0. Thus,

2 2 2 1 R0 ⇤ µbµm ((1 )⌧1 + µh)(⌘m + µm) S+⇤ > 0,I+⇤ = > 0, d⇤ 2 2 2 1 R0 ⇤ µbµm ⌧2 ((1 ) ⌧1 + µh)(⌘m + µm) ⌧2 and M+⇤ = = I+⇤ > 0. d⇤ µh µh $ Moreover, as for W+⇤ , we have that it can be expressed as W+⇤ = with d⇤ µh (µh + ⌧2)

$ = B22 ⌘ K⇤µ2 ( 1)⌧ (⌘ µ µ (⌘ + µ )) + ⇤2µ µ (⌘ + µ )(µ + ⌧ ) mh m h 2 A b m A A b m m m h 2 [( 1)(⌧ ⌧ )(B µ + µ (µ + ⌧ )) + ⌧ (B µ ( 1) + µ µ ( )+µ ( 1) ⌧ )] . ⇥ 1 2 mh h m h 2 2 mh h h m m 2

Therefore, W+⇤ is positive assuming that ⌧1 > ⌧2 and > . Finally, by using again µb⌘A >µm(µA +⌘A) and 1 > 0, we obtain that A⇤ > 0 and, moreover, m+ 2 2 (µh + ⌧2)d⇤ ⇤ R0 1 µmµb (µh + ⌧2)((1 ) ⌧1 + µh) Sm⇤ + = > 0,Em⇤ + = > 0, Bmhµhdm⇤ dm⇤ ⌘m Im⇤ + = Em⇤ + > 0. µm

After some appropriate manipulations, the matrix associated to E+ is given by

V11 V12 V12 V12 00 0V18 V µ ⌧ V V V 00 0 V 0 21 h 2 12 12 12 181 ⌧1(1 ) ⌧2(1 ) µh 00 0 0 0 B C B 0 ⌧2 0 µh 00 0 0C (32) B C B 0000V45 V56 V56 V56 C B C B V61 V62 V61 V61 ⌘a µm V76 00C B C B V V V V 0 V ⌘ µ 0 C B 61 62 61 61 76 m m C B 000000⌘ µ C B m mC @ A with

µh⇣I + ⇣S + ⇣W V11 = µh +( 1)⌧1 V12, µh⇣S)(µh + ⌧2)

Bmhd⇤⌘mµh⇣Im (µh⇣I + ⇣S + ⇣W ) (µh + ⌧2) V21 = 2 , dm⇤ µm(⌘m + µm)(⇣M + ⇣W + µh(⇣I + ⇣S(µh + ⌧2)))

µb(µm⇣Em + ⌘m⇣Im + µm(⌘m + µm⇣Sm ) V45 = ⌘a µa . dm⇤ Kµm(⌘m + µm) 54 5. SOME MODELS RELATED TO MOSQUITOES, WATER AND ENVIRONMENTS

With these notations, we have that the eigenvalues of the matrix are 1 = µ , = V µ + 4⌘ V +(µ + V )2 , 1 h 2 2 45 m a 56 m 45 1 ⇣ p ⌘ = V µ 4⌘ V +(µ + V )2 3 2 45 m a 56 m 45 ⇣ p ⌘ and the roots of a polynomial of degree five. Equivalently, we can write 2 and 3 as follows:

2 2 3 2 4 2 ⌘aµb +4µm(⌘a + µa) 2⌘aµbµm + µm ⌘aµb µm 2 = , p 2µm 2 2 3 2 4 2 ⌘aµb +4µm(⌘a + µa) 2⌘aµbµm + µm + ⌘aµb + µm 3 = . p 2µm 5 4 3 Moreover, the polynomial of degree five has leading coecient one, being given by x + 4x + 3x + 2 2x + 1x + 0 with

 = ⌘ +2µ +2µ + ⌧ V + V + V , 4 m h m 2 11 12 76  = ⌘ (2µ + µ + ⌧ V + V + V )+µ (4µ + ⌧ 2V + V +2V )+µ2 3 m h m 2 11 12 76 h m 2 11 12 76 h + V (µ + ⌧ V + V )+2⌧ µ 2V µ +2V µ + µ2 + ⌧ V ⌧ V 76 m 2 11 12 2 m 11 m 12 m m 1 12 2 11 ⌧ V + ⌧ V V V V V , 1 12 2 12 12 21 11 12  = ⌘ µ (2µ + ⌧ 2V + V +2V )+µ2 + µ (⌧ V + V )+( 1)⌧ V ⌧ V + ⌧ V 2 m h m 2 11 12 76 h m 2 11 12 1 12 2 11 2 12 2 +V76 (⌧2 V11 + V12) V11V12 V12V21 + V18(V61 V62)) + µh 2µm (⌧2 2V11 + V12 + V76)+2µ m 2 +( 1)⌧1V12 ⌧2V11 + V76 (⌧2 2V11 + V12) V11V12 V12V21)+µ (2µm V11 + V76)+2⌧1V12µm h + V (µ (⌧ V + V )+( 1)⌧ V + ⌧ (V V ) (V + V ) V ) 2⌧ V µ 2⌧ V µ 76 m 2 11 12 1 12 2 12 11 11 21 12 2 11 m 1 12 m +2⌧ V µ + ⌧ µ2 V µ2 + V µ2 2V V µ 2V V µ + ⌧ ⌧ V ⌧ ⌧ V 2 12 m 2 m 11 m 12 m 11 12 m 12 21 m 1 2 12 1 2 12 ⌧ V V ⌧ V V , 2 11 12 2 12 21  = ⌘ (µ (µ (⌧ 2V + V )+( 1)⌧ V ⌧ V + V (⌧ 2V + V ) V V V V 1 m h m 2 11 12 1 12 2 11 76 2 11 12 12 11 12 21 +V (2V V )) + µ2 (µ V + V ) µ (V ( ⌧ + ⌧ + V + V )+⌧ (V V )) 18 61 62 h m 11 76 m 12 1 1 11 21 2 11 12 +⌧ ⌧ V ⌧ V V V (V ( ⌧ + ⌧ + V + V )+⌧ (V V )) ⌧ ⌧ V ⌧ V V 1 2 12 1 18 61 76 12 1 1 11 21 2 11 12 1 2 12 2 11 12 ⌧ V V + ⌧ V V + V V V + V V V )+µ (µ (µ (⌧ 2V + V ) 2 12 21 1 18 61 11 18 62 18 21 62 m h m 2 11 12 2(V ( ⌧ + ⌧ + V + V )+⌧ V )) + µ2 (µ 2V ) µ (V ( ⌧ + ⌧ + V + V ) 12 1 1 11 21 2 11 h m 11 m 12 1 1 11 21 +⌧ (V V )) + 2⌧ V (( 1)⌧ V V )) + V (µ (µ (⌧ 2V + V )+( 1)⌧ V 2 11 12 2 12 1 11 21 76 h m 2 11 12 1 12 V (⌧ + V ) V V )+µ2 (µ V ) µ (V ( ⌧ + ⌧ + V + V ) 11 2 12 21 12 h m 11 m 12 1 1 11 21 +⌧ (V V )) + ⌧ V (( 1)⌧ V V )) , 2 11 12 2 12 1 11 21  = ⌘ ( µ (V µ ( ⌧ + ⌧ + V + V )+⌧ V µ +( 1)⌧ V V 0 m h 12 m 1 1 11 21 2 11 m 1 18 61 +V V ( ⌧ + ⌧ + V + V )+⌧ V V ⌧ V V V (V + V ) V ) 12 76 1 1 11 21 2 76 11 2 18 61 18 11 21 62 +µ2 ( (V (µ + V ) V V )) + ⌧ (V (µ + V ) V V )(( 1)⌧ V V ) h 11 m 76 18 61 2 12 m 76 18 61 1 11 21 µ (µ + ⌧ )(µ + V )(V µ + V ((1 )+V + V )) . m h 2 m 76 11 h 12 11 21 Obviously, these expressions become rather long. As a consequence, it is not possible to use the Routh– Hurwitz criterion in this general setting, but only for particular values of the parameters. Moreover, the eigenvalues can be computed numerically for the specific values given in Table 2 of the next section (realistic data from Brazil), and they are given by = 0.02, = 0.02, = 5000, = 22.3938, 1 2 3 4 = 0.0511697, = 0.044689, = 0.00919002, = 0.0079988. We can observe that local 5 6 7 8 stability of the endemic equilibrium holds, since all eigenvalues are negative real numbers. 1. MATHEMATICAL MODELLING OF ZIKA DISEASE 55

1.6. Numerical simulations: case study of Brazil. We perform numerical simulations to compare the results of our model with real data obtained from several reports published by the World Health Organization (WHO) [37], from the starting point when the first cases of Zika have been detected in Brazil and for a period of 40 weeks (from February 4, 2016 to November 10, 2016), which is assumed to be a regular pregnancy time. 1.6.1. Zika model fits well real data. According to several sources, the total population of Brazil is 206,956,000, and every year there are about 3,073,000 new born babies. As a consequence, there are about 3,000,000/52 new pregnant females every week. The number of babies with neurological disorders is taken from WHO reports [37]. See Table 2, where the values considered in this manuscript have been collected, such that the numerical experiments give good approximation of real data obtained from the WHO [37].

Symbol Description Value ⇤ new pregnant women (per week) 3000000/52 fraction of S that gets infected 0.459 B average daily biting (per day) 1

mh transmission probability from Im (per bite) 0.6

⌧1 rate at which S give birth (in weeks) 37

⌧2 rate at which I give birth (in weeks) 1/25

µh natural death rate 1/50 fraction of I that gives birth to babies with neurological disorder 0.133

hm transmission probability from Ih (per bite) 0.6

µb number of eggs at each deposit per capita (per day) 80

µA natural mortality rate of larvae (per day) 1/4

⌘A maturation rate from larvae to adult (per day) 0.5

1/⌘m extrinsic incubation period (in days) 125

1/µm average lifespan of adult mosquitoes (in days) 125 K maximal capacity of larvae 1.09034e+06

Table 2. Parameter values for system (24)–(25).

Figure 2 shows how our model fits the real data in the period from February 4, 2016 to November 10, 2016. More precisely, the `2 norm of the di↵erence between the real data and the curve produced by our model, in the full period, is 392.5591, which gives an average of about 9.57 cases of di↵erence each week. We have considered as initial values S0 =2, 180, 686 (S0 is the number of newborns corresponding to the simulation period) and the number of births in the period, I0 = 1, M0 = 0, and W0 = 0 for the women populations, and Am0 = Sm0 = Im0 = 1.0903e+06, and Em0 = 6.5421e+06 for the mosquitoes populations. The system of di↵erential equations has been solved by using the ode45 function of Matlab, in a MacBook Pro computer with a 2,8 GHz Intel Core i7 processor and 16 GB of memory 1600 MHz DDR3. 1.6.2. Local stability of the endemic equilibrium. In order to illustrate the local stability of the endemic equilibrium, we show that for di↵erent initial conditions the solution of the di↵erential system (24)–(25) tends to the endemic equilibrium point. We have done a number of numerical experiments. Precisely, we consider here four di↵erent initial conditions for the women population, C1, C2, C3, C4, defined in Table 3, giving the same value for S0 + I0 + M0 + W0. The system of di↵erential equations (24)–(25) has been solved for a period of 156 weeks, which correspond approximately to 3 years, by using the same numerical method and machine as for the comparison between our model and real data provided by the World Health Organization in Figure 2. These numerical experiments are included in Figure 3. 56 5. SOME MODELS RELATED TO MOSQUITOES, WATER AND ENVIRONMENTS

1800

1600

1400

1200

1000

800 Microcefaly babies 600

400

200

0 0 5 10 15 20 25 30 35 40 time (weeks)

Figure 2. Number of newborns with microcephaly. The red line corresponds to the real data obtained from the WHO [37] from 04/02/2016 to 10/11/2016 and the blue line has been obtained by solving numerically the system of ordinary di↵erential equations (24)–(25). The `2 norm of the di↵erence between the real data and our prediction is 992.5591, which gives an error of less than 9.57 cases per week.

S0 I0 M0 W0 C 2.18069 106 1 0 0 1 ⇥ C 2.17633 106 1454.79 1453.79 1453.79 2 ⇥ C 2.17851 106 727.896 726.896 726.896 3 ⇥ C 2.18025 106 146.379 145.379 146.379 4 ⇥ Table 3. Initial conditions used in the numerical simulations for the case study of Brazil (see Figure 3).

1.6.3. Sensitivity analysis of the basic reproduction number. In order to determine which are the most and less sensitive parameters of our model, we compute the values of expressions (31) according to the numerical values given in Table 2. The numerical results are as follows:

2 2 2 2 @R0 2 @R0 2 @R0 2 @R0 2 =4.02523R0, =2R0, =3.33333R0, = 62.5R0, @ @B @mh @⌘m 2 2 2 @R0 6 2 @R0 2 @R0 2 =1.8753 10 R0, =2.0001R0, = 312.51R0, @µb ⇥ @⌘a @µm 2 2 2 @R0 2 @R0 2 @R0 2 = 0.00020003R0, = 0.0000173333R0, = 16.6667R0. @µa @⇤ @⌧2 1. MATHEMATICAL MODELLING OF ZIKA DISEASE 57

18000

16000

14000 C1 C2 C3 12000 C4

10000

8000 Infected women

6000

4000

2000

0 0 20 40 60 80 100 120 140 160 time (weeks)

4000

3500

C1 3000 C2 C3 C4 2500

2000

1500 Cases of microcephaly

1000

500

0 0 20 40 60 80 100 120 140 160 time (weeks)

Figure 3. The solution of di↵erential system (24)–(25) tends to the endemic equilib- rium point, independently of initial conditions. In this figure, we show the evolution of the populations of infected women (I) and cases of microcephaly (M). Initial conditions are those of Table 3.

In Table 4, we present the sensitivity index of parameters ⇤, , B, mh, ⌧1, ⌧2, µb, µA, ⌘A, ⌘m, µm and K computed for the parameter values given in Table 2. From Table 4, we conclude that the most sensitive parameters are B and mh, which means that in order to decrease the basic reproduction number in x% we need to decrease these parameter values in x%. Therefore, in order to reduce the transmission of the Zika virus it is crucial to implement control measures that lead to a reduction on the number of daily biting (per day) B and the transmission probability from the infected mosquitoes, mh. The fraction of susceptible pregnant women S that get infected has a sensitive index very close 58 5. SOME MODELS RELATED TO MOSQUITOES, WATER AND ENVIRONMENTS

Parameters Sensitivity index Parameters Sensitivity index

⇤, K 1/2 µb 0.00007501125170 0.9237909865 µA -0.2500375056e-4

B, mh 1 ⌘A 0.5000250040 ⌧ 0.4995009233 ⌘ 1/4 1 m ⌧ 1/3 µ 1.250075011 2 m Table 4. The normalized forward sensitivity index of the basic reproduction number R0, for the parameter values of Table 2. to +1. This fact reinforces the importance of prevention measures, which protect susceptible pregnant women of becoming infected.

2. Modeling of Japanese Encephalitis under Aquatic Environmental E↵ects Japanese encephalitis (JE) is a mosquito-borne disease transmitted to humans through the bite of an infected mosquito, particularly a Culex tritaeniorhynchus mosquito. The mosquitoes breed where there is abundant water in rural agricultural areas, such as rice paddies, and become infected by feeding on vertebrate hosts (primarily pigs and wading birds) infected with the Japanese encephalitis virus. The virus is maintained in a cycle between those vertebrate animals and mosquitoes. Humans are dead-end hosts since usually they do not develop high enough concentrations of JE virus in their bloodstreams to infect feeding mosquitoes [19]. The infection on human occasionally causes brain’s inflammation with symptoms as headache, vomiting, fever, confusion and epileptic seizure. There is an estimate of about 68,000 clinical cases of occurrences with nearly 17,000 deaths every year in Asian’s countries [129]. Mathematical modeling in the field of biosciences is a subject of strong current research, see, e.g., [108, 93, 72]. One of the first mathematical models for the spread of JE has been proposed and analyzed in 2009, in [91]. Later, in 2012, the impact of media on the spreading and control of JE has been carried out [2], while in 2016 several control measures to JE, such as vaccination, medicine, and insecticide, have been investigated through optimal control and Pontryagin’s maximum principle. The state of the art on mathematical modeling and analysis of JE, seems to be the recent papers [98, 134] of 2018. In [98] a mathematical model on transmission of JE, described by a system of eight ordinary di↵erential equations, is proposed and studied. Main results are the basic reproduction number and a stability analysis around the interior equilibrium. The authors of [134] use mathematical modeling and likelihood-based inference techniques to try to explain the disappearance of JE human cases between 2006 and 2010 and its resurgence in 2011. Here we propose a mathematical model for the spread of JE, incorporating environmental e↵ects on the aquatic phase of mosquitoes, as the primary source of reproduction. The manuscript is organized as follows. In Section 2.1, we introduce the mathematical model. Then, in Section 2.2, the theoretical analysis of the model is investigated: the well posedness of the model is proved (see Theorem 2.1) and the meaningful disease free equilibrium and its local stability, in terms of the basic reproduction number, analyzed in detail (see Theorem 2.2). Section 2.3 is then devoted to numerical simulations. We end with Section 2.4 of conclusions.

2.1. Model Formulation. In our mathematical model, we shall consider environmental factors within three di↵erent host populations: humans, mosquitoes, and vertebrate animals (pigs or wading birds) as the reservoir host. In fact, unhygienic environmental conditions may enhance the presence and growth of vectors (mosquitoes) populations leading to fast spread of the disease. This is due to various kinds of household and other wastes, discharged into the environment in residential areas of population, and thus providing a very conducive environment for the growth of vectors [78, 103]. Since that e↵ect 2. MODELING OF JAPANESE ENCEPHALITIS UNDER AQUATIC ENVIRONMENTAL EFFECTS 59 could not be modeled as epidemiological compartments, we use the same scheme as in [47, 48] to handle that e↵ect on the JE disease, namely dE(t) (33) = Q + ✓N(t) ✓ E(t), dt 0 0 where E is the cumulative density of environmental discharges conducive to the growth rate of mosquitoes and animals. The cumulative density of environmental discharges due to human activities is given by ✓. There is also a constant influx given by Q0, and ✓0 is the depletion rate coecient of the environmental discharges. In our model, N(t) stands for the total human population, which is considered a varying function of time t. As for the reservoir animal populations, we consider its dynamics, strongly related to infected animals. Thus, the reservoir population constitute a “pool of infection”, that is a primarily source of infections and can be modeled by a single state variable, as in the framework of viruses, having free living pathogens in the environment (see, e.g., [15, 14, 31] and references therein for diseases like cholera, typhoid, or yellow fever). Therefore, we consider a single state variable, denoted by Ir,tomodelthis reservoir pool of infection:

dIr(t) Im(t) (34) = Bmr Ir(t) (µ1r + µ2rIr(t)) Ir(t) drIr(t)+0Ir(t)E(t), dt Nm(t)

Im where Bmr Ir(t) represents the force of infection due to interaction with mosquitoes through biting; Nm B is the average daily biting; is the transmission coecient from infected mosquitoes; Im is the mr Nm fraction of infected mosquitoes; µ1r the natural death rate of animals; µ2r the density dependent death rate; dr the death rate due to the disease; and 0 the per capita growth rate due to environmental discharges. Note that we are not interested on how the disease spread on animals. Our main goal is to study the transmission of infections from mosquitoes to humans as well as the related environmental e↵ects.

Remark 3. We have a serious doubt on the equation dP (35) = A (d + ↵ )P + EP dt 0 1 1 0 modeling the reservoir population given in [91]. Indeed, the authors of [91] assume a constant inflow A0 of infected reservoir population by asserting that: “we consider variable reservoir population which forms a ’pool of infection’ with constant inflow of infected individuals only, though it was assumed to be constant by taking birth and death rates equal [122]”. But this is incoherent with the cited reference [122], since equation (35) does not have a trivial fixed point (i.e., P ⇤(t) = 0) and thus the model proposed in [91] does not have a disease free equilibrium with zero infected animals population in contradiction with their cited references.

The following assumptions are made in order to build the compartmental classes for mosquitoes and humans populations: we do not consider immigration of infected humans; • the human population is not constant (we consider a disease induced death rate, due to fatality, • of 25%); we assume that the coecient of transmission of virus is constant and does not varies with • seasons; mosquitoes are assumed to be born susceptible. • Three epidemiological compartments are considered for the mosquito population, precisely, the aquatic phase, denoted by Am, and including eggs, larva and pupae stages; the susceptible mosquitoes, Sm; and the infected mosquitoes, Im. Also, there is no resistant phase due to the short lifetime of 60 5. SOME MODELS RELATED TO MOSQUITOES, WATER AND ENVIRONMENTS mosquitoes: dA (t) A (t) m = (1 m )(S (t)+I (t)) (µ + ⌘ ) A (t)+E(t)A (t), dt K m m A A m m 8 >dSm(t) (36) > = ⌘AAm(t) BrmIr(t)Sm(t) µmSm(t), > dt <> dI m = B I (t)S (t) µ I (t), > rm r m m m > dt > where parameter:> rm represents the transmission probability from infected animals Ir (per bite), B the average daily biting, stands for the number of eggs at each deposit per capita (per day), µA is the natural mortality rate of larvae (per day), ⌘A is the maturation rate from larvae to adult (per day), 1/µm denotes the average lifespan of adult mosquitoes (in days), and K is the maximal capacity of larvae. We denote by Nm the total adult mosquito populations at each instant of time t,beingdefined by Nm(t)=Sm(t)+Im(t) and with its dynamics satisfying the di↵erential equation dN (t) m = ⌘ A (t) µ N . dt A m m m The total human population, given by function N(t), is subdivided into two mutually exclusive com- partments, according to the disease status, namely susceptible individuals, S; and infected individuals, I. We do not consider a recovery state since there is no adequate treatment for the JE disease and no person to person infection exists: dN(t) = ⇤ µ N(t) d I(t), dt h h h 8 >dS(t) Im(t) (37) > = ⇤h Bmh S(t) µhS(t)+⌫hI(t), > dt Nm(t) <> ⇣ ⌘ dI(t) Im(t) = Bmh S(t) µhI(t) ⌫hI(t) dhI(t), > dt Nm(t) > ⇣ ⌘ > where parameter ⇤h denotes:> the recruitment rate of humans, mh represents the transmission probability from mosquitoes to humans, µh the natural death rate of humans, dh the disease induced death rate, and ⌫h is the rate by which infected individuals are recovered and become susceptible again. The fatality rate is estimated at 25% of the number of infected. Summarizing, our complete mathematical model for the JE disease is described by the following system of seven nonlinear ordinary di↵erential equations: dE(t) = Q + ✓N(t) ✓ E(t), dt 0 0 8 dI (t) I (t) > r m > = Bmr Ir(t) (µ1r + µ2rIr(t))Ir(t) drIr(t)+0Ir(t)E(t), > dt Nm(t) > >dAm(t) Am(t) > = (1 )Nm(t) (µA + ⌘A)Am(t)+E(t)Am(t), > dt K > > >dNm(t) (38) > = ⌘AAm(t) µmNm(t), > dt <> dIm(t) = BrmIr(t)(Nm(t) Im(t)) µmIm(t), > dt > >dN(t) > = ⇤ µ N(t) d I(t), > h h h > dt > >dI(t) Im(t) > = Bmh N(t) I(t) µhI(t) ⌫hI(t) dhI(t), > dt Nm(t) > > ⇣ ⌘⇣ ⌘ where N(t)=:>S(t)+I(t) and Nm(t)=Sm(t)+Im(t). 2. MODELING OF JAPANESE ENCEPHALITIS UNDER AQUATIC ENVIRONMENTAL EFFECTS 61

2.2. Mathematical Analysis of the JE Model. We begin by proving the positivity and bound- edness of solutions, which justifies the biological well-posedness of the proposed model.

Theorem 2.1 (positivity and boundedness of solutions). If the initial conditions

(E(0),Ir(0),Am(0),Nm(0),Im(0),N(0),I(0)) are non-negative, then the solutions (E(t),Ir(t),Am(t),Nm(t),Im(t),N(t),I(t)) of system (38) are non- 7 negative for all t>0 and the positive orthant R+ is positively invariant with respect to the flow of system (38). Furthermore, for initial conditions such that

⇤h N(0) 6 and E(0) 6 E⇤, µh one has ⇤h Nh(t) 6 ,E(t) 6 E⇤,Ir(t) 6 K, t > 0, µh 8 where ⇤h Q0 + ✓ µh µ2r E⇤ = and K = . ✓ B µ d + E 0 mr 1r r 0 ⇤ Proof. First of all, note that the right hand side of system (38) is continuous with continuous derivatives, thus local solutions exist and are unique. Next, assuming that E(0) > 0, and by continuity of the right hand side of the first equation of system (38), we have that E(t) remains non-negative on a small interval in the right hand side of t = 0. Therefore, there exists t =sup t 0:E(t) 0 . 0 m { > > } Obviously, by definition, tm > 0. To show that E(t) > 0 for all t > 0, we only need to prove that E(tm) > 0. Considering the first equation of system (38), that is, dE(t) = Q + ✓N(t) ✓ E(t), dt 0 0 it follows that d E(t)exp(✓ t) = Q + ✓N(t) exp(✓ t). dt{ 0 } 0 0 ⇣ ⌘ Hence, integrating this last equation with respect to t, from t0 =0totm,wehave

tm E(tm)exp(✓0tm) E(0) = Q0 + ✓N(t) exp(✓0t)dt, 0 Z ⇣ ⌘ which yields 1 tm E(tm)= E(0) + Q0 + ✓N(t) exp(✓0t)dt . exp(✓0tm) 0  Z ⇣ ⌘ As a consequence, E(tm) > 0 and we conclude that E(t) > 0 for all t>0. Similarly, we can prove that Ir(t), Am(t), Nm(t), Im(t), N(t), and I(t) are all non-negatives for all t>0. Moreover, because of the fact that I(t) > 0 for all t>0, it results from the sixth equation of system (38) that dN(t) ⇤ µ N(t). dt 6 h h Thus, applying Gronwall’s inequality, we obtain

⇤h N(t) 6 N(0) exp( µht)+ (1 exp( µht)) . µh Hence, N(t) ⇤h ,ifN(0) ⇤h for all t>0. Further, from the first equation of system (38) combined 6 µh 6 µh ⇤h with N(t) , and applying again Gronwall’s inequality, we get E(t) E⇤,wheneverE(0) E⇤. 6 µh 6 6 From the second equation of system (38), combined with E(t) 6 E⇤, we have that

dIr (B µ d + E⇤ µ I ) I . dt 6 mr 1r r 0 2r r r 62 5. SOME MODELS RELATED TO MOSQUITOES, WATER AND ENVIRONMENTS

Now, using the fact that Ir(t) > 0 for all t>0, we obtain that I (0)K µ I (t) r , where K = 2r . r 6 Ir(0) + (K + I (0)) exp( t) B µ d + E r mr 1r r 0 ⇤ Finally, lim sup Ir(t)=K and it follows that Ir(t) 6 K for all t>0. This concludes the proof. ⇤

The model system (38) admits two disease free equilibrium points (DFE), obtained by setting the right hand side of (38) to zero: a first DFE, E1, given by

✓⇤h + Q0µh ⇤h E =(E⇤,I⇤,A⇤ ,N ,I⇤ ,N⇤,I⇤)= , 0, 0, 0, 0, , 0 , 1 r m ⇤m m ✓ µ µ ✓ 0 h h ◆ corresponds to the DFE in the absence of mosquitoes population as well as absence of the aquatic phase, thus from a biological point of view this equilibrium is not interesting; a second DFE, E2,whichisthe biologically and ecologically meaningful steady state

✓⇤h + Q0µh ⌘A ⇤h (39) E =(E⇤,I⇤,A⇤ ,N ,I⇤ ,N⇤,I⇤)= , 0, %, %, 0, , 0 , 2 r m ⇤m m ✓ µ µ µ ✓ 0 h m h ◆ where K % = (Q0µhµm + ✓⇤hµm + ⌘Aµh✓0 ⌘A✓0µhµm ✓0µhµm). ✓0µhµm This equilibrium considers interaction with mosquito populations and, with that, aquatic phase as initial source of mosquito’s reproduction. We compute the basic reproduction number using the next generator matrix method as described in [127]. In doing so, we consider the following set of vectors:

I E + B Im I 0 r mr Nm r

= 0 BrmIr(Nm Im) 1 F B C B BmhIm(N I) C B C B N C @ A and

(µ1r + µ2rIr)Ir + drIr

= 0 µ I 1 . V m m B C B (µ + ⌫ + d )I C B h h h C @ A Then, we compute the Jacobian matrix associated to and at the DFE, E , that is, F V 2

0E⇤ 00 dr + µ1r 00

J = BrmNm⇤ 00,J= 0 µm 0 . F 2 3 V 2 3 0 Bmh 0 00µh + ⌫h + dh 4 5 4 5 1 The basic reproduction number R0 is obtained as the spectral radius of the matrix J (J ) at the F ⇥ V disease free equilibrium E2, being given by E ✓⇤ + Q µ 1 (40) R = 0 ⇤ = h 0 h . 0 d + µ 0 ✓ µ ⇥ d + µ r 1r ✓ 0 h ◆ r 1r The local stability of the DFE can be studied through an eigenvalue problem of the linearized system associated to (38) at the DFE E2. The disease free equilibrium point is locally asymptotically stable if all the eigenvalues, of the matrix representing the linearized system associated to (38) at the DFE E2, 2. MODELING OF JAPANESE ENCEPHALITIS UNDER AQUATIC ENVIRONMENTAL EFFECTS 63 have negative real parts. The aforementioned matrix is given by ✓ 00 0 00 0 0 0 M 00 000 2 22 3 Am⇤ A⇤ 0 M33 1 00 0 6 m K 7 M = 6 7 , 6 00⌘A ⇣ µm ⌘ 00 0 7 6 7 6 0 BrmNm⇤ 00 µm 007 6 7 6 000 0 0µh dh 7 6 7 6 000 0B 0 µ ⌫ d 7 6 mh h h h 7 4 5 Nm⇤ where M = E⇤ d µ and M = µ ⌘ + E⇤. The eigenvalues of this matrix are 22 0 r 1r 33 K A A = ✓ , = E⇤ d µ =(d + µ )(R 1), 1 0 2 0 r 1r r 1r 0 = µ , = µ , = µ ⌫ d 3 m 4 h 5 h h h and the other two remaining eigenvalues are of the following square matrix:

A⇤ ⌘A 1 m J = µm K . " ⌘A ⇣ µm ⌘ # ⌘A Since the trace of this matrix, Tr J = µm, is negative, and its determinant µm A A det J = ⌘ ⌘ 1 m⇤ = m⇤ A A K K positive, it follows that these two eigenvalues are both⇣ negative.⌘ In conclusion, we have just proved the following result. Theorem 2.2 (local stability of the biologically and ecologically meaningful disease free equilib- rium). The disease free equilibrium E2 with aquatic phase and in the presence of non-infected mosquitoes is locally asymptotically stable if R0 < 1 and unstable if R0 > 1, where R0 is given by (40). 2.3. Numerical Simulations. In this section, we illustrate stability and convergence of the so- lutions of the di↵erential system (38) to the disease free equilibrium (39) for di↵erent values of initial conditions considered in Table 5 (see Figures 4, 5, and 6 for the corresponding infected populations in model (38)). The following values of the parameters, borrowed from [91, 98], are considered:

Q0 = 50, ✓ =0.0002, ✓0 =0.0001, mr =0.0001,µ1r =0.1,dr =1/15,

⇤h = 150,µh =1/65,dh =1/45, ⌫h =0.45, mh =0.0003,

=0.6,K = 1000, =0.0001,µm =0.3, rm =0.00021. Moreover, the remaining parameters were estimated as follows:

µ2r =0.001, ⌘A =0.5,µA =0.25,B=1. The initial conditions were considered as in Table 5.

E(0) Ir(0) Am(0) Nm(0) Im(0) N(0) I(0) 40000 500 12000 10000 9000 7000 1000 45000 700 15000 12000 11000 10000 12000 35000 300 10000 7000 6000 5000 800

Table 5. Initial conditions considered

Our numerical simulations show that the evolution of the three infected populations are strictly decreasing curves and, all of them, converge to the disease free equilibrium (Figures 4–6). This means 64 5. SOME MODELS RELATED TO MOSQUITOES, WATER AND ENVIRONMENTS

Figure 4. The solution of model (38) tends to the disease free equilibrium. In this figure, we show the evolution of the infected animals population for di↵erent initial conditions.

Figure 5. The solution of model (38) tends to the disease free equilibrium. In this figure, we show the evolution of the infected mosquitoes population for di↵erent initial conditions. 2. MODELING OF JAPANESE ENCEPHALITIS UNDER AQUATIC ENVIRONMENTAL EFFECTS 65

Figure 6. The solution of model (38) tends to the disease free equilibrium. In this figure, we show the evolution of the infected humans population for di↵erent initial conditions. that our Japanese Encephalitis model (38) describes a situation of an epidemic disease through an interesting environmental e↵ect on the source of reproduction of mosquitoes, namely the aquatic phase of mosquitoes, which includes eggs, larva, and pupa stages. 2.4. Conclusions. In [91], a Japanese Encephalitis (JE) model is studied. Their results show persistence of disease in the population, that is, an endemic situation. In contrast, our obtained results highlight the importance of considering environmental e↵ects on the aquatic phase of mosquitoes, as the primary source of reproduction of mosquitoes. This is not considered in [91], where the environmental e↵ect is acting on the mature susceptible mosquitoes populations.

APPENDIX A

SageMath code to compute R0 and its sensitivity indexes of Chapter 4 sage: #the constant parameters values sage: beta=2.55; l=1.56; betaprim=7.65; kappa=0.25; rho_1=0.580; rho_2=0.001; gamma_a=0.94; sage: gamma_i=0.27; gamma_r=0.5; delta_i= 1/23; delta_p = 1/23; delta_h=1/23; N=11000000 sage: R_0= ((beta*gamma_a*l*rho_2 + sage: betaprim*rho_2*(gamma_r + delta_h))*(gamma_a + gamma_i + delta_i) ... + (beta*gamma_a*l*rho_1 + ... sage: beta*rho_1*(gamma_r + delta_h))*(gamma_a + gamma_i + delta_p))/((gamma_r sage: + delta_h)*(gamma_a + gamma_i + delta_i)*(gamma_a + gamma_i + delta_p)) sage: print R_0 4.37513184233091 sage: #Sentivity of beta sage: S_beta= ((delta_i + gamma_a + gamma_i)*gamma_a*l*rho_2 + (gamma_a*l*rho_1 + sage: (delta_h + gamma_r)*rho_1)*(delta_p + gamma_a + sage: gamma_i))*beta/((beta*gamma_a*l*rho_2 + betaprim*(delta_h + sage: gamma_r)*rho_2)*(delta_i + gamma_a + gamma_i) + (beta*gamma_a*l*rho_1 + sage: beta*(delta_h + gamma_r)*rho_1)*(delta_p + gamma_a + gamma_i)) sage: print S_beta 0.998605066564884 sage: # sensitivity of l sage: S_l= (beta*(delta_p + gamma_a + gamma_i)*gamma_a*rho_1 + beta*(delta_i + sage: gamma_a + gamma_i)*gamma_a*rho_2)*l/((beta*gamma_a*l*rho_2 + sage: betaprim*(delta_h + gamma_r)*rho_2)*(delta_i + gamma_a + gamma_i) + sage: (beta*gamma_a*l*rho_1 + beta*(delta_h + gamma_r)*rho_1)*(delta_p + sage: gamma_a + gamma_i)) sage: print S_l 0.728917935775866 sage: #sensitivity analysis of betaprim sage: S_betaprim= betaprim*(delta_h + gamma_r)*(delta_i + gamma_a + sage: gamma_i)*rho_2/((beta*gamma_a*l*rho_2 + betaprim*(delta_h + sage: gamma_r)*rho_2)*(delta_i + gamma_a + gamma_i) + (beta*gamma_a*l*rho_1 + sage: beta*(delta_h + gamma_r)*rho_1)*(delta_p + gamma_a + gamma_i)) sage: print S_betaprim 0.00139493343511561 sage: #sensitivity of rho_1 sage: S_rho1= (beta*gamma_a*l + beta*(delta_h + gamma_r))*(delta_p + gamma_a + sage: gamma_i)*rho_1/((beta*gamma_a*l*rho_2 + betaprim*(delta_h + sage: gamma_r)*rho_2)*(delta_i + gamma_a + gamma_i) + (beta*gamma_a*l*rho_1 + sage: beta*(delta_h + gamma_r)*rho_1)*(delta_p + gamma_a + gamma_i)) sage: print S_rho1 0.997350474592809

67 68 A. SAGEMATH CODE TO COMPUTE R0 AND ITS SENSITIVITY INDEXES OF CHAPTER 4 sage: #sensitivity of rho_2 sage: S_rho2=(beta*gamma_a*l + betaprim*(delta_h + gamma_r))*(delta_i + gamma_a + sage: gamma_i)*rho_2/((beta*gamma_a*l*rho_2 + betaprim*(delta_h + sage: gamma_r)*rho_2)*(delta_i + gamma_a + gamma_i) + (beta*gamma_a*l*rho_1 + sage: beta*(delta_h + gamma_r)*rho_1)*(delta_p + gamma_a + gamma_i)) sage: print S_rho2 0.00264952540719111 sage: #sensitivity of gamma_a sage: S_gammaa= (delta_h + gamma_r)*(delta_i + gamma_a + gamma_i)*(delta_p + gamma_a + sage: gamma_i)*gamma_a*((beta*(delta_p + gamma_a + gamma_i)*l*rho_1 + sage: beta*gamma_a*l*rho_1 + beta*(delta_i + gamma_a + gamma_i)*l*rho_2 + sage: beta*gamma_a*l*rho_2 + beta*(delta_h + gamma_r)*rho_1 + sage: betaprim*(delta_h + gamma_r)*rho_2)/((delta_h + gamma_r)*(delta_i + sage: gamma_a + gamma_i)*(delta_p + gamma_a + gamma_i)) - sage: ((beta*gamma_a*l*rho_2 + betaprim*(delta_h + gamma_r)*rho_2)*(delta_i + sage: gamma_a + gamma_i) + (beta*gamma_a*l*rho_1 + beta*(delta_h + sage: gamma_r)*rho_1)*(delta_p + gamma_a + gamma_i))/((delta_h + sage: gamma_r)*(delta_i + gamma_a + gamma_i)*(delta_p + gamma_a + gamma_i)^2) sage: - ((beta*gamma_a*l*rho_2 + betaprim*(delta_h + gamma_r)*rho_2)*(delta_i sage: + gamma_a + gamma_i) + (beta*gamma_a*l*rho_1 + beta*(delta_h + sage: gamma_r)*rho_1)*(delta_p + gamma_a + gamma_i))/((delta_h + sage: gamma_r)*(delta_i + gamma_a + gamma_i)^2*(delta_p + gamma_a + sage: gamma_i)))/((beta*gamma_a*l*rho_2 + betaprim*(delta_h + sage: gamma_r)*rho_2)*(delta_i + gamma_a + gamma_i) + (beta*gamma_a*l*rho_1 + sage: beta*(delta_h + gamma_r)*rho_1)*(delta_p + gamma_a + gamma_i)) sage: print S_gammaa -0.0209953489969400 sage: #sensitivity of Gamma_i sage: S_gammai=(delta_h + gamma_r)*(delta_i + gamma_a + gamma_i)*(delta_p + gamma_a + sage: gamma_i)*gamma_i*((beta*gamma_a*l*rho_1 + beta*gamma_a*l*rho_2 + sage: beta*(delta_h + gamma_r)*rho_1 + betaprim*(delta_h + sage: gamma_r)*rho_2)/((delta_h + gamma_r)*(delta_i + gamma_a + sage: gamma_i)*(delta_p + gamma_a + gamma_i)) - ((beta*gamma_a*l*rho_2 + sage: betaprim*(delta_h + gamma_r)*rho_2)*(delta_i + gamma_a + gamma_i) + sage: (beta*gamma_a*l*rho_1 + beta*(delta_h + gamma_r)*rho_1)*(delta_p + sage: gamma_a + gamma_i))/((delta_h + gamma_r)*(delta_i + gamma_a + sage: gamma_i)*(delta_p + gamma_a + gamma_i)^2) - ((beta*gamma_a*l*rho_2 + sage: betaprim*(delta_h + gamma_r)*rho_2)*(delta_i + gamma_a + gamma_i) + sage: (beta*gamma_a*l*rho_1 + beta*(delta_h + gamma_r)*rho_1)*(delta_p + sage: gamma_a + gamma_i))/((delta_h + gamma_r)*(delta_i + gamma_a + sage: gamma_i)^2*(delta_p + gamma_a + gamma_i)))/((beta*gamma_a*l*rho_2 + sage: betaprim*(delta_h + gamma_r)*rho_2)*(delta_i + gamma_a + gamma_i) + sage: (beta*gamma_a*l*rho_1 + beta*(delta_h + gamma_r)*rho_1)*(delta_p + sage: gamma_a + gamma_i)) sage: print S_gammai -0.215400624349636 sage: #sensitivity of gamma_r sage: S_gammar=(delta_h + gamma_r)*(delta_i + gamma_a + gamma_i)*(delta_p + gamma_a + sage: gamma_i)*gamma_r*((beta*(delta_p + gamma_a + gamma_i)*rho_1 + sage: betaprim*(delta_i + gamma_a + gamma_i)*rho_2)/((delta_h + sage: gamma_r)*(delta_i + gamma_a + gamma_i)*(delta_p + gamma_a + gamma_i)) - sage: ((beta*gamma_a*l*rho_2 + betaprim*(delta_h + gamma_r)*rho_2)*(delta_i + sage: gamma_a + gamma_i) + (beta*gamma_a*l*rho_1 + beta*(delta_h + A. SAGEMATH CODE TO COMPUTE R0 AND ITS SENSITIVITY INDEXES OF CHAPTER 4 69 sage: gamma_r)*rho_1)*(delta_p + gamma_a + gamma_i))/((delta_h + sage: gamma_r)^2*(delta_i + gamma_a + gamma_i)*(delta_p + gamma_a + sage: gamma_i)))/((beta*gamma_a*l*rho_2 + betaprim*(delta_h + sage: gamma_r)*rho_2)*(delta_i + gamma_a + gamma_i) + (beta*gamma_a*l*rho_1 + sage: beta*(delta_h + gamma_r)*rho_1)*(delta_p + gamma_a + gamma_i)) sage: print S_gammar -0.670604500913797 sage: #sensitivity of delta_i sage: S_deltai=(delta_h + gamma_r)*(delta_i + gamma_a + gamma_i)*delta_i*(delta_p + sage: gamma_a + gamma_i)*((beta*gamma_a*l*rho_2 + betaprim*(delta_h + sage: gamma_r)*rho_2)/((delta_h + gamma_r)*(delta_i + gamma_a + sage: gamma_i)*(delta_p + gamma_a + gamma_i)) - ((beta*gamma_a*l*rho_2 + sage: betaprim*(delta_h + gamma_r)*rho_2)*(delta_i + gamma_a + gamma_i) + sage: (beta*gamma_a*l*rho_1 + beta*(delta_h + gamma_r)*rho_1)*(delta_p + sage: gamma_a + gamma_i))/((delta_h + gamma_r)*(delta_i + gamma_a + sage: gamma_i)^2*(delta_p + gamma_a + gamma_i)))/((beta*gamma_a*l*rho_2 + sage: betaprim*(delta_h + gamma_r)*rho_2)*(delta_i + gamma_a + gamma_i) + sage: (beta*gamma_a*l*rho_1 + beta*(delta_h + gamma_r)*rho_1)*(delta_p + sage: gamma_a + gamma_i)) sage: print S_deltai -0.0345941891985019 sage: #sensitivity of delta_p sage: S_deltap=(delta_h + gamma_r)*(delta_i + gamma_a + gamma_i)*(delta_p + gamma_a + sage: gamma_i)*delta_p*((beta*gamma_a*l*rho_1 + beta*(delta_h + sage: gamma_r)*rho_1)/((delta_h + gamma_r)*(delta_i + gamma_a + sage: gamma_i)*(delta_p + gamma_a + gamma_i)) - ((beta*gamma_a*l*rho_2 + sage: betaprim*(delta_h + gamma_r)*rho_2)*(delta_i + gamma_a + gamma_i) + sage: (beta*gamma_a*l*rho_1 + beta*(delta_h + gamma_r)*rho_1)*(delta_p + sage: gamma_a + gamma_i))/((delta_h + gamma_r)*(delta_i + gamma_a + sage: gamma_i)*(delta_p + gamma_a + gamma_i)^2))/((beta*gamma_a*l*rho_2 + sage: betaprim*(delta_h + gamma_r)*rho_2)*(delta_i + gamma_a + gamma_i) + sage: (beta*gamma_a*l*rho_1 + beta*(delta_h + gamma_r)*rho_1)*(delta_p + sage: gamma_a + gamma_i)) sage: print S_deltap -0.0000919016790562303 sage: #sensitivity of delta_h sage: S_deltah=(delta_h + gamma_r)*delta_h*(delta_i + gamma_a + gamma_i)*(delta_p + sage: gamma_a + gamma_i)*((beta*(delta_p + gamma_a + gamma_i)*rho_1 + sage: betaprim*(delta_i + gamma_a + gamma_i)*rho_2)/((delta_h + sage: gamma_r)*(delta_i + gamma_a + gamma_i)*(delta_p + gamma_a + gamma_i)) - sage: ((beta*gamma_a*l*rho_2 + betaprim*(delta_h + gamma_r)*rho_2)*(delta_i + sage: gamma_a + gamma_i) + (beta*gamma_a*l*rho_1 + beta*(delta_h + sage: gamma_r)*rho_1)*(delta_p + gamma_a + gamma_i))/((delta_h + sage: gamma_r)^2*(delta_i + gamma_a + gamma_i)*(delta_p + gamma_a + sage: gamma_i)))/((beta*gamma_a*l*rho_2 + betaprim*(delta_h + sage: gamma_r)*rho_2)*(delta_i + gamma_a + gamma_i) + (beta*gamma_a*l*rho_1 + sage: beta*(delta_h + gamma_r)*rho_1)*(delta_p + gamma_a + gamma_i)) sage: print S_deltah -0.0583134348620693

APPENDIX B

Matlab code for the Wuhan case study of Chapter 4

In our Matlab code we use the fde12 routine freely available from MATLAB central [46]. clear all realdata=[0 6; 1 12; 2 19; 3 25; 4 31; 5 38; 6 44; 7 60; 8 80;9 131;10 131; 11 259; 12 467; 13 688; 14 776; 15 1776; 16 1460; 17 1739; 18 1984; 19 2101; 20 2590; 21 2827; 22 3233; 23 3892; 24 3697; 25 3151; 26 3387; 27 2653; 28 2984; 29 2473; 30 2022; 31 1820; 32 1998; 33 1506; 34 1278; 35 2051; 36 1772; 37 1891; 38 399; 39 894; 40 397; 41 650; 42 415; 43 518; 44 412; 45 439; 46 441; 47 435; 48 579; 49 206; 50 130; 51 120; 52 143; 53 146; 54 102; 55 46; 56 45; 57 20; 58 31; 59 26; 60 11; 61 18; 62 27; 63 29; 64 39; 65 39]; aux=size(realdata); deadpeople=[0 0; 1 0; 2 0; 3 0; 4 0; 5 0; 6 0; 7 0; 8 4; 9 4; 10 4; 11 8;12 15; 13 15; 14 25; 15 26; 16 26; 17 38; 18 43; 19 46; 20 45; 21 57; 22 64; 23 66; 24 73; 25 73; 26 86; 27 89; 28 97; 29 108; 30 97; 31 254; 32 121; 33 121; 34 142; 35 106; 36 106; 37 98; 38 115; 39 118; 40 109; 41 97; 42 150; 43 71; 44 52; 45 29; 46 44; 47 37; 48 35; 49 42; 50 31; 51 38; 52 31; 53 30; 54 28; 55 27; 56 23; 57 17; 58 22; 59 11; 60 07; 61 14; 62 10; 63 14; 64 13; 65 13];

%initial and end points in days t0=0; tend=realdata(aux(1),1); time=t0:tend;

%data for dead people t0dead=deadpeople(1,1); timedead=t0dead:tend; totaldead=deadpeople(:,2); deadpeople=[0 0; 1 0; 2 0; 3 0; 4 0; 5 0; 6 0; 7 0; 8 4; 9 4; 10 4; 11 8;12 15; 13 15; 14 25; 15 26; 16 26; 17 38; 18 43; 19 46; 20 45; 21 57; 22 64; 23 66; 24 73; 25 73; 26 86; 27 89; 28 97]; totalill=realdata(:,2);

%numerical constants of the model beta=2.55; ell=1.56; betap=3*beta; kappa=0.250;

71 72 B. MATLAB CODE FOR THE WUHAN CASE STUDY OF CHAPTER 4 rho1=0.580; rho2=0.001; gammaa=0.94; gammai=0.27; gammar=0.500;

N=11000000/(250); initialvalue=realdata(1,2); p0=5; e0=0; i0=initialvalue-p0; s0=N-i0; a0=0; h0=0; r0=0; d0=0;

%stepsize of the numerical method stepsize=0.001; delta=1/(23);

%differential system with the substitutions in system %S=X(1) %E=X(2) %I=X(3) %P=X(4) %A=X(5) %H=X(6) %R=X(7) %D=X(8) dead people system=@(t,X)[ -beta.*X(3).*X(1)./N-ell.*beta.*X(6).*X(1)./N-betap.*X(4).*X(1)./N; beta.*X(3).*X(1)./N+ell.*beta.*X(6).*X(1)./N+betap.*X(4).*X(1)./N-kappa.*X(2); kappa.*rho1.*X(2)-(gammaa+gammai).*X(3)-delta.*X(3); kappa.*rho2.*X(2)-(gammaa+gammai).*X(4)-delta.*X(4); kappa.*(1-rho1-rho2).*X(2);%-delta.*X(5); gammaa.*(X(3)+X(4))-gammar.*X(6)-delta.*X(6); gammai.*(X(3)+X(4))+gammar.*X(7); ];

[ts1,ys1]=fde12(1,system,t0,tend,[s0;e0;i0;p0;a0;h0;r0],stepsize);

% plot I+P+H figure hold on plot(time,totalill,’green-’,’LineWidth’,2.5) xlabel({’Time’,’(in days)’}) ylabel(’Confirmed cases per day’) plot(ts1(1,1:end),ys1(3,:)+ys1(4,:)+ys1(6,:),’black’,’linewidth’,2); hold off tau=9; aux2=size(ys1); aux3=size(ts1); sizetimes=aux3(2); totaltime=tend-t0; %S=X(1) %E=X(2) %I=X(3) %P=X(4) %A=X(5) %H=X(6) %R=X(7) %D=X(8) dead people B. MATLAB CODE FOR THE WUHAN CASE STUDY OF CHAPTER 4 73 for k=1:aux2(2)-tau*(sizetimes-1)/totaltime shifted(k)=delta.*(ys1(3,k+tau)+ys1(4,k+tau)+ys1(6,k+tau)); end newtime=tau:totaltime/(sizetimes-1):tend;

%plot of dead people figure hold on plot(totaldead,’red-’,’LineWidth’,2.5) xlabel({’Time’,’(in days)’}) ylabel(’Confirmed deads per day’) plot(newtime(:),shifted(:),’black’,’linewidth’,2); hold off

APPENDIX C

Python code for the Japanese encephalitis simulations

"""" Numerical simulations for japanese encephalitis disease """

# import modules for solving import scipy import scipy.integrate import numpy as np

# import module for plotting import pylab as pl

# System with substittutions #E=X[0], I_r = X[1]; A_m=X[2]; N_m=X[3]; I_m=[4]; N=X[5]; I=X[6]. def JEmodel(X, t, Q0, theta, theta0, betamr, mu1r, mu2r, dr, delta0, psi, K, muA, nuA, delta, mum, B, betarm, Lambdah, muh, nuh, dh, betamh ): z1= Q0 + theta*X[5] - theta0*X[0] z2=betamr*X[1]*X[4]/X[3] - (mu1r +mu2r*X[1] + dr)*X[1] + delta0*X[1]*X[0] z3= psi*(1- X[2]/K)*X[3] - (muA + nuA)*X[2] + delta*X[0]*X[2] z4= nuA*X[2]-mum*X[3] z5= B*betarm*X[1]*(X[3]-X[4])-mum*X[4] z6= Lambdah - muh*X[5]-dh*X[6] z7= (B*betamh*X[4]/X[3])*(X[5]-X[6])-nuh*X[6] - muh*X[6] - dh*X[6] return (z1, z2, z3, z4, z5, z6,z7) if __name__== "__main__":

X0= [40000, 500, 12000, 10000, 9000, 7000, 1000]; X1= [45000, 700, 15000, 12000, 11000, 10000, 1200]; X2= [35000, 300, 10000, 7000, 6000, 5000, 800]; t = np.arange(0, 20, 0.1) #t1 = np.arange(0, 200, 0.1) Q0= 50 theta=0.0002 theta0=0.0001 betamr= 0.0001 mu1r=0.1 dr=1/15.0 delta0=0.000001 psi=0.6 K=1000 muA=0.25 nuA=0.5 delta=0.0001 mum=0.3 B=1; mu2r= 0.001

75 76 C. PYTHON CODE FOR THE JAPANESE ENCEPHALITIS SIMULATIONS betarm=0.00021 Lambdah=150 muh=1.0/65 dh=1.0/45 nuh=0.45 betamh=0.0003 r=scipy.integrate.odeint(JEmodel, X0, t, args=(Q0, theta, theta0, betamr, mu1r, mu2r, dr, delta0, psi, K, muA, nuA, delta, mum, B, betarm, Lambdah, muh, dh, nuh, betamh)) r1=scipy.integrate.odeint(JEmodel, X1, t, args=(Q0, theta, theta0, betamr, mu1r, mu2r, dr, delta0, psi, K, muA, nuA, delta, mum, B, betarm, Lambdah, muh, dh, nuh, betamh)) r2=scipy.integrate.odeint(JEmodel, X2, t, args=(Q0, theta, theta0, betamr, mu1r, mu2r, dr, delta0, psi, K, muA, nuA, delta, mum, B, betarm, Lambdah, muh, dh, nuh, betamh))

#pl.plot(t,r[:,0], t,r1[:,0], t,r2[:,0]) #pl.legend([’E’, ’Ir’, ’Am’, ’Nm’, ’Im’, ’N’, ’I’],loc=’upper right’) #pl.xlabel(’Time’) #pl.ylabel(’Environmental growth’) #pl.title(’Japanese model’) #pl.savefig(’environment.eps’) #pl.show(); pl.plot(t,r[:,1], t,r1[:,1], t,r2[:,1]) #pl.legend([’E’, ’Ir’, ’Am’, ’Nm’, ’Im’, ’N’, ’I’],loc=’upper right’) pl.xlabel(’Time’) pl.ylabel(’Reservoir population’) #pl.title(’Japanese model’) pl.savefig(’reservoir.eps’) pl.show();

#pl.plot(t,r[:,2], t,r1[:,2], t,r2[:,2]) #pl.legend([’E’, ’Ir’, ’Am’, ’Nm’, ’Im’, ’N’, ’I’],loc=’upper right’) #pl.xlabel(’Time’) #pl.ylabel(’Aquatic phase’) #pl.title(’Japanese model’) #pl.savefig(’aquatic.eps’) #pl.show();

#pl.plot(t,r[:,3], t,r1[:,3], t,r2[:,3]) #pl.legend([’E’, ’Ir’, ’Am’, ’Nm’, ’Im’, ’N’, ’I’],loc=’upper right’) #pl.xlabel(’Time’) #pl.ylabel(’Total mosquitoes population’) #pl.title(’Japanese model’) #pl.savefig(’total_mosquitoes.eps’) #pl.show(); pl.plot(t,r[:,4],t,r1[:,4],t,r2[:,4]) pl.xlabel(’Time’) pl.ylabel(’Infected mosquitoes’) #pl.title(’Japanese model’) pl.savefig(’mosquitoes.eps’) pl.show(); C. PYTHON CODE FOR THE JAPANESE ENCEPHALITIS SIMULATIONS 77

#pl.plot(t,r[:,5], t,r[:,5], t,r2[:,5]) #pl.legend([’E’, ’Ir’, ’Am’, ’Nm’, ’Im’, ’N’, ’I’],loc=’upper right’) #pl.xlabel(’Time’) #pl.ylabel(’Total humans population’) #pl.title(’Japanese model’) #pl.savefig(’total_humans.eps’) #pl.show(); pl.plot(t,r[:,6],t,r1[:,6],t,r2[:,6]) pl.xlabel(’Time’) pl.ylabel(’Infected humans’) #pl.title(’Japanese model’) pl.savefig(’infected_human.eps’) pl.show()

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mathematics

Article Mathematical Modeling of Japanese Encephalitis under Aquatic Environmental Effects

Faïçal Ndaïrou 1,2,† , Iván Area 1,† and Delfim F. M. Torres 2,*,† 1 E. E. Aeronáutica e do Espazo, Campus de Ourense, Universidade de Vigo, 32004 Ourense, Spain; [email protected] (F.N.); [email protected] (I.A.) 2 Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal * Correspondence: delfi[email protected]; Tel.: +351-234-370-668 † These authors contributed equally to this work.

!"#!$%&'(! Received: 10 September 2020; Accepted: 19 October 2020; Published: 30 October 2020 !"#$%&'

Abstract: We propose a mathematical model for the spread of Japanese encephalitis with emphasis on the environmental effects on the aquatic phase of mosquitoes. The model is shown to be biologically well-posed and to have a biologically and ecologically meaningful disease-free equilibrium point. Local stability is analyzed in terms of the basic reproduction number and numerical simulations presented and discussed.

Keywords: mathematical modeling; Japanese encephalitis; environment; numerical simulations

MSC: 92D25; 92D30

1. Introduction Japanese encephalitis (JE) is a mosquito-borne disease transmitted to humans through the bite of an infected mosquito, particularly a Culex tritaeniorhynchus mosquito. The mosquitoes breed where there is abundant water in rural agricultural areas, such as rice paddies, and become infected by feeding on vertebrate hosts (primarily pigs and wading birds) infected with the Japanese encephalitis virus. The virus is maintained in a cycle between those vertebrate animals and mosquitoes. Humans are dead-end hosts since they usually do not develop high enough concentrations of JE virus in their bloodstreams to infect feeding mosquitoes [1]. Human infection occasionally causes brain inflammation with symptoms such as headache, vomiting, fever, confusion, and epileptic seizure. There is an estimate of about 68,000 clinical cases of occurrences with nearly 17,000 deaths every year in Asian countries [2]. The first case of Japanese encephalitis viral disease was documented in 1871 in Japan, but the virus itself was first isolated in 1935 and has subsequently been found across most of Asia. There is uncertainty on the origin of the name of that virus; however, phylogenetic comparisons with other flaviviruses suggest that it evolved from an African ancestral virus, perhaps as recently as a few centuries ago (see [3] and references therein). Note that, despite its name, Japanese encephalitis is now relatively rare in Japan as a result of a mass immunization program. Mathematical modeling in the field of biosciences is a subject of strong current research (see, e.g., [4–6]). One of the first mathematical models for the spread of JE was proposed and analyzed in 2009 in [7]. Later, in 2012, a study of the impact of media on the spreading and control of JE was carried out [8], while in 2016, several measures to control JE, such as vaccination, medicine, and insecticide, were investigated through optimal control and Pontryagin’s maximum principle. The state of the art of mathematical modeling and analysis of JE seems to be found in recent papers [9,10] from 2018.

Mathematics 2020, 8, 1880; doi:10.3390/math8111880 www.mdpi.com/journal/mathematics Mathematics 2020, 8, 1880 2 of 14

In [9], a mathematical model of transmission of JE, described by a system of eight ordinary differential equations, is proposed and studied. The main results are the basic reproduction number and a stability analysis around the interior equilibrium. The authors of [10] use mathematical modeling and likelihood-based inference techniques to try to explain the disappearance of JE human cases between 2006 and 2010 and its resurgence in 2011. Here, we propose a mathematical model for the spread of JE, incorporating environmental effects on the aquatic phase of mosquitoes as the primary source of reproduction. The manuscript is organized as follows. In Section 2, we introduce the mathematical model. Then, in Section 3, the theoretical analysis of the model is investigated: the well-posedness of the model is proved (see Theorem 1), and the meaningful disease-free equilibrium and its local stability, in terms of the basic reproduction number, are analyzed in detail (see Theorem 2). Section 4 is then devoted to numerical simulations. We end with Section 5 of conclusions, where we also point out some possible directions of future research.

2. Model Formulation In our mathematical model, we shall consider environmental factors within three different host populations: humans, mosquitoes, and vertebrate animals (pigs or wading birds) as the reservoir host. In fact, unhygienic environmental conditions may enhance the presence and growth of vectors (mosquitoes) populations, leading to fast spread of the disease. This is due to the discharge of various kinds of household and other wastes into the environment in residential areas of population, thus providing a very conducive environment for the growth of vectors [11,12]. Since that effect could not be modeled as epidemiological compartments, we use the same scheme as in [13,14] to handle that effect on the JE disease, namely [7]

dE(t) = Q + qN(t) q E(t), (1) dt 0 0 where E is the cumulative density of environmental discharges conducive to the growth rate of mosquitoes and animals. The cumulative density of environmental discharges due to human activities is given by q. There is also a constant influx given by Q0, and q0 is the depletion rate coefficient of the environmental discharges. In our model, N(t) stands for the total human population, which is considered a varying function of time t. As for the reservoir animal populations, we consider their dynamics, strongly related to infected animals. Thus, the reservoir population constitutes a “pool of infection” that is a primarily source of infections and can be modeled by a single state variable, as in the framework of viruses, having free living pathogens in the environment (see, e.g., [15–17] and references therein for diseases like cholera, typhoid, or yellow fever). Therefore, we consider a single state variable, denoted by Ir, to model this reservoir pool of infection:

dIr(t) Im(t) = Bbmr Ir(t) (µ1r + µ2r Ir(t)) Ir(t) dr Ir(t)+d0 Ir(t)E(t), (2) dt Nm(t)

Im where Bbmr Ir(t) represents the force of infection due to interaction with mosquitoes through biting; Nm B is the average daily biting; b is the transmission coefficient from infected mosquitoes; Im is the mr Nm fraction of infected mosquitoes; µ1r is the natural death rate of animals; µ2r is the density dependent death rate; dr is the death rate due to the disease; and d0 is the per capita growth rate due to environmental discharges. Note that we are not interested in how the disease spreads to other animals. Our main goal is to study the transmission of infections from mosquitoes to humans as well as the related environmental effects. The following assumptions are made in order to build the compartmental classes for mosquitoes and human populations: Mathematics 2020, 8, 1880 3 of 14

we do not consider immigration of infected humans; • the human population is not constant (we consider a disease-induced death rate, due to fatality, • of 25%); we assume that the coefficient of transmission of the virus is constant and does not vary • with seasons, which is reasonable due to the short course of the disease; mosquitoes are assumed to be born susceptible. • Three epidemiological compartments are considered for the mosquito population—precisely, the aquatic phase, denoted by Am, and including eggs, larva, and pupae stages; the susceptible mosquitoes, Sm; and the infected mosquitoes, Im. There is also no resistant phase due to the short lifetime of mosquitoes:

dAm(t) Am(t) = y(1 )(Sm(t)+Im(t)) (µA + hA) Am(t)+dE(t)Am(t), 8 dt K > dSm(t) > = hA Am(t) BbrmIr(t)Sm(t) µmSm(t), (3) > dt <> dIm > = BbrmIr(t)Sm(t) µm Im(t), > dt > > where parameter: brm represents the transmission probability from infected animals Ir (per bite), B is the average daily biting, y stands for the number of eggs at each deposit per capita (per day), µA is the natural mortality rate of larvae (per day), hA is the maturation rate from larvae to adult (per day), and d is the per capita growth rate in the level of aquatic phase due to conducive environmental discharge. Here, 1 denotes the average lifespan of adult mosquitoes (in days), and K is the maximal µm capacity of larvae. We denote by Nm the total adult mosquito populations at each instant of time t, being defined by Nm(t)=Sm(t)+Im(t) and with its dynamics satisfying the differential equation

dN (t) m = h A (t) µ N . dt A m m m

The total human population, given by function N(t), is subdivided into two mutually exclusive compartments, according to the disease status, namely susceptible individuals, S, and infected individuals, I. We do not consider a recovery state since there is no adequate treatment for the JE disease and no person-to-person infection exists:

dN(t) = L µ N(t) d I(t), dt h h h 8 > dS(t) Im(t) > = Lh Bbmh S(t) µhS(t)+nh I(t), (4) > dt Nm(t) <> ⇣ ⌘ dI(t) Im(t) = Bbmh S(t) µh I(t) nh I(t) dh I(t), > dt Nm(t) > > ⇣ ⌘ :> where parameter Lh denotes the recruitment rate of humans, bmh represents the transmission probability from mosquitoes to humans, µh is the natural death rate of humans, dh is the disease-induced death rate, and nh is the rate by which infected individuals are recovered and become susceptible again. The fatality rate is estimated at 25% of the number of infected. In summary, our complete mathematical model for the JE disease is described by the following system of seven nonlinear ordinary differential equations: Mathematics 2020, 8, 1880 4 of 14

dE(t) = Q + qN(t) q E(t), dt 0 0 8 > dI (t) I (t) > r = Bb m I (t) (µ + µ I (t))I (t) d I (t)+d I (t)E(t), > mr r 1r 2r r r r r 0 r > dt Nm(t) > > dAm(t) Am(t) > = y(1 )Nm(t) (µ + h )Am(t)+dE(t)Am(t), > dt K A A > > > dNm(t) > = hA Am(t) µm Nm(t), (5) > dt <> dIm(t) = BbrmIr(t)(Nm(t) Im(t)) µm Im(t), > dt > > ( ) > dN t > = Lh µh N(t) dh I(t), > dt > > dI(t) I (t) > = Bb m N(t) I(t) µ I(t) n I(t) d I(t), > dt mh N (t) h h h > m > ⇣ ⌘⇣ ⌘ > where N(t)=:S(t)+I(t) and Nm(t)=Sm(t)+Im(t).

3. Mathematical Analysis of the JE Model We begin by proving the positivity and boundedness of solutions, which justifies the biological well-posedness of the proposed model.

Theorem 1 (positivity and boundedness of solutions). If the initial conditions (E(0), Ir(0), Am(0), Nm(0), Im(0), N(0), I(0)) are non-negative, then the solutions (E(t), Ir(t), Am(t), Nm(t), Im(t), N(t), I(t)) of System (5) are non-negative for all t > 0 and the 7 positive orthant R+ is positively invariant with respect to the flow of System (5). Furthermore, for initial conditions such that Lh N(0) 6 and E(0) 6 E⇤, µh one has Lh N(t) 6 , E(t) 6 E⇤, Ir(t) 6 L, t > 0, µh 8 where Lh Q0 + q µh Bbmr µ1r dr + d0E⇤ E⇤ = and L = . q0 µ2r

Proof. First of all, note that the right hand side of System (5) is continuous with continuous derivatives; thus, local solutions exist and are unique. Next, assuming that E(0) > 0, and by continuity of the right hand side of the first equation of System (5), we have that E(t) remains non-negative on a small interval in the right hand side of t = 0. Therefore, there exists t = sup t 0 : E(t) 0 . 0 m { > > } Obviously, by definition, tm > 0. To show that E(t) > 0 for all t > 0, we only need to prove that E(tm) > 0. Considering the first equation of System (5), that is,

dE(t) = Q + qN(t) q E(t), dt 0 0 it follows that d E(t) exp(q t) = Q + qN(t) exp(q t). dt{ 0 } 0 0 ⇣ ⌘ Hence, integrating this last equation with respect to t, from t0 = 0 to tm, we have

tm E(tm) exp(q0tm) E(0)= Q0 + qN(t) exp(q0t)dt, 0 Z ⇣ ⌘ Mathematics 2020, 8, 1880 5 of 14 which yields 1 tm E(tm)= E(0)+ Q0 + qN(t) exp(q0t)dt . exp(q0tm) 0  Z ⇣ ⌘ As a consequence, E(tm) > 0 and we conclude that E(t) > 0 for all t > 0. Similarly, we can prove that Ir(t), Am(t), Nm(t), Im(t), N(t), and I(t) are all non-negatives for all t > 0. Moreover, because of the fact that I(t) > 0 for all t > 0, it results from the sixth equation of System (5) that

dN(t) 6 Lh µh N(t). dt Thus, applying Gronwall’s inequality, we obtain

Lh N(t) 6 N(0) exp( µht)+ (1 exp( µht)) . µh

Hence, N(t) Lh , if N(0) Lh for all t > 0. Furthermore, from the first equation of 6 µh 6 µh Lh System (5) combined with N(t) , and applying Gronwall’s inequality again, we get E(t) E⇤, 6 µh 6 whenever E(0) 6 E⇤. From the second equation of System (5), combined with E(t) 6 E⇤, we have that

dIr 6 (A µ2r Ir) Ir, with A = Bbmr µ1r dr + d0E⇤. dt

dIr 1 dIr A 1 Note that 6 (A µ2r Ir) Ir implies 2 6 µ2r + and, by setting z(t)= , we get dt Ir dt Ir Ir

dz(t) 6 µ2r Az(t). dt Then, we follow Gronwall’s inequality to obtain that

µ2r z(t) 6 z(0) exp( At) (1 exp( At)) , A meaning that AIr(0) Ir(t) 6 . A exp( At)+µ I (0) (1 exp( At)) 2r r Finally, lim sup I (t)= A , and it follows that I (t) A for all t > 0. This concludes r µ2r r 6 µ2r the proof.

The model System (5) admits two disease-free equilibrium points (DFE), obtained by setting the right hand side of (5) to zero. The first DFE, E1, given by

qLh + Q0µh Lh E = (E⇤, I⇤, A⇤ , N⇤ , I⇤ , N⇤, I⇤) = , 0, 0, 0, 0, ,0 , 1 r m m m q µ µ ✓ 0 h h ◆ corresponds to the DFE in the absence of mosquitoes population as well as absence of the aquatic phase; thus, from a biological point of view, this equilibrium is not interesting. There is a second DFE, E2, which is the biologically and ecologically meaningful steady state

qLh + Q0µh hA Lh E = (E⇤, I⇤, A⇤ , N⇤ , I⇤ , N⇤, I⇤) = , 0, $, $, 0, ,0 , (6) 2 r m m m q µ µ µ ✓ 0 h m h ◆ where K $ = (dQ0µhµm + dqLhµm + yhAµhq0 hAq0µhµm q0µhµmµA), yq0µhhA Mathematics 2020, 8, 1880 6 of 14 which can be rewritten as K $ = (dµm(qLh + Q0µh)+yhAµhq0 hAq0µhµm q0µhµmµA) yq0µhhA K dµm µmµA = E⇤ + y µ . y h m h ✓ A A ◆ Therefore, hA K hAy N⇤ = $ = dE⇤ + h µ . (7) m µ y µ A A m ✓ m ◆ The equilibrium E2 considers interaction with mosquito populations and, with that, the aquatic phase as the initial source of mosquito reproduction. We compute the basic reproduction number using the next generator matrix method as described in [18]. In doing so, we consider the following set of vectors:

d I E + Bb Im I 0 r mr Nm r 0 Bb I (N I ) 1 = rm r m m F B C B C B Bbmh Im(N I) C B C B N C @ A and (µ1r + µ2r Ir)Ir + dr Ir

= 0 µ I 1 . V m m B C B (µ + n + ) C B h h dh I C @ A Then, we compute the Jacobian matrix associated to and at the DFE, E , that is, F V 2

d0E⇤ 00 dr + µ1r 00 J = BbrmNm⇤ 00, J = 0 µm 0 . F 2 3 V 2 3 0 Bb 0 00µ + n + d 6 mh 7 6 h h h 7 4 5 4 5 1 The basic reproduction number R0 is obtained as the spectral radius of the matrix J (J ) at F ⇥ V the disease-free equilibrium E2, being given by

d E qL + Q µ 1 R = 0 ⇤ = d h 0 h . (8) 0 d + µ 0 q µ ⇥ d + µ r 1r ✓ 0 h ◆ r 1r The local stability of the disease-free equilibrium (DFE) can be studied through an eigenvalue problem of the linearized system associated with System (5) at the DFE E2. The DFE point is locally asymptotically stable if all the eigenvalues of the matrix representing the linearized system associated to System (5) at the DFE E2 have negative real parts [19]. The aforementioned matrix is given by

q 00 0 00 0 0 0 M 00 000 2 22 3 Am⇤ dA⇤ 0 M33 y 1 00 0 6 m K 7 M = 6 7 , 6 00hA ⇣ µm ⌘ 00 0 7 6 7 6 0 BbrmN⇤ 00 µm 007 6 m 7 6 000 0 0µ d 7 6 h h 7 6 7 6 000 0Bbmh 0 µh nh dh 7 4 5 Mathematics 2020, 8, 1880 7 of 14

yNm⇤ hAy where M22 = d0E⇤ dr µ1r and M33 = µ h + dE⇤ = , using Equation (7). K A A µm The eigenvalues of this matrix are

l = q , l = d E⇤ d µ =(d + µ )(R 1), 1 0 2 0 r 1r r 1r 0 l = µ , l = µ , l = µ n d 3 m 4 h 5 h h h and the other two remaining eigenvalues are of the following square matrix:

hAy y 1 Am⇤ J = µm K . " h ⇣ µ ⌘ # A m

hAy Since the trace of this matrix, Tr J = µm, is negative, and its determinant µm

A A det J = h y h y 1 m⇤ = m⇤ A A K K ⇣ ⌘ positive, it follows that these two eigenvalues are both negative. In conclusion, we have just proved the following result.

Theorem 2 (local stability of the biologically and ecologically meaningful disease free equilibrium). The disease-free equilibrium E2 with aquatic phase and in the presence of non-infected mosquitoes is locally asymptotically stable if R0 < 1 and unstable if R0 > 1, where R0 is given by Equation (8).

4. Numerical Simulations In this section, we illustrate stability and convergence of the solutions of the differential System (5) to the disease-free equilibrium (Equation (6)) for different values of initial conditions considered in Table 1 (see Figures 1–3 for the corresponding infected populations in Model (5)). We perform numerical simulations to solve the model system (Model (5)) using the Python programming language, precisely the freely available routine integrate.odeint of library SciPy. The following values of the parameters, borrowed from [7,9], are considered:

Q0 = 50, q = 0.0002, q0 = 0.0001, bmr = 0.0001, µ1r = 0.1, dr = 1/15,

d0 = 0.000001, Lh = 150, µh = 1/65, dh = 1/45, nh = 0.45, bmh = 0.0003,

y = 0.6, K = 1000, d = 0.0001, µm = 0.3, brm = 0.00021.

Figure 1. The solution of Model (5) tends toward the disease-free equilibrium. In this figure, we show the evolution of the infected animals population for different initial conditions. Mathematics 2020, 8, 1880 8 of 14

Figure 2. The solution of Model (5) tends toward the disease-free equilibrium. In this figure, we show the evolution of the infected mosquitoes population for different initial conditions.

Figure 3. The solution of Model (5) tends toward the disease-free equilibrium. In this figure, we show the evolution of the infected humans population for different initial conditions.

Table 1. Initial conditions considered.

E(0) Ir(0) Am(0) Nm(0) Im(0) N(0) I(0)

X1(0) 40,000 500 12,000 10,000 9000 7000 1000 X2(0) 45,000 700 15,000 12,000 11,000 10,000 12,000 X3(0) 35,000 300 10,000 7000 6000 5000 800

Moreover, the remaining parameters were estimated as follows:

µ2r = 0.001, hA = 0.5, µA = 0.25, B = 1.

The value of the DFE, E2 is computed as below:

E2 = (E⇤, Ir⇤, Am⇤ , Nm⇤ , Im⇤ , N⇤, I⇤) = (122.959, 0, 262.296, 437.160, 0, 9750, 0) .

The matrices J and J are obtained as follows F V Mathematics 2020, 8, 1880 9 of 14

0.000123 0 0 0.167 0 0 J = 0.0918 0 0 , J = 0 0.3 0 , F 2 3 V 2 3 0 0.0003 0 0 0 0.488 6 7 6 7 4 5 4 5 which leads to the value of R0 = 0.000738. Furthermore, we have that the matrix M is equal to

0.0001 0 0 0 0 0 0 0 0.167 0 0 0 0 0 2 3 0.0262 0 1 0.443 0 0 0 6 7 M = 6 0 0 0.5 0.3 0 0 0 7 , 6 7 6 0 0.0918 0 0 0.3 0 0 7 6 7 6 000000.0154 0.0222 7 6 7 6 7 6 0 0 0 0 0.0003 0 0.487 7 4 5 and its eigenvalues are

1.236, 0.0636, 0.0001, 0.0154, 0.488, 0.3, 0.167 all negatives in accordance with Theorem 2, since R0 < 1. The initial conditions were considered as in Table 1 and the evolution of the three infected populations are strictly decreasing curves with all of them converging to the disease-free equilibrium (Figures 1–3) for these specific parameter values. Our numerical simulations show that the evolution of the three infected populations are strictly decreasing curves, and all of them converge to the disease-free equilibrium (Figures 1–3). This means that our Japanese Encephalitis model (Model (5)) describes a situation of an epidemic disease through an interesting environmental effect on the source of reproduction of mosquitoes, namely the aquatic phase of mosquitoes, which includes eggs, larva, and pupa stages. Furthermore, in Figures 4 and 5, the variation of the evolution of the infected animals population and infected mosquitoes population is shown, respectively, with respect to different values in the level of environmental discharge due to constant influx (Q0). It is found that with the decrease in the level of environmental discharge due to constant influx (Q0), the infected animals population and infected mosquitoes population decrease and approach the disease-free equilibrium state.

Figure 4. Variation of animals population with respect to Q0. Mathematics 2020, 8, 1880 10 of 14

Figure 5. Variation of infected mosquitoes population with respect to Q0.

We observe in Figures 6 and 7 that the decrease of the per capita growth rate d0 of animals due to environmental discharges results in the decrease of the infected animals population as well as for the infected mosquitoes population.

Figure 6. Variation of animals population with respect to d0.

Figure 7. Variation of infected mosquitoes population with respect to d0. Mathematics 2020, 8, 1880 11 of 14

The role of conducive environmental discharge d on the infected mosquitoes population is shown in Figure 8. We found that when the value of d is smaller than 0.0001, there is then a strict decrease in the number of infected mosquitoes population. However, when d becomes larger, the infected mosquitoes population increases up to a certain optimum value and then decreases to the disease-free equilibrium state.

Figure 8. Variation of infected mosquitoes population with respect to d.

5. Conclusions In [7], a Japanese Encephalitis model is studied. Its results show persistence of disease in the population—that is, an endemic situation. In contrast, our obtained results highlight the importance of considering environmental effects on the aquatic phase of mosquitoes as the primary source of reproduction of mosquitoes. This is not considered in [7], where the environmental effect is acting on the mature susceptible mosquitoes populations. Here, we have shown that the basic reproduction number is a linear dependent function with respect to the equilibrium state of the cumulative density of environmental discharges, conducive to the growth rate of mosquitoes and animals. All our computational experiments were carried out using the free and open-source scientific computing Python library SciPy. To make our results reproducible, we provide the main computer code in Appendix A. As future work, it would be interesting to validate the model with real data and take into account possible control measures, e.g., vaccination of the population and vector or environmental controls.

Author Contributions: The authors equally contributed to this paper, and read and approved the final manuscript: formal analysis, F.N., I.A. and D.F.M.T.; investigation, F.N., I.A. and D.F.M.T.; writing—original draft, F.N., I.A. and D.F.M.T.; writing—review & editing, F.N., I.A. and D.F.M.T. All authors have read and agreed to the published version of the manuscript. Funding: This research was partially funded by the Portuguese Foundation for Science and Technology (FCT) through CIDMA, grant number UIDB/04106/2020 (F.N. and D.F.M.T.); and by the Agencia Estatal de Investigación (AEI) of Spain under Grant MTM2016-75140-P, cofinanced by the European Community fund FEDER (I.A.). F.N. was also supported by FCT through the PhD fellowship PD/BD/150273/2019. Acknowledgments: The authors are grateful to four anonymous reviewers for several pertinent questions and comments. Conflicts of Interest: The authors declare no conflict of interest.

Appendix A. Python Code for Figures 1–3

"""" Numerical simulations for Japanese Encephalitis disease~"""

#importmodulesforsolving Mathematics 2020, 8, 1880 12 of 14 import scipy import scipy.integrate import numpy as~np

#importmoduleforplotting import pylab as~pl

#Systemwithsubstitutions #E=X[0], I_r = X[1]; A_m=X[2]; N_m=X[3]; I_m=[4]; N=X[5]; I=X[6]. def JEmodel(X, t, Q0, theta, theta0, betamr, mu1r, mu2r, dr, delta0, psi, K, muA, nuA, delta, mum, B, betarm, Lambdah, muh, nuh, dh, betamh ): z1= Q0 + theta*X[5] - theta0*X[0] z2=betamr*X[1]*X[4]/X[3] - (mu1r +mu2r*X[1] + dr)*X[1] + delta0*X[1]*X[0] z3= psi*(1- X[2]/K)*X[3] - (muA + nuA)*X[2] + delta*X[0]*X[2] z4= nuA*X[2]-mum*X[3] z5= B*betarm*X[1]*(X[3]-X[4])-mum*X[4] z6= Lambdah - muh*X[5]-dh*X[6] z7= (B*betamh*X[4]/X[3])*(X[5]-X[6])-nuh*X[6] - muh*X[6] - dh*X[6] return (z1, z2, z3, z4, z5, z6, z7) if __name__== "__main__":

X0= [40000, 500, 12000, 10000, 9000, 7000, 1000]; X1= [45000, 700, 15000, 12000, 11000, 10000, 1200]; X2= [35000, 300, 10000, 7000, 6000, 5000, 800]; t=np.arange(0,20,0.1)

Q0= 50 theta=0.01 theta0=0.0001 betamr= 0.0001 mu1r=0.1 dr=1/15.0 delta0=0.000001 psi=0.6 K=1000 muA=0.25 nuA=0.5 delta=0.0001 mum=0.3 B=1; mu2r= 0.001 betarm=0.00021 Lambdah=150 muh=1.0/65 dh=1.0/45 nuh=0.45 betamh=0.0003 r=scipy.integrate.odeint(JEmodel, X0, t, args=(Q0, theta, theta0, betamr, mu1r, mu2r, dr, delta0, psi, K, muA, nuA, delta, mum, B, betarm, Lambdah, muh, dh, nuh, betamh)) r1=scipy.integrate.odeint(JEmodel, X1, t, args=(Q0, theta, theta0, betamr, mu1r, mu2r, dr, delta0, psi, K, muA, nuA, delta, mum, B, betarm, Lambdah, muh, dh, nuh, betamh)) r2=scipy.integrate.odeint(JEmodel, X2, t, args=(Q0, theta, theta0, betamr, mu1r, mu2r, dr, delta0, psi, K, muA, nuA, delta, mum, B, betarm, Lambdah, muh, dh, nuh, betamh)) Mathematics 2020, 8, 1880 13 of 14

pl.plot(t,r[:,1], t,r1[:,1], t,r2[:,1]) pl.legend([’$X_1(0)$’, ’$X_2(0)$’, ’$X_3(0)$’],loc=’upper right’) pl.xlabel(’Time (weeks)’) pl.ylabel(’Reservoir population’) #pl.title(’Japaneese model’) pl.savefig(’reservoir.eps’) pl.show(); pl.plot(t,r[:,4], t,r1[:,4],t,r2[:,4]) pl.xlabel(’Time (weeks)’) pl.ylabel(’Infected mosquitoes’) pl.legend([’$X_1(0)$’, ’$X_2(0)$’, ’$X_3(0)$’],loc=’upper right’) pl.savefig(’mosquitoes.eps’) pl.show(); pl.plot(t,r[:,6], t,r1[:,6],t,r2[:,6]) pl.xlabel(’Time (weeks)’) pl.ylabel(’Infected humans’) pl.legend([’$X_1(0)$’, ’$X_2(0)$’, ’$X_3(0)$’],loc=’upper right’) pl.savefig(’infected_human.eps’) pl.show()

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c 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). Received: 11 August 2017 DOI: 10.1002/mma.4702

SPECIAL ISSUE PAPER

Mathematical modeling of Zika disease in pregnant women and newborns with microcephaly in Brazil

Faïçal Ndaïrou1,2 Iván Area2 Juan J. Nieto3 Cristiana J. Silva4 Delfim F. M. Torres4

1African Institute for Mathematical Sciences, AIMS-Cameroon, P.O. Box 608 We propose a new mathematical model for the spread of Zika virus. Special Limbé Crystal Gardens, South West attention is paid to the transmission of microcephaly. Numerical simulations Region, Cameroon show the accuracy of the model with respect to the Zika outbreak occurred in 2Departamento de Matemática Aplicada II, E.E. Aeronáutica e do Espazo, Brazil. Universidade de Vigo, Campus As Lagoas s/n Ourense, 32004, Spain KEYWORDS 3Instituto de Matemáticas, Universidade Brazil, epidemiology, mathematical modeling, positivity and boundedness of solutions, stability, de Santiago de Compostela, Santiago de Zika virus and microcephaly Compostela, 15782, Spain 4Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, Aveiro, 3810-193, Portugal

Correspondence Delfim F. M. Torres, Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, Aveiro, 3810-193, Portugal. Email: [email protected]

Communicated by: A. Debbouche

Funding information Agencia Estatal de Innovación, Grant/Award Number: MTM2016-75140-P; Xunta de Galicia, Grant/Award Number: GRC 2015-004 and R 2016/022; FCT, Grant/Award Number: CIDMA UID/MAT/04106/2013, TOCCATA PTDC/EEI-AUT/2933/2014 and SFRH/BPD/72061/2010

MSC Classification: 34D20; 92D30 1 INTRODUCTION

Zika virus infection on humans is mainly caused by the bite of an infected Aedes mosquito, either Aedes aegypti or Aedes albopictus. The infection on human usually causes rash, mild fever, conjunctivitis, and muscle pain. These symp- toms are quite similar to dengue and chikungunya diseases, which can be transmitted by the same mosquitoes. Other modes of transmission of Zika disease have been observed, as sexual transmission, though less common.1 Such modes of

Math Meth Appl Sci. 2018;41:8929–8941. wileyonlinelibrary.com/journal/mma Copyright © 2017 John Wiley & Sons, Ltd. 8929 8930 NDAÏROU ET AL. transmission are included in mathematical models found in the recent literature: see van den Driessche2, Section 8 for a good state of the art. The name of the virus comes from the Zika forest in Uganda, where the virus was isolated for the first time in 1947. Up to very recent times, most of the Zika outbreaks have occurred in Africa with some sporadic outbreaks in Southeast Asia and also in the Pacific Islands. Since May 2015, Zika virus infections have been confirmed in Brazil, and since Octo- ber 2015, other countries and territories of the Americas have reported the presence of the virus: see Worobey,3 where evolutionary trees, constructed using both newly sequenced and previously available Zika virus genomes, reveal how the recent outbreak arose in Brazil and spread across the Americas. The subject attracted a lot of attention and is now under strong current investigations. In Agusto et al,4 a deterministic model, based on a system of ordinary differential equations, was proposed for the study of the transmission dynamics of the Zika virus. The model incorporates mother-to-child transmission as well as the development of microcephaly in newly born babies. The analysis shows that the disease-free equilibrium of the model is locally and globally asymptotically stable, whenever the associated reproduction number is less than one, and unstable otherwise. A sensitivity analysis was performed showing that the mosquito biting, recruit- ment, and death rates, are among the parameters with the strongest impact on the basic reproduction number. Then, some control strategies were proposed with the aim to reduce such values.4 A 2-patch model, where host-mobility is mod- eled using a Lagrangian approach, is used in Moreno et al,1 to understand the role of host-movement on the transmission dynamics of Zika virus in an idealized environment. Here, we are concerned with the situation in Brazil and its conse- quences on brain anomalies, in particular microcephaly, which occur in fetuses of infected pregnant woman. This is a crucial question as far as the main problem related with Zika virus is precisely the number of neurological disorders and neonatal malformations.5 Our study is based on the Zika virus situation reports for Brazil, as publicly available at the World Health Organization (WHO) web page. Based on a systematic review of the literature up to May 30, 2016, the WHO has concluded that Zika virus infection during pregnancy is a cause of congenital brain abnormalities, including microcephaly. Moreover, another important conclusion of the WHO is that the Zika virus is a trigger of Guillain-Barré syndrome.6 Our analysis is focused on the number of confirmed cases of Zika in Brazil. For this specific case, an estimate of the population of the country is known, as well as the number of newborns. Moreover, from WHO data, it is possible to have an estimation of the number of newborn babies with neurological disorder. Our mathematical model allows to predict the number of cases of newborn babies with neurological disorder. The manuscript is organized as follows. In Section 2, we introduce the model. Then, in Section 3, we prove that the model is biologically well-posed, in the sense that the solutions belong to a biologically feasible region (see Theorem 1). In Section 4, we give analytical expressions for the 2 disease-free equilibria of our dynamical system. We compute the basic reproduction number R0 of the system and study the relevant disease-free equilibrium point of interaction between women and mosquitoes, showing its local asymptotically stability when R0 is less than one (see Theorem 2). The sensitivity of the basic reproduction number R0,withrespecttotheparametersofthesystem,isinvestigatedinSection5intermsof the normalized forward sensitivity index. The possibility of occurrence of an endemic equilibrium is discussed in Section 6. We end with Sections 7 and 8 of numerical simulations and conclusions, respectively.

2 THE ZIKA MODEL

We consider women as the population under study. The total women population, given by N,issubdividedinto4mutu- ally exclusive compartments, according to disease status, namely, susceptible pregnant women (S), infected pregnant women (I), women who gave birth to babies without neurological disorder (W), and women who gave birth to babies with neurological disorder due to microcephaly (M). As for the mosquitoes population, since the Zika virus is transmitted by the same virus as Dengue disease, we shall 7 use the same scheme as in Rodrigues et al. There are 4 state variables related to the (female) mosquitoes, namely, Am(t), which corresponds to the aquatic phase, that includes the egg, larva, and pupa stages; Sm(t), for the mosquitoes that might contract the disease (susceptible); Em(t), for the mosquitoes that are infected but are not able to transmit the Zika virus to humans (exposed); and Im(t), for the mosquitoes capable of transmitting the Zika virus to humans (infected). The following assumptions are considered in our model: (A.1) there is no immigration of infected humans; (A.2) the total human populations N is constant; (A.3) the coefficient of transmission of Zika virus is constant and does not varies with seasons; NDAÏROU ET AL. 8931

(A.4) after giving birth, pregnant women are no more pregnant and they leave the population under study at a rate !h equal to the rate of humans birth; (A.5) death is neglected, as the period of pregnancy is much smaller than the mean humans lifespan; (A.6) there is no resistant phase for the mosquito, due to its short lifetime. Note that the male mosquitoes are not considered in this study because they do not bite humans and consequently they do not influence the dynamics of the disease. The differential system that describes the model is composed by compartments of pregnant women and women who gave birth:

dS =Λ−( B Im +(1 − ) + )S dt " #mh N " $1 !h ,

dI Im ⎧ = "B#mh S −($2 + !h)I, dt N (1) ⎪ dW ⎪ =(1 − ")$1S +(1 − %)$2I − !hW, ⎪ dt dM ⎨ = %$2I − !hM, ⎪ dt ⎪ where N = S + I + W + M is the total⎪ population (women). The parameter Λ denotes the new pregnant women per ⎩ week, " stands for the fraction of susceptible pregnant women that gets infected, B is the average daily biting (per day), #mh represents the transmission probability from infected mosquitoes Im (per bite), $1 is the rate at which susceptible pregnant women S give birth (in weeks), $2 is the rate at which infected pregnant women I give birth (in weeks), !h is the natural death rate for pregnant women, and % denotes the fraction of infected pregnant women I that give birth babies with neurological disorder due to microcephaly. The above system (1) is coupled with the dynamics of the mosquitoes8:

dAm = 1 − Am (S + E + I ) − ( + ) A dt !b K m m m !A &A m, % & dSm = A − B I + S ⎧ dt &A m #hm N !m m, ⎪ % & (2) ⎪ dEm I = B#hm Sm −(&m + !m)Em, ⎪ dt N ⎨ % & dIm = E − I ⎪ dt &m m !m m, ⎪ ⎪ where parameter #hm represents the⎩ transmission probability from infected humans Ih (per bite), !b stands for the number of eggs at each deposit per capita (per day), !A is the natural mortality rate of larvae (per day), &A is the maturation rate from larvae to adult (per day), 1∕&m represents the extrinsic incubation period (in days), 1∕!m denotes the average lifespan of adult mosquitoes (in days), and K is the maximal capacity of larvae. See Table 1 for the description of the state variables and parameters of the Zika model (1)-(2). In Figure 1, we describe the behavior of the movement of individuals among these compartments. We consider system (1)-(2) with given initial conditions

S(0)=S0, I(0)=I0, W(0)=W0, M(0)=M0, Am(0)=Am0, Sm(0)=Sm0, Em(0)=Em0, Im(0)=Im0,

8 with (S0, I0, W0, M0, Am0, Sm0, Em0, Im0) ∈ R+.Inwhatfollows,weassume#mh = #hm.

3 POSITIVITY AND BOUNDEDNESS OF SOLUTIONS

Since the system of equations (1) and (2) represent, respectively, human and mosquitoes populations, and all parameters in the model are nonnegative, we prove that, given nonnegative initial values, the solutions of the system are nonnegative. More precisely, let us consider the biologically feasible region

8 Λ Ω= (S, I, W, M, Am, Sm, Em, Im)∈R+ ∶ S + I + W + M ≤ , Am ≤ kNh , Sm + Em + Im ≤ mNh . (3) ' !h (

The following result holds.

8 Theorem 1. The region Ω defined by (3) is positively invariant for model (1)-(2) with initial conditions in R+. 8932 NDAÏROU ET AL.

TABLE 1 Variables and parameters of the Zika model (1)-(2) Variable/Symbol Description S(t) Susceptible pregnant women I(t) Infected pregnant women W(t) Women who gave birth to babies without neurological disorder M(t) Women who gave birth to babies with neurological disorder due to microcephaly

Am(t) Mosquitoes in the aquatic phase

Sm(t) Susceptible mosquitoes

Em(t) Exposed mosquitoes

Im(t) Infected mosquitoes Λ New pregnant women (per week) ! Fraction of S that gets infected B Average daily biting (per day)

"mh Transmission probability from Im (per bite)

#1 Rate at which S give birth (in weeks)

#2 Rate at which I give birth (in weeks)

$h Natural death rate % Fraction of I that gives birth to babies with neurological disorder

"hm Transmission probability from Ih (per bite)

$b Number of eggs at each deposit per capita (per day)

$A Natural mortality rate of larvae (per day)

&A Maturation rate from larvae to adult (per day)

1∕&m Extrinsic incubation period (in days)

1∕$m Average lifespan of adult mosquitoes (in days) K Maximal capacity of larvae

FIGURE 1 Flowchart presentation of the compartmental model (1)-(2) for Zika

Proof. Our proof is inspired by Rodrigues et al.8 System (1)-(2) can be rewritten in the following way: dX = M(X)X +F, dt where X =(S, I, W, M, Am, Sm, Em, Im), M(X)= M1 M2 with ! " NDAÏROU ET AL. 8933

− B Im −(1 − ) − 000 ! "mh N ! #1 $h

Im !B"mh −#2 − $h 00 ⎛ N ⎞ ⎜ (1 − !)#1 −(1 − %)#2 −$h 0 ⎟ ⎜ ⎟ M = ⎜ 0 %#2 0 −$h ⎟ 1 ⎜ ⎟ , ⎜ 0000⎟ ⎜ 0000⎟ ⎜ ⎟ ⎜ 0000⎟ ⎜ ⎟ ⎜ 0000⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 0000 0000 ⎛ ⎞ ⎜ 0000⎟ ⎜ 0000⎟ ⎜ ⎟ M2 = ⎜ Sm+Em+Im ⎟ , −$b − $A − &A $b $b $b ⎜ K ⎟ ⎜ −B I − 00⎟ ⎜ &A "hm N $m ⎟ ⎜ ⎟ 0 B I − − 0 ⎜ "hm N &m $m ⎟ ⎜ ⎟ ⎜ 00&m −$m ⎟ ⎜ ⎟ ⎜ ⎟ and F = (Λ, 0, 0, 0, 0, 0, 0, 0)T. Matrix⎝ M(X) is Metzler, ie, the off-diagonal elements of A are⎠ nonnegative. Using the fact that F 0, system dX = M(X)X + F is positively invariant in R8 ,9 which means that any trajectory with initial ≥ dt + 8 conditions in R+ remains in Ω for all t > 0.

4 EXISTENCE AND LOCAL STABILITY OF THE DISEASE-FREE EQUILIBRIA

System (1)-(2) admits 2 disease-free equilibrium points (DFE), obtained by setting the right-hand sides of the equations in the model to zero: the DFE E1,givenby

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Λ #1 Λ (1 − !) E1 =(S , I , W , M , Am, Sm, Em, Im)= , 0, , 0, 0, 0, 0, 0 , (1 − )+ ( (1 − )+ ) '#1 ! $h $h #1 ! $h ( which corresponds to the DFE in the absence of mosquitoes, and the DFE in the presence of mosquitoes, E2,givenby

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Λ #1 Λ (1 − !) K' K' E2 =(S , I , W , M , Am, Sm, Em, Im)= , 0, , 0, − , − , 0, 0 , (1 − )+ ( (1 − )+ ) '#1 ! $h $h #1 ! $h $b&A $b$m ( where

' = &A($m − $b)+$A$m. (4)

In what follows, we consider only the DFE E2,becausethisequilibriumpointconsidersinteractionbetweenhumans and mosquitoes, being therefore more interesting from the biological point of view. The local stability of E2 can be established using the next-generation operator method on (1)-(2). Following the approach 10 of van den Driessche and Watmough, we compute the basic reproduction number R0 of system (1)-(2) writing the right-hand side of (1)-(2) as  −  with 8934 NDAÏROU ET AL.

0 −Λ + ( B Im +(1 − ) + )S ! "mh N ! #1 $h Im B S ( 2 + h)I ! "mh N # $ ⎛ ⎞ ⎛ ⎞ 0 ⎜ −(1 − !)#1S +(1 − %)#2I + $hW ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟ $hM − %#2I = = ⎜ ⎟  ⎜ ⎟ ,  Am . 0 ⎜ −$b(1 − )(Sm + Em + Im)+($A + &A)Am ⎟ ⎜ ⎟ ⎜ K ⎟ ⎜ 0 ⎟ ⎜ I ⎟ ⎜ ⎟ −&AAm +(B"hm + $m)Sm I ⎜ N ⎟ ⎜ B"hm Sm ⎟ ⎜ N ⎟ ⎜ (&m + $m)Em ⎟ ⎜ ⎟ ⎜ ⎟ 0 ⎜ −&mEm + $mIm ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ Then we consider the Jacobian⎝ matrices⎠ associated⎝ with  and : ⎠ 00000000 ( + + ) !'Im I W M − !'Im S − !'Im S − !'Im S 000 S N N N N !' ⎛ ⎞ ⎜ 00000000⎟ ⎜ 00000000⎟ J = ⎜ ⎟ , ⎜ 00000000⎟ ⎜ 00000000⎟ ⎜ ⎟ ' ISm ' Sm(S+W+M) 'ISm ' ISm ⎜ − − − 0 ' I 0 ' Sm ⎟ ⎜ N N N N ⎟ ⎜ 00000000⎟ ⎜ ⎟ ⎜ ⎟ where ' = B"mh∕N and J = J1 ⎝J2 with ⎠ ' ( A − !'ImS − !'ImS − !'ImS 000!'S N N N 0000 0 #2 + $h 00 ⎛ ⎞ ⎛ 0000⎞ #1 (! − 1)(% − 1) #2 $h 0 ⎜ ⎟ ⎜ 0000⎟ ⎜ 0 −%#2 0 $h ⎟ ⎜ ⎟ J1 = ⎜ ⎟ , J2 = ⎜ $b(Sm+Em+Im)+($A+&A)K $b (Am−K) $b (Am−K) $b (Am−K) ⎟ 0000 ⎜ ⎟ ⎜ K K K K ⎟ IS S (S+W+M) IS IS ⎜ − ' m ' m − ' m − ' m ⎟ ⎜ − 'I + 00⎟ ⎜ N N N N ⎟ ⎜ &A N $m ⎟ ⎜ 0000⎟ ⎜ 00&M + $m 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0000⎟ ⎜ 00−&M $m ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ B I (N−S) ⎟ ⎜ ⎟ and A = (1 − )+ + ! "mh m .ThebasicreproductionnumberR is obtained as the spectral radius of the matrix #1 ⎝ ! $h N2 ⎠ ⎝ 0 ⎠ −1 J ×(J ) at the disease-free equilibrium E2 and is given by

−$bΛ ($h + #2)(#1(1 − !)+$h)(&m + $m) K!&m (&A($m − $b)+$A$m)"mh$hB R0 = (5) √ $bΛ ($h + #2)(#1(1 − !)+$h)(&m + $m) $m or 2 2 2 !B " K $ &m ($b&A − $m($A + &A)) 2 = mh h R0 2 . (6) $m$b Λ ($h + #2)(&m + $m)(#1(1 − !)+$h)

The disease-free equilibrium E2 is locally asymptotically stable if all the roots of the characteristic equation of the lin- earized system associated to (1)-(2) at the DFE E2 have negative real parts. The characteristic equation associated with E2 is given by

p1(()p2(()p3(()p4(()=0(7)

2 2 2 with p1(()=( + $h + #1(1 − !), p2(()=(( + $h) , p3(()=−( $m + −$b&A − $m ( + $m($m($A + &A)−$b&A) and p ( )=a 3 + a 2 + a + a ,where 4 ( 3( 2( 1( 0 ' ( NDAÏROU ET AL. 8935

2 2 2 2 !m!b Λ (!h + "2)(#m + !m)("1(1 − $)+!h) − $B % K ! #m (!b#A − !m(!A + #A)) =− mh h a0 2 , Λ!m!b ("1(1 − $)+!h) 2 !m + (2"2 + 2!h + #m) !m + #m("2 + !h) !h + #m + "2 + 2!m 1 a1 =− , a2 =− , a3 =− . !m !m !m

By the Routh–Hurwitz criterion, all the roots of the characteristic equation (7) have negative parts whenever R0 < 1. We have just proved the following result.

Theorem 2. The disease-free equilibrium in the presence of noninfected mosquitoes, E2,islocallyasymptoticallystable if R0 < 1 and unstable if R0 > 1.

5 SENSITIVITY OF THE BASIC REPRODUCTION NUMBER

The sensitivity of the basic reproduction number R0 is an important issue because it determines the model robustness to parameter values. The sensitivity of R0 with respect to model parameters is here measured by the so-called sensitivity index. Definition 3. (See Chitnis et al11 and Kong et al12) The normalized forward sensitivity index of a variable & that depends differentiability on a parameter p is defined by Υ& ∶= '& × p . p 'p &

! ! & & Remark 4. If Υp =+1, then an increase (decrease) of p by x% increases (decreases) & by x%;ifΥp =−1, then an increase (decrease) of p by x% decreases (increases) & by x%.

From (5) and Definition 3, it is easy to derive the normalized forward sensitivity index of R0 with respect to the average daily biting B and to the transmission probability %mh from infected mosquitoes Im per bite.

R0 R0 Proposition 5. The normalized forward sensitivity index of R0 with respect to B and %mh is one: Υ = 1 and Υ = 1. B %mh

Proof. It is a direct consequence of (5) and Definition 3.

The sensitivity index of R0 with respect to $, #m, !h, #A, !m, "1, "2, !b, !A, K,andΛ is given, respectively, by

R !h + "1 R !m R ("1(!h + 2"2))(1 − $)+"2!h R (!m − !b) #A Υ 0 = Υ 0 = Υ 0 = Υ 0 = $ , #m , !h , #A , 2("1(1 − $)+!h) 2(#m + !m) 2 (!h + "2)("1(1 − $)+!h) 2(!m(#A + !A)−#A!b)

R0 !m ((#A + !A)(−#m − 2!m)+3#A!b) + 2#A#m!b R0 "1 (−1 + $) R0 "2 Υ!m = , Υ"1 = , Υ"2 =− , 2 (#M + !M)(!M(#A + !A)−#A!b) 2("1(1 − $)+!h) 2(!h + "2) R !M (!A + #A) R !A!m R 1 R 1 Υ 0 =− Υ 0 = Υ 0 = Υ 0 =− !b , !A , K , Λ . 2 (!M(#A + !A)−#A!b) 2(!m(#A + !A)−#A!b) 2 2 In Section 7, we compute the previous sensitivity indexes for data from Brazil. 2 To analyze the sensitivity of R0 with respect to all the parameters involved, we compute appropriate derivatives:

2 2 2 2 'R0 !h + "1 2 'R0 2 2 'R0 2 2 'R0 !m 2 = R0, = R0, = R0, = R0, '$ $ (!h +(1 − $)"1) 'B B '%mh %mh '#m #m(!m + #m) 2 2 2 'R0 !m (#a + !a) 2 'R0 !b − !m 2 'R0 !m 2 = R0, = R0, = R0, (8) '!b !b (#a!b − !m (#a + !a)) '#A #a!b − !m (#a + !a) '!A !m (#a + !a) − #a!b 2 2 2 'R0 #a + !a 1 2 'R0 1 'R0 1 = − − R2, =− R2, =− R2. ( + ) − + 0 Λ Λ 0 + 0 '!m "!m #a !a #a!b #m !m !m # ' '"2 !h "2

In Section 7, we compute the values of these expressions according with the numerical values given in Table 2, up to 2 the R0 factor, which appears in all the right-hand expressions of (8), to determine which are the most and less sensitive parameters. 8936 NDAÏROU ET AL.

TABLE 2 Parameter values for system (1)-(2) Symbol Description Value Λ New pregnant women (per week) 3000000∕52 ! Fraction of S that gets infected 0.459 B Average daily biting (per day) 1

"mh Transmission probability from Im (per bite) 0.6

#1 Rate at which S give birth (in weeks) 37

#2 Rate at which I give birth (in weeks) 1∕25

$h Natural death rate 1∕50 % Fraction of I that gives birth to babies with neurological disorder 0.133

"hm Transmission probability from Ih (per bite) 0.6

$b Number of eggs at each deposit per capita (per day) 80

$A Natural mortality rate of larvae (per day) 1∕4

&A Maturation rate from larvae to adult (per day) 0.5

1∕&m Extrinsic incubation period (in days) 125

1∕$m Average lifespan of adult mosquitoes (in days) 125 K Maximal capacity of larvae 1.09034e+06

6 EXISTENCE AND STABILITY ANALYSIS OF THE ENDEMIC EQUILIBRIUM POINT

The system (1)-(2) has one endemic equilibrium (EE) with biologic meaning whenever R0 > 1. This EE is given by + ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 'S ∗ 'I ∗ 'W ∗ 'M ∗ −K( E = S+, I+, W+, M+, Am+, Sm+, Em+, Im+ with S+ = ∗ , I+ = ∗ , W+ = ∗ , M+ = ∗ , Am+ = , d d ($h+#2) d $h ($h+#2) d $h ($h+#2) $b&A ∗ 'Sm ∗ 'Em ∗ ∗ &m Sm+ =! ∗ , Em+ = ∗ , Im+ = Em+ ,where" ( is defined in (4) and dm (&m+$m)dm $m

∗ d = [BK"mh &m !$h ( −Λ$m$b (&m #1 (1 − !) + $m #1 (1 − !) + $h (&m + $m))] $h "mh B, ∗ dm = $b (B"mh$h + $m$h + $m#2) &m!B"mh$h > 0, 2 'S =−$b (&m + $m)(B"mh $h + $m($h + #2)) $mΛ , 2 2 2 2 'I =Λ B K"mh&m !$h( +Λ$m$b ($h + #2)((1 − !) #1 + $h)(&m + $m) , 2 2 2 2 'W =ΛK!"mh&m$h!#2 (1 − %) (B −Λ "mh$m$b$h#1 (1 − !)($h + #2)(&m" + $m) B 2 2 −(Λ $m$b ($h + #2)($h#2 (−1 + %) + %#1#2 (1 − !) + $h#1 (1 − !)) (&m + $m)), 2 2 2 2 'M =Λ $h(K"mhB &m! +Λ$m$b ($h + #2)(#1 (1 − !) + $h)(&m + $m) %#2,

'Sm =−(!$h + #2)(BK"mh&m!$h( −Λ$m$b ((1 − !)#1 + $h)(&m + $m)) ," 2 2 2 2 'Em =−$h(K"mhB &m! − ($h + #2) Λ$m$b (&m + $m)((1 − !)#1 + $h) .

2 2 2 ∗ (1−R )Λ $b$ ((1−!)#1+$h)(&m+$m) From (6), ( + ), 1, and d 0. Thus, S∗ 0, I∗ = 0 m 0, and M∗ = $b&A >$m $A &A !< < + > + d∗ > + 2 2 2 (1−R0)Λ $b$m %#2 ((1−!)#1+$h)(&m+$m) %#2 ∗ ∗ ∗ ) ∗ = I+ > 0. Moreover, as for W+,wehavethatitcanbeexpressedasW+ = ∗ d $h $h d $h ($h+#2) with

2 2 2 2 ) = B "mh&mKΛ$h!(% − 1)#2(&A$b − $m(&A + $A)) + Λ $b$m(&m + $m)($h + #2) × [(! − 1)(#1 − #2)(B"mh$h + $m($h + %#2)) + #2(B"mh$h(! − 1)+$h$m(! − %)+$m(! − 1)%#2)] .

∗ Therefore, W+ is positive assuming that #1 >#2 and %>!. Finally, by using again $b&A >$m($A + &A) and 1 − !>0, ∗ we obtain that Am+ > 0 and, moreover,

∗ 2 2 ∗ −($h + #2)d ∗ Λ R0 − 1 $m$b ($h + #2)((1 − !) #1 + $h) ∗ ∗ &m Sm+ = ∗ > 0, Em+ = ∗ > 0, Im+ = Em+ > 0. B"mh$hdm ! " dm $m

After some appropriate manipulations, the matrix associated to E+ is given by NDAÏROU ET AL. 8937

V11 V12 V12 V12 00 0V18 V21 −!h − "2 − V12 −V12 −V12 00 0−V18 "1(1 − #) "2(1 − $)−!h 00 0 0 0 ⎛ 0 $"2 0 −!h 00 0 0⎞ ⎜ 0000V V V V ⎟ (9) ⎜ 55 56 56 56 ⎟ ⎜ V61 V62 V61 V61 %a −!m − V76 00⎟ ⎜ −V61 −V62 −V61 −V61 0 V76 −%m − !m 0 ⎟ ⎜ 000000%m −!m ⎟ ⎜ ⎟ with ⎜ ⎟ ⎝ ∗ ⎠ !h&I + &S + &W d V18 &Im %m B#!h 'mh &S (!h + "2) V11 =−!h +(# − 1)"1 − , V12 =−V21 − ∗ , V18 =− , !h&S(!h + "2) dm !m &S (%m + !m) !h (&S (!h + "2) + &I) + &M + &W ∗ B'mhd %m!h&I (!h&I + &S + &W ) #(!h + "2) !b(!m&E + %m&I + !m(%m + !m)&S ) = m =− − − m m m V21 ∗ 2 , V55 %a !a ∗ , dm!m(%m + !m)(&M + &W + !h(&I + &S(!h + "2))) dmK!m(%m + !m) ∗ 2 ∗ !a Bd !h 'hm (!h + "2) &Sm (!h &S (!h + "2) + &M + &W ) d V18 'hm &Sm V56 = !m + 1 , V62 =− , V61 = V62 − , d∗ ( ( ( + ) + ) + + ) d∗ ' %a ( m !h &S !h "2 &I &M &W # m 'mh &S B&I !h 'hm V76 = . !h (&S (!h + "2) + &I) + &M + &W With these notations, we have that the eigenvalues of the matrix are

1 2 1 2 (1 =−!h,(2 = V55 − !m + 4%aV56 +(!m + V55) ,(3 = V55 − !m − 4%aV56 +(!m + V55) 2 2 ) √ + ) √ + and the roots of a polynomial of degree 5. Equivalently, we can write (2 and (3 as follows:

2 2 3 2 4 2 2 2 3 2 4 2 %a!b + 4!m(%a + !a)−2%a!b!m + !m − %a!b − !m %a!b + 4!m(%a + !a)−2%a!b!m + !m + %a!b + !m (2 = , ,(3 =−, . 2!m 2!m 5 4 3 2 Moreover, the polynomial of degree 5 has leading coefficient one, being given by x + )4x + )3x + )2x + )1x + )0 with

)4 = %m + 2!h + 2!m + "2 − V11 + V12 + V76, 2 )3 = %m (2!h + !m + "2 − V11 + V12 + V76) + !h (4!m + "2 − 2V11 + V12 + 2V76) + !h 2 + V76 (!m + "2 − V11 + V12) + 2"2!m − 2V11!m + 2V12!m + !m + #"1V12 − "2V11

− "1V12 + "2V12 − V12V21 − V11V12, 2 )2 = %m !h (2!m + "2 − 2V11 + V12 + 2V76) + !h + !m ("2 − V11 + V12) +(# − 1)"1V12 − "2V11 + "2V12 2 +V-76 ("2 − V11 + V12) − V11V12 − V12V21 + V18(V61 − V62)) + !h 2!m ("2 − 2V11 + V12 + V76) + 2!m 2 +(# − 1)"1V12 − "2V11 + V76 ("2 − 2V11 + V12) − V11V12 − V12V21-) + !h (2!m − V11 + V76) + 2#"1V12!m + V76 (!m ("2 − V11 + V12) +(# − 1)"1V12 + "2 (V12 − V11) − (V11 + V21) V12) − 2"2V11!m − 2"1V12!m 2 2 2 + 2"2V12!m + "2!m − V11!m + V12!m − 2V11V12!m − 2V12V21!m + #"1"2V12 − "1"2V12

− "2V11V12 − "2V12V21,

)1 = %m (!h (!m ("2 − 2V11 + V12) +(# − 1)"1V12 − "2V11 + V76 ("2 − 2V11 + V12) − V12V11 − V12V21 2 +V18 (2V61 − V62)) + !h (!m − V11 + V76) − !m (V12 (−#"1 + "1 + V11 + V21) + "2 (V11 − V12)) +#"1"2V12 − #"1V18V61 − V76 (V12 (−#"1 + "1 + V11 + V21) + "2 (V11 − V12)) − "1"2V12 − "2V11V12

−"2V12V21 + "1V18V61 + V11V18V62 + V18V21V62) + !m (!h (!m ("2 − 2V11 + V12) 2 −2 (V12 (−#"1 + "1 + V11 + V21) + "2V11)) + !h (!m − 2V11) − !m (V12 (−#"1 + "1 + V11 + V21) +"2 (V11 − V12)) + 2"2V12 ((# − 1)"1 − V11 − V21)) + V76 (!h (!m ("2 − 2V11 + V12) +(# − 1)"1V12 2 −V11 ("2 + V12) − V21V12) + !h (!m − V11) − !m (V12 (−#"1 + "1 + V11 + V21) +"2 (V11 − V12)) + "2V12 ((# − 1)"1 − V11 − V21)) ,

)0 = %m (−!h (V12!m (−#"1 + "1 + V11 + V21) + "2V11!m +(# − 1)"1V18V61

+V12V76 (−#"1 + "1 + V11 + V21) + "2V76V11 − "2V18V61 − V18 (V11 + V21) V62) 2 +!h (− (V11 (!m + V76) − V18V61)) + "2 (V12 (!m + V76) − V18V61)((# − 1)"1 − V11 − V21) − !m (!h + "2)(!m + V76)(V11!h + V12 ((1 − #)+V11 + V21)) . . 8938 NDAÏROU ET AL.

Obviously, these expressions become rather long. As a consequence, it is not possible to use the Routh–Hurwitz crite- rion in this general setting, but only for particular values of the parameters. Moreover, the eigenvalues can be computed numerically for the specific values given in Table 2 of the next section (realistic data from Brazil), and they are given by !1 =−0.02, !2 =−0.02, !3 =−5000, !4 =−22.3938, !5 =−0.0511697, !6 =−0.044689, !7 =−0.00919002, and !8 =−0.0079988. We can observe that local stability of the endemic equilibrium holds, since all eigenvalues are negative real numbers.

7 NUMERICAL SIMULATIONS: CASE STUDY OF BRAZIL

We perform numerical simulations to compare the results of our model with real data obtained from several reports published by the WHO,13 from the starting point when the first cases of Zika have been detected in Brazil and for a period of 40 weeks (from February 4, 2016, to November 10, 2016), which is assumed to be a regular pregnancy time.

7.1 Zika model fits well real data According to several sources, the total population of Brazil is 206 956 000, and every year, there are about 3 073 000 new born babies. As a consequence, there are about 3 000 000/52 new pregnant females every week. The number of babies with neurological disorders is taken from WHO reports.13 See Table 2, where the values considered in this manuscript have been collected, such that the numerical experiments give good approximation of real data obtained from the WHO. 13 Figure 2 shows how our model fits the real data in the period from February 4, 2016, to November 10, 2016. More precisely, the !2 norm of the difference between the real data and the curve produced by our model, in the full period, is 392.5591, which gives an average of about 9.57 cases of difference each week. We have considered as initial values S0 = 2180686 (S0 is the number of newborns corresponding to the simulation period) and the number of births in the period, I0 = 1, M0 = 0, and W0 = 0forthewomenpopulations,andAm0 = Sm0 = Im0 = 1.0903e+06, and Em0 = 6.5421e+06 for the mosquitoes populations. The system of differential equations has been solved by using the ode45 function of Matlab, in a MacBook Pro computer with a 2.8 GHz Intel Core i7 processor and 16 GB of memory 1600 MHz DDR3.

7.2 Local stability of the endemic equilibrium To illustrate the local stability of the endemic equilibrium, we show that for different initial conditions the solution of the differential system (1)-(2) tends to the endemic equilibrium point. We have done a number of numerical experiments. Precisely, we consider here 4 different initial conditions for the women population, C1, C2, C3, C4,definedinTable3,giving

1800

1600

1400

1200

1000

800

Microcefaly babies 600

400

200

0 0 5 10 15 20 25 30 35 40 time (weeks)

FIGURE 2 Number of newborns with microcephaly. The red line corresponds to the real data obtained from the WHO13 from 04/02/2016 to 10/11/2016, and the blue line has been obtained by solving numerically the system of ordinary differential equations (1) and (2). The !2 norm of the difference between the real data and our prediction is 992.5591, which gives an error of less than 9.57 cases per week [Colour figure can be viewed at wileyonlinelibrary.com] NDAÏROU ET AL. 8939

TABLE 3 Initial conditions used in the numerical simulations for the case study of Brazil (see Figure 3)

S0 I0 M0 W0 6 C1 2.18069 × 10 100 6 C2 2.17633 × 10 1454.79 1453.79 1453.79 6 C3 2.17851 × 10 727.896 726.896 726.896 6 C4 2.18025 × 10 146.379 145.379 146.379

18000 4000

16000 3500

14000 3000

12000 2500 10000 2000 8000 1500 Infected women 6000

Cases of microcephaly 1000 4000

2000 500

0 0 0 20 40 60 80 100 120 140 160 0 20 40 60 80 100 120 140 160 time (weeks) time (weeks)

FIGURE 3 The solution of differential system (1)-(2) tends to the endemic equilibrium point, independently of initial conditions. In this figure, we show the evolution of the populations of infected women (I)andcasesofmicrocephaly(M). Initial conditions are those of Table 3 [Colour figure can be viewed at wileyonlinelibrary.com]

TABLE 4 The normalized forward sensitivity index of the basic reproduction number R0,fortheparametervaluesofTable2 Parameters Sensitivity index Parameters Sensitivity index

Λ, K −1∕2 !b 0.00007501125170

" 0.9237909865 !A −0.2500375056e-4

B, #mh 1 $A 0.5000250040

%1 −0.4995009233 $m 1∕4

%2 −1∕3 !m −1.250075011

the same value for S0 + I0 + M0 + W0.Thesystemofdifferentialequations(1)and(2)hasbeensolvedforaperiodof 156 weeks, which correspond approximately to 3 years, by using the same numerical method and machine as for the comparison between our model and real data provided by the WHO in Figure 2. These numerical experiments are included in Figure 3.

7.3 Sensitivity analysis of the basic reproduction number To determine which are the most and less sensitive parameters of our model, we compute the values of expressions (8) according to the numerical values given in Table 2. The numerical results are as follows:

2 2 2 2 2 &R0 2 &R0 2 &R0 2 &R0 2 &R0 −6 2 = 4.02523R0, = 2R0, = 3.33333R0, = 62.5R0, = 1.8753 × 10 R0, &" &B &#mh &$m &!b 2 2 2 2 2 &R0 2 &R0 2 &R0 2 &R0 2 &R0 2 = 2.0001R0, =−0.00020003R0, =−312.51R0, =−0.0000173333R0, =−16.6667R0. &$a &!a &!m &Λ &%2

In Table 4, we present the sensitivity index of parameters Λ, ", B, #mh, %1, %2, !b, !A, $A, $m, !m,andK computed for the parameter values given in Table 2. From Table 4, we conclude that the most sensitive parameters are B and #mh,which 8940 NDAÏROU ET AL. means that to decrease the basic reproduction number in x%,weneedtodecreasetheseparametervaluesinx%.Therefore, to reduce the transmission of the Zika virus, it is crucial to implement control measures that lead to a reduction on the number of daily biting (per day) B and the transmission probability from the infected mosquitoes, !mh.Thefraction" of susceptible pregnant women S that get infected has a sensitive index very close to +1. This fact reinforces the importance of prevention measures, which protect susceptible pregnant women of becoming infected.

8 CONCLUSION

We proposed a new mathematical model for the transmission of Zika disease taking into account newborns with micro- cephaly. It has been shown that the proposed model fits well the recent reality of Brazil from February 4, 2016, to November 10, 2016.13 From a sensitivity analysis, we conclude that to reduce the number of new infection by Zika virus, it is impor- tant to implement control measures that reduce the average number of daily biting and the transmission probability from infected mosquitoes to susceptible pregnant women.

ACKNOWLEDGEMENTS Area and Nieto were supported by Agencia Estatal de Innovación of Spain, project MTM2016-75140-P, co-financed by FEDER and Xunta de Galicia,grantsGRC2015–004andR2016/022.SilvaandTorresweresupportedby FCT and CIDMA within project UID/MAT/04106/2013; by PTDC/EEI-AUT/2933/2014 (TOCCATA), co-funded by 3599-PPCDT and FEDER funds through COMPETE 2020, POCI. Silva is also grateful to the FCT post-doc fellowship SFRH/BPD/72061/2010.

ORCID

Faïçal Ndaïrou http://orcid.org/0000-0002-0119-6178 Iván Area http://orcid.org/0000-0003-0872-5017 Juan J. Nieto http://orcid.org/0000-0001-8202-6578 Cristiana J. Silva http://orcid.org/0000-0002-7238-546X Delfim F. M. Torres http://orcid.org/0000-0001-8641-2505

REFERENCES 1. Moreno VM, Espinoza B, Bichara D, Holechek SA, Castillo-Chavez C. Role of short-term dispersal on the dynamics of Zika virus in an extreme idealized environment. Infect Dis Model.2017;2:21-34. 2. van den Driessche P. Reproduction numbers of infectious disease models. Infect Dis Model.2017;2(3):288-303. 3. Worobey M. Epidemiology: molecular mapping of Zika spread. Nature.2017;546:355-357. 4. Agusto FB, Bewick S, Fagan WF. Mathematical model of Zika virus with vertical transmission. Infect Dis Model.2017;2:244-267. 5. Tang H, Hammack C, Ogden SC, et al. Zika virus infects human cortical neural progenitors and attenuates their growth. Cell Stem Cell. 2016;18(5):587-590. 6. WHO Zika Causality Working Group. Zika virus infection as a cause of congenital brain abnormalities and Guillain Barré syndrome: systematic review. PLOS Med.2017:27.URLhttps://doi.org/10.1371/journal.pmed.1002203. 7. Rodrigues HS, Monteiro MTT, Torres DFM. Seasonality effects on dengue: basic reproduction number, sensitivity analysis and optimal control. Math Methods Appl Sci.2016;39(16):4671-4679. 8. Rodrigues HS, Monteiro MTT, Torres DFM, Zinober A. Dengue disease, basic reproduction number and control. Int J Comput Math. 2012;89(3):334-346. 9. Abate A, Tiwari A, Sastry S. Box invariance in biologically-inspired dynamical systems. Automatica.2009;45(7):1601-1610. 10. van den Driessche P, Watmough J. Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission. Math Biosc.2002;180:29-48. 11. Chitnis N, Hyman JM, Cushing JM. Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model. Bull Math Biol.2008;70(5):1272-1296. 12. Kong Q, Qiu Z, Sang Z, Zou Y. Optimal control of a vector-host epidemics model. Math Control Relat Fields.2011;1:493-508. 13. http://www.who.int/emergencies/zika-virus/situation-report/en/, accessed July 13, 2017. NDAÏROU ET AL. 8941

How to cite this article: Ndaïrou F, Area I, Nieto JJ, Silva CJ, Torres DFM. Mathematical modeling of Zika disease in pregnant women and newborns with microcephaly in Brazil. Math Meth Appl Sci.2018;41:8929–8941. https://doi.org/10.1002/mma.4702 Chaos, Solitons and Fractals 135 (2020) 109846

Contents lists available at ScienceDirect

Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena

journal homepage: www.elsevier.com/locate/chaos

Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan

a , b b c a , Faïçal Ndaïrou , Iván Area , Juan J. Nieto , Delfim F.M. Torres ∗ a Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, Aveiro 3810-193, Portugal b Departamento de Matemática Aplicada II, E. E. Aeronáutica e do Espazo, Campus de Ourense, Universidade de Vigo, Ourense 32004, Spain c Instituto de Matematicas, Universidade de Santiago de Compostela, Santiago de Compostela 15782, Spain

a r t i c l e i n f o a b s t r a c t

Article history: We propose a compartmental mathematical model for the spread of the COVID-19 disease with special Received 18 April 2020 focus on the transmissibility of super-spreaders individuals. We compute the basic reproduction number

Accepted 22 April 2020 threshold, we study the local stability of the disease free equilibrium in terms of the basic reproduction Available online 27 April 2020 number, and we investigate the sensitivity of the model with respect to the variation of each one of its 2010 MSC: parameters. Numerical simulations show the suitability of the proposed COVID-19 model for the outbreak 34D05 that occurred in Wuhan, China. 92D30 ©2020 Elsevier Ltd. All rights reserved. Keywords: Mathematical modeling of COVID-19 pandemic Wuhan case study Basic reproduction number Stability Sensitivity analysis Numerical simulations

1. Introduction 1,353,361 confirmed cumulative cases with 79,235 deaths, accord- ing to the report dated by April 8, 2020, by the Word Health Mathematical models of infectious disease transmission dynam- Organization. ics are now ubiquitous. Such models play an important role in The global problem of the outbreak has attracted the interest of helping to quantify possible infectious disease control and mitiga- researchers of different areas, giving rise to a number of proposals tion strategies [1–3] . There exist a number of models for infectious to analyze and predict the evolution of the pandemic [6,7] . Our diseases; as for compartmental models, starting from the very clas- main contribution is related with considering the class of super- sical SIR model to more complex proposals [4] . spreaders, which is now appearing in medical journals (see, e.g., Coronavirus disease 2019 (COVID-19) is an infectious disease [8,9] ). This new class, as added to any compartmental model, im- caused by severe acute respiratory syndrome coronavirus 2 (SARS- plies a number of analysis about disease free equilibrium points, CoV-2). The disease was first identified December 2019 in Wuhan, which is also considered in this work. the capital of Hubei, China, and has since spread globally, re- The manuscript is organized as follows. In Section 2 , we pro- sulting in the ongoing 2020 pandemic outbreak [5] . The COVID- pose a new model for COVID-19. A qualitative analysis of the 19 pandemic is considered as the biggest global threat world- model is investigated in Section 3 : in Section 3.1 , we compute the wide because of thousands of confirmed infections, accompanied basic reproduction number R 0 of the COVID-19 system model; in by thousands deaths over the world. Notice, by March 26, 2020, Section 3.2 , we study the local stability of the disease free equi- report 503,274 confirmed cumulative cases with 22,342 deaths. librium in terms of R 0 . The sensitivity of the basic reproduction At the time of this revision, the numbers have increased to number R 0 with respect to the parameters of the system model is given in Section 4 . The usefulness of our model is then illustrated in Section 5 of numerical simulations, where we use real data from Corresponding author. ∗ E-mail addresses: [email protected] (F. Ndaïrou), [email protected] (I. Area), Wuhan. We end with Section 6 of conclusions, discussion, and fu- [email protected] (J.J. Nieto), delfi[email protected] (D.F.M. Torres). ture research. https://doi.org/10.1016/j.chaos.2020.109846 0960-0779/© 2020 Elsevier Ltd. All rights reserved. 2 F. Ndaïrou, I. Area and J.J. Nieto et al. / Chaos, Solitons and Fractals 135 (2020) 109846

Fig. 1. Flowchart of model (1) .

2. The proposed COVID-19 compartment model the disease induced death rates due to infected, super-spreaders, and hospitalized individuals, respectively. At each instant of time, Based on a 2016 model [10] , and taking into account the ex- dF (t ) istence of super-spreaders in the family of corona virus [11] , we D(t) : δi I(t) δp P(t) δh H(t) (2) propose a new epidemiological compartment model that takes = + + = dt into account the super-spreading phenomenon of some individu- gives the number of death due to the disease. The transmissibility als. Moreover, we consider a fatality compartment, related to death from asymptomatic individuals has been modeled in this way since due to the virus infection. In doing so, the constant total popula- it was not apparent their behavior. Indeed, at present, this question tion size N is subdivided into eight epidemiological classes: sus- is a controversial issue for epidemiologists. A flowchart of model ceptible class ( S ), exposed class ( E ), symptomatic and infectious (1) is presented in Fig. 1 . class ( I ), super-spreaders class ( P ), infectious but asymptomatic class ( A ), hospitalized ( H ), recovery class ( R ), and fatality class ( F ). 3. Qualitative analysis of the model The model takes the following form:

One of the most significant thresholds when studying infectious dS I H " P disease models, which quantifies disease invasion or extinction in β N S lβ N S β N S, dt = − − − a population, is the basic reproduction number [12] . In this section

 dE we obtain the basic reproduction number for our model (1) and β I S lβ H S β " P S κE,  N N N study the locally asymptotically stability of its disease free equilib-  dt = + + −   dI rium (see Theorem 1).  κρ E ( γa γ )I δ I,  dt = 1 − + i − i  3.1. The basic reproduction number   dP  κρ E ( γa γ )P δp P,  dt 2 i  = − + − (1) The basic reproduction number, as a measure for disease spread  dA  in a population, plays an important role in the course and control  κ(1 ρ1 ρ2 )E,  dt = − − of an ongoing outbreak. It can be understood as the average num- dH ber of cases one infected individual generates, over the course of  γa ( I P) γr H δh H,  dt = + − − its infectious period, in an otherwise uninfected population. Using   dR  the next generation matrix approach outlined in van den Driess-  γi ( I P) γr H,  dt = + + che and Watmough [13] to our model (1) , the basic reproduction   dF number can be computed by considering the below generation ma-   δi I δp P δh H,  dt = + + trices F and V , that is, the Jacobian matrices associated to the rate  of appearance of new infections and the net rate out of the corre- with β quantifying the human-to-human transmission coefficient  sponding compartments, respectively, per unit time (days) per person, β" quantifies a high transmis- sion coefficient due to super-spreaders, and l quantifies the rela- 0 ββl β " tive transmissibility of hospitalized patients. Here κ is the rate at 0 0 0 0 J and which an individual leaves the exposed class by becoming infec- F = 0 0 0 0  tious (symptomatic, super-spreaders or asymptomatic); ρ1 is the 0 0 0 0   proportion of progression from exposed class E to symptomatic in-  κ 0 0 0 fectious class I; ρ2 is a relative very low rate at which exposed in- 0 0 κρ1 &i dividuals become super-spreaders while 1 ρ ρ is the progres- J − , − 1 − 2 V =  κρ2 0 & p 0  sion from exposed to asymptomatic class; γ a is the average rate at − 0 γa γa &h which symptomatic and super-spreaders individuals become hos-  − − 

where  pitalized; γ i is the recovery rate without being hospitalized; γ r is the recovery rate of hospitalized patients; and δ , δp , and δ are & γa γ δ , & p γa γ δp and & γr δ . (3) i h i = + i + i = + i + h = + h F. Ndaïrou, I. Area and J.J. Nieto et al. / Chaos, Solitons and Fractals 135 (2020) 109846 3

Table 1 Values of the model parameters corresponding to the situation of Wuhan, as discussed in Section 5 , for which R 0 . 945 . 0 = Name Description Value Units

1 β Transmission coefficient from infected individuals 2.55 day − l Relative transmissibility of hospitalized patients 1.56 dimensionless

1 β " Transmission coefficient due to super-spreaders 7.65 day − 1 κ Rate at which exposed become infectious 0.25 day − ρ1 Rate at which exposed people become infected I 0.580 dimensionless ρ2 Rate at which exposed people become super-spreaders 0.001 dimensionless 1 γ a Rate of being hospitalized 0.94 day − 1 γ i Recovery rate without being hospitalized 0.27 day − 1 γ r Recovery rate of hospitalized patients 0.5 day − 1 δi Disease induced death rate due to infected class 3.5 day − 1 δp Disease induced death rate due to super-spreaders 1 day − 1 δh Disease induced death rate due to hospitalized class 0.3 day −

The basic reproduction number R 0 is obtained as the spectral radius of F V 1 , precisely, Next, by using the Liénard–Chipard test [14,15] , all the roots of · − Z ( λ) are negative or have negative real part if, and only if, the fol- " βρ1 (γa l &h ) (βγa l β &h )ρ2 R + + . (4) lowing conditions are satisfied: 0 = &i &h + &p &h

1. ai > 0, i 1, 2, 3, 4; For the parameters used in our simulations (see Table 1 ), one = 2. a 1 a 2 > a 3 . computes this basic reproduction number to obtain R 0 . 945 . 0 = This means that the epidemic outbreak that has occurred in In order to check these conditions of the Liénard–Chipard test,

Wuhan was well controlled by the Chinese authorities. we rewrite the coefficients a1 , a2 , a3 , and a4 of the characteristic polynomial in terms of the basic reproduction number given by 3.2. Local stability in terms of the basic reproduction number (4) :

a1 κ &h &i &p , Noting that the two last equations and the fifth of system = + + + " βρ1 β ρ2 (1) are uncoupled to the remaining equations of the system, we a (1 R )(κ& κ& p ) κ& p κ& 2 0 i i can easily obtain, by direct integration, the following analytical re- = − + + &i + &p sults: 1 1 &p &i βγa lρ κ βγa lρ κ t 1 2 + &h + &h &i + &h + &h &p A(t) κ(1 ρ1 ρ2 ) 0 E(s)ds = − − . / t t & ' (κ &i )&h (&h &i )&p , R(t) γi 0 I(s) P(s%) ds γr 0 H(s)ds (5)  = + + + + + + t t t βρ & F t I s ds P s ds H s ds. 1 h  ( ) δi %0 &( ) δp 0' ( ) % δh 0 ( ) a3 κ(1 R0 )(&h &p &h &i &i &p ) κ&p = + + = − + + + & i Furthermore,% since the% total population% size N is constant, one "  β ρ2 &h 1 1 has κ& κ& p βγa lρ i 1 + &p + &h + &i S(t) N [E (t) I(t) P (t) A (t) H(t) R (t) F (t) ]. (6) & ' = − + + + + + + 1 1 κ& βγa lρ & & & p , i 2 i h Therefore, the local stability of model (1) can be studied + &h + &p + through the remaining coupled system of state variables, namely, . / a κ& & & p (1 R ) . the variables E, I, P , and H in (1) . The Jacobian matrix associated to 4 = i h − 0 these variables of (1) is the following one:

Moreover, we also compute, in terms of R 0 , the following ex- " κβ lββ pression: − κρ1 & i 0 0 J , (7) M − = κρ 0 & p 0  a1 a2 a3 (1 R0 )(κ &i )κ&i (1 R0 )(κ &h &p)κ&p 2 − − = − + + − + +

0 γa γa &h βρ1 βγa lρ1  −  (κ & p & i ) κ& p   + + + & p + & where ϖi , ϖp , and ϖh are defined in (3). The eigenvalues of the ma- i

& ' trix JM are the roots of the following characteristic polynomial: β " ρ2 βγa lρ2 (κ & p & i ) κ& i 4 3 2 Z(λ) λ a λ a λ a λ a , + + + &p + &p = + 1 + 2 + 1 + 4 &βγa lρ1 κ ' βγa lρ2 κ where (κ & & ) (κ & & p ) h i h + + + &h + + + &h a1 κ &h &i &p , (κ & i ) & (& & i ) & p . = + + + + + h + h + a βκρ β " κρ κ& κ& & & κ& p 2 = − 1 − 2 + h + i + h i + & & p & & p , From these previous expressions, it is clear that if R < 1, then + h + i 0 the conditions of the Liénard–Chipard test are satisfied and, as a a βγ κlρ βγ κlρ βκρ & β " κρ & βκρ & 3 a 1 a 2 1 h 2 h 1 p = − − − − − consequence, the disease free equilibrium is stable. In the case " β κρ2 &i κ&h &i κ&h &p κ&i &p &h &i &p , when R 0 > 1, we have that a 4 < 0 and, by using Descartes’ rule − + + + + " of signs, we conclude that at least one of the eigenvalues is posi- a 4 βγa κlρ2 & βγa κlρ1 & p β κρ2 & & = − i − − i h tive. Therefore, the system is unstable. In conclusion, we have just βκρ & & p κ& & & p . − 1 h + h i proved the following result: 4 F. Ndaïrou, I. Area and J.J. Nieto et al. / Chaos, Solitons and Fractals 135 (2020) 109846

Table 2 because a small perturbation in that parameter leads to small

Sensitivity of R0 evaluated for the changes. parameter values given in Table 1 . From Table 2 , we conclude that the most sensitive parameters Parameter Sensitivity index to the basic reproduction number R 0 of the COVID-19 model (1) are

β 0.963 β, ρ1 and δi . In concrete, an increase of the value of β will in- l 0.631 crease the basic reproduction number by 96.3% and this happens, ! β 0.366 in a similar way, for the parameter ρ . In contrast, an increase of 1 κ 0.000 the value of δi will decrease R0 by 69.9%. ρ1 0.941 ρ2 0.059

γ a 0.418 5. Numerical simulations: the case study of Wuhan γ i 0.061 − γ r 0.395 − We perform numerical simulations to compare the results of δ 0.699 i − our model with the real data obtained from several reports pub- δp 0.027 − lished by WHO [20,21] and worldometer [5] . δh 0.238 − The starting point of our simulations is 4 January 2020 (day 0), when the Chinese authorities informed about the new virus [20] , with already 6 confirmed cases in one day. From this period up to Theorem 1. The disease free equilibrium of system (1) , that is, ( N , 0, January 19, there is less information about the number of people 0, 0, 0, 0, 0, 0), is locally asymptotically stable if R < 1 and unstable 0 contracting the disease. Only on January 20, we have the report if R > 1 . 0 [21] , with 1460 new reported cases in that day and 26 the dead. Next we investigate the sensitiveness of the COVID-19 model Thus, the infection gained much more attention from 21 January (1) , with respect to the variation of each one of its parameters, for 2020, with 1739 confirmed cases and 38 the dead, up to 4 March the endemic threshold (4) . 2020, when the numbers in that day were as low as 11 and 7, re- spectively infected and dead, after a pick of 3892 confirmed cases 4. Sensitivity analysis on 27 January 2020 and a pick of 254 dead on 4 February 2020. Here we follow the data of the daily reports published by [5] . We As we saw in Section 3 , the basic reproduction number for the show that our COVID-19 model describes well the real data of daily COVID-19 model (1) , which we propose in Section 2 , is given by confirmed cases during the 2 months outbreak (66 days to be pre- (4) . The sensitivity analysis for the endemic threshold (4) tells us cise, from January 4 to March 9, 2020). how important each parameter is to disease transmission. This in- The total population of Wuhan is about 11 million. During the formation is crucial not only for experimental design, but also to COVID-19 outbreak, there was a restriction of movements of in- data assimilation and reduction of complex models [16] . Sensi- dividuals due to quarantine in the city. As a consequence, there tivity analysis is commonly used to determine the robustness of was a limitation on the spread of the disease. In agreement, in our model we consider, as the total population under study, N model predictions to parameter values, since there are usually er- = rors in collected data and presumed parameter values. It is used 110 0 0 0 0 0 / 250 . This denominator has been determined in the first to discover parameters that have a high impact on the threshold days of the outbreak and later has been proved to be a correct value: according to the real data published by the WHO, it is an R 0 and should be targeted by intervention strategies. More accu- rately, sensitivity indices’ allows us to measure the relative change appropriate value for the restriction of movements of individu- in a variable when a parameter changes. For that purpose, we use als. As for the initial conditions, the following values have been fixed: S N 6 , E 0 , I 1 , P 5 , A 0 , H 0 , R 0 , the normalized forward sensitivity index of a variable with respect 0 = − 0 = 0 = 0 = 0 = 0 = 0 = and F 0 . to a given parameter, which is defined as the ratio of the relative 0 = change in the variable to the relative change in the parameter. If We would like to mention that there exist gaps in the reports of such variable is differentiable with respect to the parameter, then the WHO at the beginning of the outbreak. For completeness, we the sensitivity index is defined as follows. give here the list LC of the number of confirmed cases in Wuhan per day, corresponding to the green line of Fig. 2 , and the list L D of Definition 1.1 (See [17,18] ) . The normalized forward sensitivity in- the number of dead individuals in Wuhan per day, corresponding dex of R 0 , which is differentiable with respect to a given parameter to the red line of Fig. 3 : θ, is defined by LC [6 , 12 , 19 , 25 , 31 , 38 , 44 , 60 , 80 , 131 , 131 , 259 , 467 , 688 , 776 ,

R ∂R0 θ = 0 ϒθ . 1776, 1460, 1739, 1984, 2101, 2590, 2827, 3233, 3892, 3697, 3151, = ∂θ R 0 3387 , 2653 , 2984 , 2473 , 2022 , 1820 , 1998 , 1506 , 1278 , 2051 , 1772 , The values of the sensitivity indices for the parameters values 1891 , 399 , 894 , 397 , 650 , 415 , 518 , 412 , 439 , 441 , 435 , 579 , 206 , of Table 1 , are presented in Table 2 . These values have been determined experimentally in such a 130 , 120 , 143 , 146 , 102 , 46 , 45 , 20 , 31 , 26 , 11 , 18 , 27 , 29 , 39 , 39] , way the mathematical model describes well the real data, giving rise to Figs. 2 and 3 . Other values for the parameters can be found, L [0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 4 , 4 , 4 , 8 , 15 , 15 , 25 , 26 , 26 , 38 , 43 , 46 , 45 , e.g., in Aguilar et al. [19]. D = Note that the sensitivity index may depend on several param- 57 , 64 , 66 , 73 , 73 , 86 , 89 , 97 , 108 , 97 , 254 , 121 , 121 , 142 , 106 , 106 , eters of the system, but also can be constant, independent of any 98, 115, 118, 109, 97, 150, 71, 52, 29, 44, 37, 35, 42, 31, 38, 31, 30, parameter. For example, ϒ R0 1 means that increasing (decreas- θ = + 28 , 27 , 23 , 17 , 22 , 11 , 7 , 14 , 10 , 14 , 13 , 13] . ing) θ by a given percentage increases (decreases) always R 0 by that same percentage. The estimation of a sensitive parameter Lists L C and L D have 66 numbers, where L C (0) represents the should be carefully done, since a small perturbation in such pa- number of confirmed cases 04 January 2020 (day 0) and L C (65) rameter leads to relevant quantitative changes. On the other hand, the number of confirmed cases 09 March 2020 (day 65) and, anal- the estimation of a parameter with a rather small value for the ogously, L D (0) represents the number of dead on January 4, and sensitivity index does not require as much attention to estimate, L D (65) the number of dead on March 9, 2020. F. Ndaïrou, I. Area and J.J. Nieto et al. / Chaos, Solitons and Fractals 135 (2020) 109846 5

Fig. 2. Number of confirmed cases per day. The green line corresponds to the real data obtained from reports [5,20,21] while the black line ( I P H) has been obtained + + by solving numerically the system of ordinary differential Eq. (1) , by using the Matlab code ode45 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 3. Number of confirmed deaths per day. The red line corresponds to the real data obtained from reports [5,20,21] while the black line has been obtained by solving numerically, using the Matlab code ode45 , our system of ordinary differential Eq. (1) to derive D ( t ) given in (2) . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

6. Conclusions and discussion reality of the Wuhan outbreak (see Fig. 2 ) and predicting a di- minishing on the daily number of confirmed cases of the disease. Classical models consider SIR populations. Here we have taken This is in agreement with our computations of the basic reproduc- into consideration the super-spreaders ( P ), hospitalized ( H ), and fa- tion number in Section 4 that, surprisingly, is obtained less than 1. tality class ( F ), so that its derivative (see formula (2) ) gives the Moreover, it is worth to mention that our model fits also enough number of deaths ( D ). Our model is an ad hoc compartmental well the real data of daily confirmed deaths, as shown in Fig. 3 . model of the COVID-19, taking into account its particularities, some Our theoretical findings and numerical results adapt well to the of them still not well-known, giving a good approximation of the real data and it reflects or reflected the reality in Wuhan, China. 6 F. Ndaïrou, I. Area and J.J. Nieto et al. / Chaos, Solitons and Fractals 135 (2020) 109846

The number of hospitalized persons is relevant to give an estimate [2] Ndaïrou F, Area I, Nieto JJ, Silva CJ, Torres DFM. Mathematical modeling of Zika of the Intensive Care Units (ICU) needed. Some preliminary simula- disease in pregnant women and newborns with microcephaly in Brazil. Math Methods Appl Sci 2018;41:8929–41. doi: 10.1002/mma.4702 . tions indicate that this would be useful for the health authorities. [3] Rachah A, Torres DFM. Dynamics and optimal control of Ebola transmission. Our model can also be used to study the reality of other coun- Math Comput Sci 2016;10:331–42. doi: 10.1007/s11786- 016- 0268- y . tries, whose outbreaks are currently on the rise. We claim that [4] Brauer F , Castillo-Chavez C , Feng Z . Mathematical models in epidemiology.

New York: Springer-Verlag; 2019. some mathematical models like the one we have proposed here [5] COVID-19 Coronavirus Pandemic. 2020. https://www.worldometers.info/ will contribute to reveal some important aspects of this pandemia. coronavirus/repro , Accessed March 26. Of course, this investigation has some limitations, being the [6] Chen T-M, Rui J, Wang Q-P, Cui J-A, Yin L. A mathematical model for simulat-

first on the relative recent spread of the new coronavirus and ing the phase-based transmissibility of a novel coronavirus. Infect Dis Poverty 2020;9(1):24. doi: 10.1186/s40249- 020- 00640- 3 . therefore the limited data accessible at the beginning of this study. [7] Maier BF, Brockmann D. Effective containment explains subexponential growth In the future, we can develop further this prototype. Even with in recent confirmed COVID-19 cases in China. Science 2020;8. doi: 10.1126/ these shortcomings, the model can be useful due to the high rel- science.abb4557. [8] Trilla A. One world, one health: the novel coronavirus COVID-19 epidemic. Med evance of the topic. Finally, we suggest new directions for further Clin (Barc) 2020;154(5):175–7. doi: 10.1016/j.medcle.2020.02.001 . research: [9] Wong G, Liu W, Liu Y, Zhou B, Bi Y, Gao GF. MERS, SARS, and Ebola: the role of super-spreaders in infectious disease. Cell Host Microbe 2015;18(4):398–401. 1. the transmissibility from asymptomatic individuals; doi: 10.1016/j.chom.2015.09.013 .

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3. consider sub-populations related to age, gender, etc.; rea. Osong Public Health Res Perspect 2016;7:49–55. doi: 10.1016/j.phrp.2016. 4. introduce preventive measures in this COVID-19 epidemic and 01.001 . for future viruses; [11] Alasmawi H , Aldarmaki N , Tridane A . Modeling of a super-spreading event of the MERS-corona virus during the Hajj season using simulation of the existing

5. integrate into the model some imprecise data by using fuzzy data. Int J Stat Med Biol Res 2017;1:24–30 . differential equations; [12] van den Driessche P. Reproduction numbers of infectious disease models. In- 6. include the viral load of the infectious into the model. fect Dis Model 2017;2:288–303. doi: 10.1016/j.idm.2017.06.002 . [13] van den Driessche P, Watmough J. Reproduction numbers and sub-threshold These and other questions are under current investigation and endemic equilibria for compartmental models of disease transmission. Math Biosci 2002;180:29–48. doi: 10.1016/S0025- 5564(02)00108- 6 . will be addressed elsewhere. [14] Gantmacher FR . The theory of matrices, 1. Providence, RI: AMS Chelsea Pub- lishing; 1998 . Funding [15] Liénart A , Chipart H . Sur le signe de la partie réelle des racines d’une équation algébrique. J Math Pures Appl (6 ème série) 1914;10:291–346 . [16] Powell DR , Fair J , LeClaire RJ , Moore LM , Thompson D . Sensitivity analysis of an This research was funded by the Portuguese Foundation for infectious disease model. In: Boston M, editor. Proceedings of the international Science and Technology (FCT) within project UIDB/04106/2020 system dynamics conference; 2005 .

[17] Chitnis N, Hyman JM, Cushing JM. Determining important parameters in the (CIDMA). Ndaïrou is also grateful to the support of FCT through spread of malaria through the sensitivity analysis of a mathematical model. the Ph.D. fellowship PD/BD/150273/2019. The work of Area and Ni- Bull Math Biol 2008;70:1272–96. doi: 10.1007/s11538- 008- 9299- 0 . eto has been partially supported by the Agencia Estatal de Inves- [18] Rodrigues HS, Monteiro MTT, Torres DFM. Sensitivity analysis in a dengue epi- tigación (AEI) of Spain, cofinanced by the European Fund for Re- demiological model. Conference papers in mathematics. Hindawi, editor; 2013. doi: 10.1155/2013/721406 . Vol. 2013, Art. ID 721406 gional Development (FEDER) corresponding to the 2014–2020 mul- [19] Aguilar J.B., Faust G.S.M., Westafer L.M., Gutierrez J.B.. Investigating the tiyear financial framework, project MTM2016-75140-P . Moreover, impact of asymptomatic carriers on COVID-19 transmission. Preprint.

Nieto also thanks partial financial support by Xunta de Galicia un- 10.1101/2020.03.18.20037994 [20] de la Salud O.P.. Alerta epidemiológica nuevo coronavirus (ncov). der grant ED431C 2019/02. 2020a. https://www.paho.org/hq/index.php?option=com _docman&view= download&category _slug=coronavirus- alertas- epidemiologicas&alias= Declaration of Competing Interest 51351- 16- de- enero- de- 2020- nuevo- coronavirus- ncov- alerta- epidemiologica- 1 &Itemid=270&lang=es , accessed on January 16. [21] de la Salud O.P.. Actualización epidemiológica nuevo coronavirus (2019-ncov). The authors declare that they have no known competing finan- 2020b. https://www.paho.org/hq/index.php?option=com _docman&view= cial interests or personal relationships that could have appeared to download&category _slug=coronavirus- alertas- epidemiologicas&alias= 51355- 20- de- enero- de- 2020- nuevo- coronavirus- ncov-actualizacion- influence the work reported in this paper. epidemiologica-1&Itemid=270&lang=es , accessed on January 20. References

[1] Djordjevic J, Silva CJ, Torres DFM. A stochastic SICA epidemic model for HIV transmission. Appl Math Lett 2018;84:168–75. doi: 10.1016/j.aml.2018.05.005 . Chaos, Solitons and Fractals 141 (2020) 110311

Contents lists available at ScienceDirect

Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena

journal homepage: www.elsevier.com/locate/chaos

Letter to the editor

Corrigendum to “Mathematical modeling of COVID-19 Parameter Sensitivity index transmission dynamics with a case study of Wuhan” β 0.999 [Chaos Solitons Fractals 135 (2020), 109846] l 0.729

β ! 0.00139 κ 0.000

ρ1 0.997 The following corrections need to be done to Ndaïrou et al. [1] : ρ2 0.00265

γa 0.0210 − γ 0.215 1. In Section 3.1, the Jacobian matrix J has the entry in the first i − F row and the third column interchanged with the entry in the γr 0.671 − first row and the fourth column. In other words, the correct δi 0.0346 matrix is − δp 0.0000919 − δ 0.0583 h − 0 ββ! βl Please see Appendix A for all details, where the computations 0 0 0 0 J . to obtain such values are carried out in the free and open- F = 0 0 0 0  source computer algebra system . 0 0 0 0 SageMath   6. Figure 3 needs to be substituted by the following one:  

2. In Table 1, where the values of the model parameters used in the numerical simulations are given, the values associated with

δi , δp and δh are wrong. The correct values are:

1 1 δ δp δ day− . i = = h = 23

3. For the parameters used in the simulations, the basic reproduc- tion number is not R 0 . 945 , as wrongly given in Ndaïrou 0 = et al. [1] , but should be corrected to R 4 . 375 . Please see 0 = Appendix A for all details, where computations are carried out in the free and open-source computer algebra system Sage- Math . 4. In (7), the Jacobian matrix J M has the entry in the first row and the third column interchanged with the entry in the first row and the fourth column. In other words, the correct matrix is

κβ β! lβ This figure was obtained in Matlab, and the full code is given − in Appendix B . κρ1 % i 0 0 J M − . = κρ 0 % p 0  2 − 0 γa γa %  − h   The authors of [1] would like to apologize for any inconve- 5. The values given in Table 2, of the sensitivity of R 0 evaluated nience caused. for the parameter values used in the simulations, need to be corrected as follows:

https://doi.org/10.1016/j.chaos.2020.110311 0960-0779/© 2020 Elsevier Ltd. All rights reserved. F. Ndaïrou, I. Area, G. Bader et al. Chaos, Solitons and Fractals 141 (2020) 110311

Appendix A. SageMath code to compute R 0 and its sensitivity indexes

sage:#the constant parameters values sage: beta 2.55; l 1.56; betaprim 7.65; kappa 0.25; rho_1 0.580; rho_2 0.001; gamma_a 0.94;======sage: =gamma_i 0.27; gamma_r 0.5; delta_i 1/23; delta_p 1/23; delta_h 1/23; N 11000000 ======sage: R_0 ((beta ∗gamma_a ∗l ∗rho_2 + = sage: betaprim ∗rho_2 ∗(gamma_r + delta_h)) ∗(gamma_a + gamma_i + delta_i) . . . + (beta ∗gamma_a ∗l ∗rho_1 + . . . sage: beta ∗rho_1 ∗(gamma_r + delta_h)) ∗(gamma_a + gamma_i + delta_p))/((gamma_r sage: + delta_h) ∗(gamma_a + gamma_i + delta_i) ∗(gamma_a + gamma_i + delta_p)) sage: print R_0 4.37513184233091 sage: #Sentivity of beta sage: S_beta ((delta_i + gamma_a + gamma_i) ∗gamma_a ∗l ∗rho_2 + (gamma_a ∗l ∗rho_1 + = sage: (delta_h + gamma_r) ∗rho_1) ∗(delta_p + gamma_a + sage: gamma_i)) ∗beta/((beta ∗gamma_a ∗l ∗rho_2 + betaprim ∗(delta_h + sage: gamma_r) ∗rho_2) ∗(delta_i + gamma_a + gamma_i) + (beta ∗gamma_a ∗l ∗rho_1 + sage: beta ∗(delta_h + gamma_r) ∗rho_1) ∗(delta_p + gamma_a + gamma_i)) sage: print S_beta 0.998605066564884 sage: # sensitivity of l sage: S_l (beta ∗(delta_p + gamma_a + gamma_i) ∗gamma_a ∗rho_1 + beta ∗(delta_i + = sage: gamma_a + gamma_i) ∗gamma_a ∗rho_2) ∗l/((beta ∗gamma_a ∗l ∗rho_2 + sage: betaprim ∗(delta_h + gamma_r) ∗rho_2) ∗(delta_i + gamma_a + gamma_i) + sage: (beta ∗gamma_a ∗l ∗rho_1 + beta ∗(delta_h + gamma_r) ∗rho_1) ∗(delta_p + sage: gamma_a + gamma_i)) sage: print S_l 0.728917935775866 sage: #sensitivity analysis of betaprim sage: S_betaprim betaprim ∗(delta_h + gamma_r) ∗(delta_i + gamma_a + = sage: gamma_i) ∗rho_2/((beta ∗gamma_a ∗l ∗rho_2 + betaprim ∗(delta_h + sage: gamma_r) ∗rho_2) ∗(delta_i + gamma_a + gamma_i) + (beta ∗gamma_a ∗l ∗rho_1 + sage: beta ∗(delta_h + gamma_r) ∗rho_1) ∗(delta_p + gamma_a + gamma_i)) sage: print S_betaprim 0.00139493343511561 sage: #sensitivity of rho_1 sage: S_rho1 (beta ∗gamma_a ∗l + beta ∗(delta_h + gamma_r)) ∗(delta_p + gamma_a + = sage: gamma_i) ∗rho_1/((beta ∗gamma_a ∗l ∗rho_2 + betaprim ∗(delta_h + sage: gamma_r) ∗rho_2) ∗(delta_i + gamma_a + gamma_i) + (beta ∗gamma_a ∗l ∗rho_1 + sage: beta ∗(delta_h + gamma_r) ∗rho_1) ∗(delta_p + gamma_a + gamma_i)) sage: print S_rho1 0.997350474592809 sage: #sensitivity of rho_2 sage: S_rho2 (beta ∗gamma_a ∗l + betaprim ∗(delta_h + gamma_r)) ∗(delta_i + gamma_a + = sage: gamma_i) ∗rho_2/((beta ∗gamma_a ∗l ∗rho_2 + betaprim ∗(delta_h + sage: gamma_r) ∗rho_2) ∗(delta_i + gamma_a + gamma_i) + (beta ∗gamma_a ∗l ∗rho_1 + sage: beta ∗(delta_h + gamma_r) ∗rho_1) ∗(delta_p + gamma_a + gamma_i)) sage: print S_rho2 0.00264952540719111 sage: #sensitivity of gamma_a sage: S_gammaa (delta_h + gamma_r) ∗(delta_i + gamma_a + gamma_i) ∗(delta_p + gamma_a + = sage: gamma_i) ∗gamma_a ∗((beta ∗(delta_p + gamma_a + gamma_i) ∗l ∗rho_1 + sage: beta ∗gamma_a ∗l ∗rho_1 + beta ∗(delta_i + gamma_a + gamma_i) ∗l ∗rho_2 + sage: beta ∗gamma_a ∗l ∗rho_2 + beta ∗(delta_h + gamma_r) ∗rho_1 + sage: betaprim ∗(delta_h + gamma_r) ∗rho_2)/((delta_h + gamma_r) ∗(delta_i + sage: gamma_a + gamma_i) ∗(delta_p + gamma_a + gamma_i)) - sage: ((beta ∗gamma_a ∗l ∗rho_2 + betaprim ∗(delta_h + gamma_r) ∗rho_2) ∗(delta_i + sage: gamma_a + gamma_i) + (beta ∗gamma_a ∗l ∗rho_1 + beta ∗(delta_h + sage: gamma_r) ∗rho_1) ∗(delta_p + gamma_a + gamma_i))/((delta_h + sage: gamma_r) ∗(delta_i + gamma_a + gamma_i) ∗(delta_p + gamma_a + gamma_i) ^ 2) sage: - ((beta ∗gamma_a ∗l ∗rho_2 + betaprim ∗(delta_h + gamma_r) ∗rho_2) ∗(delta_i sage: + gamma_a + gamma_i) + (beta ∗gamma_a ∗l ∗rho_1 + beta ∗(delta_h + sage: gamma_r) ∗rho_1) ∗(delta_p + gamma_a + gamma_i))/((delta_h + sage: gamma_r) ∗(delta_i + gamma_a + gamma_i) ^ 2 ∗(delta_p + gamma_a +

2 F. Ndaïrou, I. Area, G. Bader et al. Chaos, Solitons and Fractals 141 (2020) 110311

sage: gamma_i)))/((beta ∗gamma_a ∗l ∗rho_2 + betaprim ∗(delta_h + sage: gamma_r) ∗rho_2) ∗(delta_i + gamma_a + gamma_i) + (beta ∗gamma_a ∗l ∗rho_1 + sage: beta ∗(delta_h + gamma_r) ∗rho_1) ∗(delta_p + gamma_a + gamma_i)) sage: print S_gammaa -0.0209953489969400 sage: #sensitivity of Gamma_i sage: S_gammai (delta_h + gamma_r) ∗(delta_i + gamma_a + gamma_i) ∗(delta_p + gamma_a + = sage: gamma_i) ∗gamma_i ∗((beta ∗gamma_a ∗l ∗rho_1 + beta ∗gamma_a ∗l ∗rho_2 + sage: beta ∗(delta_h + gamma_r) ∗rho_1 + betaprim ∗(delta_h + sage: gamma_r) ∗rho_2)/((delta_h + gamma_r) ∗(delta_i + gamma_a + sage: gamma_i) ∗(delta_p + gamma_a + gamma_i)) - ((beta ∗gamma_a ∗l ∗rho_2 + sage: betaprim ∗(delta_h + gamma_r) ∗rho_2) ∗(delta_i + gamma_a + gamma_i) + sage: (beta ∗gamma_a ∗l ∗rho_1 + beta ∗(delta_h + gamma_r) ∗rho_1) ∗(delta_p + sage: gamma_a + gamma_i))/((delta_h + gamma_r) ∗(delta_i + gamma_a + sage: gamma_i) ∗(delta_p + gamma_a + gamma_i) ^ 2) - ((beta ∗gamma_a ∗l ∗rho_2 + sage: betaprim ∗(delta_h + gamma_r) ∗rho_2) ∗(delta_i + gamma_a + gamma_i) + sage: (beta ∗gamma_a ∗l ∗rho_1 + beta ∗(delta_h + gamma_r) ∗rho_1) ∗(delta_p + sage: gamma_a + gamma_i))/((delta_h + gamma_r) ∗(delta_i + gamma_a + sage: gamma_i) ^ 2 ∗(delta_p + gamma_a + gamma_i)))/((beta ∗gamma_a ∗l ∗rho_2 + sage: betaprim ∗(delta_h + gamma_r) ∗rho_2) ∗(delta_i + gamma_a + gamma_i) + sage: (beta ∗gamma_a ∗l ∗rho_1 + beta ∗(delta_h + gamma_r) ∗rho_1) ∗(delta_p + sage: gamma_a + gamma_i)) sage: print S_gammai -0.215400624349636 sage: #sensitivity of gamma_r sage: S_gammar (delta_h + gamma_r) ∗(delta_i + gamma_a + gamma_i) ∗(delta_p + gamma_a + = sage: gamma_i) ∗gamma_r ∗((beta ∗(delta_p + gamma_a + gamma_i) ∗rho_1 + sage: betaprim ∗(delta_i + gamma_a + gamma_i) ∗rho_2)/((delta_h + sage: gamma_r) ∗(delta_i + gamma_a + gamma_i) ∗(delta_p + gamma_a + gamma_i)) - sage: ((beta ∗gamma_a ∗l ∗rho_2 + betaprim ∗(delta_h + gamma_r) ∗rho_2) ∗(delta_i + sage: gamma_a + gamma_i) + (beta ∗gamma_a ∗l ∗rho_1 + beta ∗(delta_h + sage: gamma_r) ∗rho_1) ∗(delta_p + gamma_a + gamma_i))/((delta_h + sage: gamma_r) ^ 2 ∗(delta_i + gamma_a + gamma_i) ∗(delta_p + gamma_a + sage: gamma_i)))/((beta ∗gamma_a ∗l ∗rho_2 + betaprim ∗(delta_h + sage: gamma_r) ∗rho_2) ∗(delta_i + gamma_a + gamma_i) + (beta ∗gamma_a ∗l ∗rho_1 + sage: beta ∗(delta_h + gamma_r) ∗rho_1) ∗(delta_p + gamma_a + gamma_i)) sage: print S_gammar -0.670604500913797 sage: #sensitivity of delta_i sage: S_deltai (delta_h + gamma_r) ∗(delta_i + gamma_a + gamma_i) ∗delta_i ∗(delta_p + = sage: gamma_a + gamma_i) ∗((beta ∗gamma_a ∗l ∗rho_2 + betaprim ∗(delta_h + sage: gamma_r) ∗rho_2)/((delta_h + gamma_r) ∗(delta_i + gamma_a + sage: gamma_i) ∗(delta_p + gamma_a + gamma_i)) - ((beta ∗gamma_a ∗l ∗rho_2 + sage: betaprim ∗(delta_h + gamma_r) ∗rho_2) ∗(delta_i + gamma_a + gamma_i) + sage: (beta ∗gamma_a ∗l ∗rho_1 + beta ∗(delta_h + gamma_r) ∗rho_1) ∗(delta_p + sage: gamma_a + gamma_i))/((delta_h + gamma_r) ∗(delta_i + gamma_a + sage: gamma_i) ^ 2 ∗(delta_p + gamma_a + gamma_i)))/((beta ∗gamma_a ∗l ∗rho_2 + sage: betaprim ∗(delta_h + gamma_r) ∗rho_2) ∗(delta_i + gamma_a + gamma_i) + sage: (beta ∗gamma_a ∗l ∗rho_1 + beta ∗(delta_h + gamma_r) ∗rho_1) ∗(delta_p + sage: gamma_a + gamma_i)) sage: print S_deltai -0.0345941891985019 sage: #sensitivity of delta_p sage: S_deltap (delta_h + gamma_r) ∗(delta_i + gamma_a + gamma_i) ∗(delta_p + gamma_a + = sage: gamma_i) ∗delta_p ∗((beta ∗gamma_a ∗l ∗rho_1 + beta ∗(delta_h + sage: gamma_r) ∗rho_1)/((delta_h + gamma_r) ∗(delta_i + gamma_a + sage: gamma_i) ∗(delta_p + gamma_a + gamma_i)) - ((beta ∗gamma_a ∗l ∗rho_2 + sage: betaprim ∗(delta_h + gamma_r) ∗rho_2) ∗(delta_i + gamma_a + gamma_i) + sage: (beta ∗gamma_a ∗l ∗rho_1 + beta ∗(delta_h + gamma_r) ∗rho_1) ∗(delta_p + sage: gamma_a + gamma_i))/((delta_h + gamma_r) ∗(delta_i + gamma_a + sage: gamma_i) ∗(delta_p + gamma_a + gamma_i) ^ 2))/((beta ∗gamma_a ∗l ∗rho_2 + sage: betaprim ∗(delta_h + gamma_r) ∗rho_2) ∗(delta_i + gamma_a + gamma_i) + sage: (beta ∗gamma_a ∗l ∗rho_1 + beta ∗(delta_h + gamma_r) ∗rho_1) ∗(delta_p + sage: gamma_a + gamma_i)) sage: print S_deltap -0.0000919016790562303

3 F. Ndaïrou, I. Area, G. Bader et al. Chaos, Solitons and Fractals 141 (2020) 110311

sage: #sensitivity of delta_h sage: S_deltah (delta_h + gamma_r) ∗delta_h ∗(delta_i + gamma_a + gamma_i) ∗(delta_p + = sage: gamma_a + gamma_i) ∗((beta ∗(delta_p + gamma_a + gamma_i) ∗rho_1 + sage: betaprim ∗(delta_i + gamma_a + gamma_i) ∗rho_2)/((delta_h + sage: gamma_r) ∗(delta_i + gamma_a + gamma_i) ∗(delta_p + gamma_a + gamma_i)) - sage: ((beta ∗gamma_a ∗l ∗rho_2 + betaprim ∗(delta_h + gamma_r) ∗rho_2) ∗(delta_i + sage: gamma_a + gamma_i) + (beta ∗gamma_a ∗l ∗rho_1 + beta ∗(delta_h + sage: gamma_r) ∗rho_1) ∗(delta_p + gamma_a + gamma_i))/((delta_h + sage: gamma_r) ^ 2 ∗(delta_i + gamma_a + gamma_i) ∗(delta_p + gamma_a + sage: gamma_i)))/((beta ∗gamma_a ∗l ∗rho_2 + betaprim ∗(delta_h + sage: gamma_r) ∗rho_2) ∗(delta_i + gamma_a + gamma_i) + (beta ∗gamma_a ∗l ∗rho_1 + sage: beta ∗(delta_h + gamma_r) ∗rho_1) ∗(delta_p + gamma_a + gamma_i)) sage: print S_deltah -0.0583134348620693

Appendix B. Matlab code for Figures 2 and 3 of [1]

In our Matlab code we use the fde12 routine freely available from MATLAB central [2] . clear all realdata [0 6; 1 12; 2 19; 3 25; 4 31; 5 38; 6 44; 7 60; 8 80;9 131;10 131; 11 259; 12= 467; 13 688; 14 776; 15 1776; 16 1460; 17 1739; 18 1984; 19 2101; 20 2590; 21 2827; 22 3233; 23 3892; 24 3697; 25 3151; 26 3387; 27 2653; 28 2984; 29 2473; 30 2022; 31 1820; 32 1998; 33 1506; 34 1278; 35 2051; 36 1772; 37 1891; 38 399; 39 894; 40 397; 41 650; 42 415; 43 518; 44 412; 45 439; 46 441; 47 435; 48 579; 49 206; 50 130; 51 120; 52 143; 53 146; 54 102; 55 46; 56 45; 57 20; 58 31; 59 26; 60 11; 61 18; 62 27; 63 29; 64 39; 65 39]; aux size(realdata); deadpeople= [0 0; 1 0; 2 0; 3 0; 4 0; 5 0; 6 0; 7 0; 8 4; 9 4; 10 4; 11 8;12 15; 13= 15; 14 25; 15 26; 16 26; 17 38; 18 43; 19 46; 20 45; 21 57; 22 64; 23 66; 24 73; 25 73; 26 86; 27 89; 28 97; 29 108; 30 97; 31 254; 32 121; 33 121; 34 142; 35 106; 36 106; 37 98; 38 115; 39 118; 40 109;41 97; 42 150; 43 71; 44 52; 45 29; 46 44; 47 37; 48 35; 49 42; 50 31;51 38; 52 31; 53 30; 54 28; 55 27; 56 23; 57 17; 58 22; 59 11; 60 07;61 14; 62 10; 63 14; 64 13; 65 13]; t0 0; tend= realdata(aux(1),1); time = t0:tend; t0dead= deadpeople(1,1); timedead= t0dead:tend; totaldead= deadpeople(:,2); deadpeople= [0 0; 1 0; 2 0; 3 0; 4 0; 5 0; 6 0; 7 0; 8 4; 9 4; 10 4; 11 8;12 15; 13= 15; 14 25; 15 26; 16 26; 17 38; 18 43; 19 46; 20 45; 21 57; 22 64; 23 66; 24 73; 25 73; 26 86; 27 89; 28 97]; totalill realdata(:,2); beta 2.55;= ell = 1.56; = betap 3 ∗beta; kappa = 0.250; rho1 = 0.580; rho2 = 0.001; gammaa= 0.94; gammai = 0.27; gammar = 0.500; N 11000000/(250);= initialvalue= realdata(1,2); p0 5; = e0 = 0; i0 = initialvalue-p0; s0 = N-i0; a0 = 0; h0 = 0; r0 = 0; d0 = 0; stepsize= 0.001; delta 1/(23);= system= @(t,X)[ =

4 F. Ndaïrou, I. Area, G. Bader et al. Chaos, Solitons and Fractals 141 (2020) 110311

-beta. ∗X(3). ∗X(1)./N-ell. ∗beta. ∗X(6). ∗X(1)./N-betap. ∗X(4). ∗X(1)./N; beta. ∗X(3). ∗X(1)./N+ell. ∗beta. ∗X(6). ∗X(1)./N+betap. ∗X(4). ∗X(1)./N-kappa. ∗X(2); kappa. ∗rho1. ∗X(2)-(gammaa+gammai). ∗X(3)-delta. ∗X(3); kappa. ∗rho2. ∗X(2)-(gammaa+gammai). ∗X(4)-delta. ∗X(4); kappa. ∗(1-rho1-rho2). ∗X(2); gammaa. ∗(X(3)+X(4))-gammar. ∗X(6)-delta. ∗X(6); gammai. ∗(X(3)+X(4))+gammar. ∗X(7); ]; [ts1,ys1] fde12(1,system,t0,tend,[s0;e0;i0;p0;a0;h0;r0],stepsize); figure = hold on plot(time,totalill,’green-’,’LineWidth’,2.5) xlabel({’Time’,’(in days)’}) ylabel(’Confirmed cases per day’) plot(ts1(1,1:end),ys1(3,:)+ys1(4,:)+ys1(6,:),’black’,’linewidth’,2); hold off tau 9; aux2= size(ys1); aux3 = size(ts1); sizetimes= aux3(2); totaltime = tend-t0; = for k 1:aux2(2)-tau ∗(sizetimes-1)/totaltime = shifted(k) delta. ∗(ys1(3,k+tau)+ys1(4,k+tau)+ys1(6,k+tau)); end = newtime tau:totaltime/(sizetimes-1):tend; figure = hold on plot(totaldead,’red-’,’LineWidth’,2.5) xlabel({’Time’,’(in days)’}) ylabel(’Confirmed deads per day’) plot(newtime(:),shifted(:),’black’,’linewidth’,2); hold off

5 F. Ndaïrou, I. Area, G. Bader et al. Chaos, Solitons and Fractals 141 (2020) 110311

Funding Faïçal Ndaïrou Center for Research and Development in Mathematics and This research was funded by the Portuguese Foundation for Applications (CIDMA), Department of Mathematics, University of Science and Technology (FCT) within project UIDB/04106/2020 Aveiro, Aveiro 3810-193, Portugal (CIDMA). Ndaïrou is also grateful to the support of FCT through Iván Area the Ph.D. fellowship PD/BD/150273/2019. The work of Area and Ni- Departamento de Matemática Aplicada II, E. E. Aeronáutica e do eto has been partially supported by the Agencia Estatal de Inves- Espazo, Campus de Ourense, Universidade de Vigo, Ourense 32004, tigación (AEI) of Spain, cofinanced by the European Fund for Re- Spain gional Development (FEDER) corresponding to the 2014–2020 mul- tiyear financial framework, project MTM2016-75140-P. Moreover, Georg Bader Nieto also thanks partial financial support by Xunta de Galicia un- Institute for Applied Mathematics and Scientific Computing, der grant ED431C 2019/02 . Brandenburgische Technische Universtität Cottbus, Konrad-Wachsmann-Allee 1, Cottbus D-03046, Germany Declaration of Competing Interest Juan J. Nieto Instituto de Matematicas, Universidade de Santiago de Compostela, The authors declare that they have no known competing finan- Santiago de Compostela 15782, Spain cial interests or personal relationships that could have appeared to influence the work reported in this paper. Delfim F.M. Torres ∗ Center for Research and Development in Mathematics and Acknowledgments Applications (CIDMA), Department of Mathematics, University of Aveiro, Aveiro 3810-193, Portugal Typos found in [1] were first communicated by Georg Bader and Andreas Kleefeld. We are very grateful to both of them: to ∗Corresponding author. E-mail addresses: [email protected] (F. Ndaïrou), [email protected] (I. Area), Georg for having accepted to join us in this corrigendum, as an [email protected] (G. Bader), [email protected] (J.J. author; and to Andreas, who kindly also accepted to review and Nieto), delfi[email protected] (D.F.M. Torres) validate this corrigendum.

References

[1] Ndaïrou F , Area I , Nieto JJ , Torres DFM . Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan. Chaos Solitons Fractals 2020;135(109846):6 . [2] Garrappa R. Predictor-corrector PECE method for fractional differential equations MATLAB central file exchange. 2020. https://www.mathworks. com/matlabcentral/fileexchange/32918-predictor-corrector-pece-method-for- fractional- differential- equations .

6 JOURNAL OF INDUSTRIAL AND doi:10.3934/jimo.2017054 MANAGEMENT OPTIMIZATION

EBOLA MODEL AND OPTIMAL CONTROL WITH VACCINATION CONSTRAINTS

Ivan´ Area

Departamento de Matem´aticaAplicada II, E. E. Aeron´auticae do Espazo Campus As Lagoas, Universidade de Vigo, 32004 Ourense, Spain

Fa¨ıc¸al Nda¨ırou

African Institute for Mathematical Sciences (AIMS–Cameroon) P.O. Box 608, Limbe Crystal Gardens, South West Region, Cameroon

Juan J. Nieto

Departamento de An´alise Matem´atica, Estat´ısticae Optimizaci´on Facultade de Matem´aticas, Universidade de Santiago de Compostela 15782 Santiago de Compostela, Spain

Cristiana J. Silva and Delfim F. M. Torres⇤

Center for Research and Development in Mathematics and Applications (CIDMA) Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

(Communicated by Cheng-Chew Lim)

Abstract. The Ebola virus disease is a severe viral haemorrhagic fever syn- drome caused by Ebola virus. This disease is transmitted by direct contact with the body fluids of an infected person and objects contaminated with virus or infected animals, with a death rate close to 90% in humans. Recently, some mathematical models have been presented to analyse the spread of the 2014 Ebola outbreak in West Africa. In this paper, we introduce vaccination of the susceptible population with the aim of controlling the spread of the disease and analyse two optimal control problems related with the transmission of Ebola disease with vaccination. Firstly, we consider the case where the total number of available vaccines in a fixed period of time is limited. Secondly, we analyse the situation where there is a limited supply of vaccines at each instant of time for a fixed interval of time. The optimal control problems have been solved analytically. Finally, we have performed a number of numerical simulations in order to compare the models with vaccination and the model without vacci- nation, which has recently been shown to fit the real data. Three vaccination scenarios have been considered for our numerical simulations, namely: unlim- ited supply of vaccines; limited total number of vaccines; and limited supply of vaccines at each instant of time.

2010 Mathematics Subject Classification. Primary: 49J15, 92D30; Secondary: 34C60, 49N90. Key words and phrases. Ebola virus, mathematical modelling, transmission of Ebola, control of the spread of the Ebola disease, optimal control with vaccination constraints, vaccination scenarios. ⇤Corresponding author: delfi[email protected] (D. F. M. Torres).

1 2I.AREA,F.NDA¨IROU, J. J. NIETO, C. J. SILVA AND D. F. M. TORRES

1. Introduction. Ebola is a lethal virus for humans that is currently under strong research due to the recent outbreak in West Africa and its socioeconomic impact (see, e.g., [5, 17, 18, 21, 24, 25, 30, 34, 37] and references therein). World Health Organization (WHO) has declared Ebola virus disease epidemic as a public health emergency of international concern with severe global economic burden. At fatal Ebola infection stage, patients usually die before the antibody response. Mainly after the 2014 Ebola outbreak in West Africa, some attempts to obtain a vaccine for Ebola disease have been realized. According to the WHO, results in July 2015 from an interim analysis of the Guinea Phase III ecacy vaccine trial show that VSV-EBOV (Merck, Sharp & Dohme) is highly e↵ective against Ebola [2]. Since 2014, di↵erent mathematical models to analyze the spread of the 2014 Ebola outbreak have been presented (see, e.g., [3, 4, 19, 32, 33] and references therein). In these models the populations under study are divided into compart- ments, and the rates of transfer between compartments are expressed mathemat- ically as derivatives with respect to time of the size of the compartments. In a recent work [4], a system of eight nonlinear (fractional) di↵erential equations for a population divided into eight mutually exclusive groups was considered: suscep- tible, exposed, infected, hospitalized, asymptomatic but still infectious, dead but not buried, died, and completely recovered. By comparing the numerical results of this mathematical model and the real data provided by WHO, the di↵erence in the period of 438 days analyzed is about 7 cases per day. Note that in the day 438 after the beginning of the outbreak, the number of confirmed cases is 15018. There exist di↵erent models for the spreading of Ebola, beginning with the sim- plest SIR and SEIR models [2, 9, 10] and later more complex but also more re- alistic models have been considered [4, 24, 29]. In [25], a stochastic discrete-time Susceptible-Exposed-Infectious-Recovered (SEIR) model for infectious diseases is developed with the aim of estimating parameters from daily incidence and mortal- ity time series for an outbreak of Ebola in the Democratic Republic of Congo in 1995. In [24], the authors use data from two epidemics (in Democratic Republic of Congo in 1995 and in Uganda in 2000) and built a SEIHFR (Susceptible-Exposed- Infectious-Hospitalized-F(dead but not yet buried)-Removed) mathematical model for the spread of Ebola haemorrhagic fever epidemics taking into account transmis- sion in di↵erent epidemiological settings (in the community, in the hospital, during burial ceremonies). In [5], the authors propose a SIRD (Susceptible-Infectious- Recovered-Dead) mathematical model using classical and beta derivatives. In this model, the class of susceptible individuals does not consider new born or immigra- tion. The study shows that, for small portion of infected individuals, the whole country could die out in a very short period of time in case there is no good pre- vention. In [3], a fractional order SEIR Ebola epidemic model is proposed and the authors show that the model gives a good approximation to real data published by WHO, starting from March 27th, 2014. Optimal control is a mathematical theory that emerged after the Second World War with the formulation of the celebrated Pontryagin maximum principle, re- sponding to practical needs of engineering, particularly in the field of aeronautics and flight dynamics [31]. In the last decade, optimal control has been largely ap- plied to biomedicine, namely to models of cancer chemotherapy (see, e.g., [23]), and recently to epidemiological models [15, 26, 36]. EBOLA MODEL AND OPTIMAL CONTROL 3

In [32], the authors present a comparison between SIR and SEIR mathematical models used in the description of the Ebola virus propagation. They applied op- timal control techniques in order to understand how the spread of the virus may be controlled, e.g., through education campaigns, immunization or isolation. In [1], the authors introduce a deterministic SEIR type model with additional hospital- ization, quarantine and vaccination components in order to understand the disease dynamics. Optimal control strategies, both in the case of hospitalization (with and without quarantine) and vaccination, are used to predict the possible future out- come in terms of resource utilization for disease control and the e↵ectiveness of vaccination on sick populations. Both in [1] and [32], the authors study optimal control problems with L2 cost functionals without any state or control constraints. Here, we modify the model analyzed in [4] in order to consider optimal control problems with vaccination constraints. More precisely, we introduce an extra vari- able for the number of vaccines used, and we compare the hypothetical results if the vaccine were available at the beginning of the outbreak with the results of the model without vaccines. Firstly, we consider an optimal control problem with an end-point state constraint, that is, the total number of available vaccines, in a fixed period of time, is limited. Secondly, we analyze an optimal control problem with a mixed state constraint, in which there is a limited supply of vaccines at each instant of time for a fixed interval of time. Both optimal control problems have been an- alytically solved. Moreover, we have performed a number of numerical simulations in three di↵erent scenarios: unlimited supply of vaccines; limited total number of vaccines to be used; and limited supply of vaccines at each instant of time. From the results obtained in the first two cases, when there is no limit in the supply of vaccines or when the total number of vaccines used is limited, the optimal vaccina- tion strategy implies a vaccination of 100% of the susceptible population in a very short period of time (smaller than one day). In practice, this is a very dicult task because limitations in the number of vaccines and also in the number of humani- tarian and medical teams in the a↵ected regions are common. In this direction, the third analyzed case is extremely important since we consider a limited supply of vaccines at each instant of time. The paper is organized as follows. In Section 2, we recall a mathematical model for Ebola virus. In Section 3, the introduction of e↵ective vaccination for Ebola virus is modeled. An optimal control problem with an end-point state constraint is formulated and solved analytically in Section 4, which models the case where the total number of available vaccines in a fixed period of time is limited. In Section 5, the limited supply of vaccines at each instant of time for a fixed interval of time is mathematically translated into an optimal control problem with a mixed state control constraint. A closed form of the unique optimal control is given. In Section 6, we solve numerically the optimal control problems proposed in Sections 4 and 5. Finally, we end with Section 7 of discussion of the results. 2. Initial mathematical model for Ebola. The total population N under study is subdivided into eight mutually exclusive groups: susceptible (S), exposed (E), infected (I), hospitalized (H), asymptomatic but still infectious (R), dead but not buried (D), buried (B), and completely recovered (C). This model is adapted from [20] and analyzed in [4], where the birth and death rate are assumed to be equal and are denoted by µ, and the contact rate of susceptible individuals with infective, dead, hospitalized and asymptomatic individuals are denoted by i, d, h and r, respectively. Exposed individuals become infectious at a rate . The per capita 4I.AREA,F.NDA¨IROU, J. J. NIETO, C. J. SILVA AND D. F. M. TORRES rate of progression of individuals from the infectious class to the asymptomatic and hospitalized classes are denoted by 1 and ⌧, respectively. Individuals in the dead class progress to the buried class at a rate 1. Hospitalized individuals progress to the buried class and to the asymptomatic class at rates 2 and 2,respectively. Asymptomatic individuals become completely recovered at a rate 3. Infectious individuals progress to the dead class at a fatality rate ✏. Dead and buried bodies are incinerated at a rate ⇠. We assume that the total population, N = S + E + I + R + H + D + B + C, is constant, that is, the birth and death rates, both denoted by µ, are equal to the incineration rate ⇠. The model is mathematically described by the following system of eight nonlinear ordinary di↵erential equations: dS = µN i SI h SH d SD r SR µS, dt N N N N 8dE > i h d r > = SI + SH + SD + SR E µE, > dt N N N N >dI > = E ( + ✏ + ⌧ + µ)I, > 1 > dt >dR > = I + H ( + µ)R, > 1 2 3 > dt (1) >dD > = ✏I D ⇠D, < dt 1 dH > = ⌧I ( + + µ)H, > dt 2 2 > >dB > = D + H ⇠B > dt 1 2 > >dC > = R µC. > dt 3 > > In Fig. 1, we:> give a flowchart presentation of model (1). In this flowchart, we identify the compartmental classes as well as the parameters appearing in the model. Moreover, the values of the parameters are given in Table 1.

µN

i d 1 3 S E I R C h r µ µ µ µ ✏ ⌧ µ 2 ⇠ D H µ

1

⇠ 2 B

Figure 1. Flowchart presentation of the compartmental model (1) for Ebola.

The basic reproduction number (that is, the number of cases one case generates on average over the course of its infectious period, in an otherwise uninfected pop- ulation) of model (1) can be computed using the associated next-generation matrix method [13]. It is obtained as the spectral radius of the following matrix, known as EBOLA MODEL AND OPTIMAL CONTROL 5

Symbol Description Value per capita rate at which exposed individuals become infectious 1/11.4 µ death rate 14/1000

i contact rate of infective and susceptible individuals 0.14

d contact rate of infective and dead individuals 0.40

h contact rate of infective and hospitalized individuals 0.29 r contact rate of infective and asymptomatic individuals 0.185 1 per capita rate of progression of individuals from the infectious class to the asymptomatic class 1/10 ✏ fatality rate 1/9.6

1 per capita rate of progression of individuals from the dead class to the buried class 1/2

2 per capita rate of progression of individuals from the hospitalized class to the buried class 1/4.6

2 per capita rate of progression of individuals from the hospitalized class to the asymptomatic class 1/5 ⌧ per capita rate of progression of individuals from the infectious class to the hospitalized class 1/5

3 per capita rate of progression of individuals from the asymptomatic class to the completely recovered class 1/30 ⇠ incineration rate 14/1000 Table 1. Parameter values for model (1), corresponding to a basic reproduction number R0 =2.287. The values of the parameters come from [7, 14, 22, 24, 28, 34, 35].

the next-generation-matrix:

A11 A12 A13 A14 A15 00000 0 1 1 00000 FV = , B 00000C B C B 00000C B C B 00000C B C where @ A a3ia1a4 + a3r (a41 + ⌧2)+d✏a1a4 + a3h⌧a1 A11 = , a1a2a3a4a5

r r(a41 + ⌧2) d✏ h⌧ A12 = + + + , a1 a1a2a3 a2a3 a2a4 r d r2 h A13 = ,A14 = ,A15 = + a1 a3 a1a4 a4 with a1 = 2+2+µ, a2 = 3+µ, a3 = 1+⇠, a4 = +µ, and a5 = 1+✏+⌧ +µ.

Therefore, the basic reproduction number R0 is given by

2 R0 = a3 i(3a1 + µ(2 + 2)+µ )+r(1a1 + 2⌧)+h⌧a2 a1a2a3a4a5 2 ⇥ +d✏ (3a1 + µ(2 + 2)) + µ .

As it is well-known, if the basic reproduction number R0 < 1, then the infection⇤ will stop in the long run; but if R0 > 1, then the infection will spread in population. 6I.AREA,F.NDA¨IROU, J. J. NIETO, C. J. SILVA AND D. F. M. TORRES

In this section, we have recalled a model for describing the Ebola virus transmis- sion. Now we want to address the question about how to introduce vaccination as a prevention measure. This is analyzed in the next section.

3. Mathematical model for Ebola with vaccination. We now introduce vac- cination of the susceptible population with the aim of controlling the spread of the disease. We assume that the vaccine is e↵ective so that all vaccinated susceptible individuals become completely recovered (see, e.g., [6, 27] for vaccination in a SEIR model that corresponds to a system of four nonlinear ordinary di↵erential equa- tions). Let us introduce in model (1) a control function u(t), which represents the percentage of susceptible individuals being vaccinated at each instant of time t with t [0,tf ]. Unfortunately, in many situations, the number of available vaccines does not2 fulfill the necessities in order to eradicate the disease. In this paper, we consider limitations on the total number of vaccines during a fixed interval of time [0,tf ] and on the number of available vaccines at each instant of time t with t [0,tf ]. In order to translate this real situation mathematically, we introduce an extra2 variable W that denotes the number of vaccines used: dW (t)=u(t)S(t), subject to the initial condition W (0) = 0. dt Hence, the model with vaccination is given by the following system of nine nonlinear ordinary di↵erential equations: dS (t)=µN i S(t)I(t) h S(t)H(t) d S(t)D(t) dt N N N 8 > r S(t)R(t) µS(t) S(t)u(t), > > N > >dE i h d > (t)= S(t)I(t)+ S(t)H(t)+ S(t)D(t) > dt N N N > > r > + S(t)R(t) E(t) µE(t), > N > > >dI > (t)=E(t) (1 + ✏ + ⌧ + µ)I(t), > dt > >dR > (t)=1I(t)+2H(t) (3 + µ)R(t), (2) > dt >

dt > >dH > (t)=⌧I(t) (2 + 2 + µ)H(t), > dt > >dB > (t)= D(t)+ H(t) ⇠B(t), > 1 2 > dt > >dC > (t)=3R(t) µC(t)+S(t)u(t), > dt > >dW > (t)=S(t)u(t). > dt > > In model:> (2), the vaccination parameter is fixed. In the next section, we address the question of how to choose this parameter in an optimal way along time. EBOLA MODEL AND OPTIMAL CONTROL 7

4. Optimal control with an end-point state constraint. We start by consid- ering the case where the total number of available vaccines, in a fixed period of time, is limited. We formulate and solve analytically such optimal control problem with end-point state constraint, which will be then solved numerically in Section 6. We consider the model with vaccination (2) and formulate the optimal control problem with the aim to determine the vaccination strategy u over a fixed interval of time [0,tf ] that minimizes the cost functional

tf 2 J(u)= w1I(t)+w2u (t) dt, (3) Z0 ⇥ ⇤ where the constants w1 and w2 represent the weights associated with the number of infected individuals and on the cost associated with the vaccination program, respectively. We assume that the control function u takes values between 0 and 1. When u(t) = 0, no susceptible individual is vaccinated at time t;ifu(t) = 1, then all susceptible individuals are vaccinated at t. Let # denote the total amount of available vaccines in a fixed period of time [0,tf ]. This constraint is represented by W (t ) #. (4) f  Let 9 x(t)=(x1(t),...,x9(t)) = (S(t),E(t),I(t),R(t),D(t),H(t),B(t),C(t),W(t)) R . 2 The optimal control problem consists to find the optimal trajectoryx ˜ associated with the optimal controlu ˜, satisfying the control system (2), the initial conditions x(0) = (18000, 0, 15, 0, 0, 0, 0, 0, 0) (5) (see [4]), the constraint (4), and where the controlu ˜ ⌦ minimizes the objective functional (3)with 2

⌦= u( ) L1(0,t ) 0 u(t) 1 . (6) · 2 f |   ⇢ The existence of an optimal controlu ˜ and associated optimal trajectoryx ˜ comes from the convexity of the integrand of the cost function (3)withrespecttothe control u and the Lipschitz property of the state system with respect to state variables x (see, e.g., [8, 16] for existence results of optimal solutions). According to the Pontryagin Maximum Principle [31], ifu ˜ ⌦ is optimal for the problem 2 (2), (3) with initial conditions (5) and fixed final time tf ,thenthereexists 9 2 AC([0,tf ]; R ), (t)=(1(t),...,9(t)), called the adjoint vector, such that @ @ x˙ = H1 and ˙ = H1 , @ @x where the Hamiltonian is defined by H1 (x, u, )=w x + w u2 + (f(x)+Ax + Bxu) H1 1 3 2 with

f =(f1 f2 0 0 0 0 0 0 0) , f (x)=µN i SI h SH d SD r SR, 1 N N N N f (x)= i SI + h SH + d SD + r SR, 2 N N N N 8I.AREA,F.NDA¨IROU, J. J. NIETO, C. J. SILVA AND D. F. M. TORRES

µ 00 0 0 0 00 0 µ 00 0 0 00 0 0 ⇤ 00 0001 B 00 ( + µ)0 00C B 1 3 2 C A = B 00✏ 0 ( + ⇠)0 00C B 1 C B 00⌧ 00( + + µ)0 0C B 2 2 C B 000 0 ⇠ 0 C B 1 2 C B 000 000µC B 3 C B 000 0 0 0 00C B C with ⇤ @= ( + ✏ + ⌧ + µ), and A 1 B =(bZ) , where b =( 1 0 0 0 0 0 0 1 1)T and Z =0with0the8 9 null matrix. The mini- mization condition ⇥

1 x˜(t), u˜(t), ˜(t) =min 1 x˜(t),u,˜(t) (7) H u ⌦ H ⇣ ⌘ 2 ⇣ ⌘ holds almost everywhere on [0,tf ]. Moreover, transversality conditions ˜i(tf ) = 0, i =1,...,8, hold. Solving the minimality condition (7) on the interior of the set of admissible controls ⌦ gives ˜ ˜ ˜ 1 1(t)+8(t)+9(t) x˜1(t) u˜(t)= , 2 ⇣ !2 ⌘ where the adjoint functions satisfy

˙ i h d r ˜ = ˜ x˜ x˜ x˜ x˜ µ u˜ 1 1 N 3 N 6 N 5 N 4 8 ✓ ◆ ˜ i h d r ˜ ˜ > 2 x˜3 + x˜6 + x˜5 + x˜4 8u˜ 9u,˜ > N N N N > ✓ ◆ >˜˙ ˜ ˜ >2 = 2 ( µ) 3 , > > >˜˙ ˜ i ˜ i ˜ ˜ ˜ ˜ >3 = !1 + 1 x˜1 2 x˜1 3 ( 1 ✏ ⌧ µ) 41 5✏ 6⌧, > N N > >˜˙ ˜ r ˜ r ˜ ˜ >4 = 1 x˜1 2 x˜1 4 ( 3 µ) 83, > N N > > ˙ d d <>˜ = ˜ x˜ ˜ x˜ ˜ ( ⇠) ˜ , 5 1 N 1 2 N 1 5 1 7 1 >˜˙ ˜ h ˜ h ˜ ˜ ˜ >6 = 1 x˜1 2 x˜1 42 6 ( 2 2 µ) 72, > N N > >˜˙ ˜ >7 = 7⇠, > > ˙ >˜8 = ˜8µ, > > >˜˙ >9 =0. > > (8) :> Since W has initial and terminal conditions, the adjoint function ˜9, associated with the state variable W , has no transversality condition. From (8), ˜9(t) k, where the constant k must be such that the end point conditions W (0) = 0⌘ and W (tf )=# are satisfied. As the optimal controlu ˜ can take values on the boundary EBOLA MODEL AND OPTIMAL CONTROL 9 of the control set [0, 1], the optimal controlu ˜ must satisfy

˜ ˜ ˜ 1 1(t) 8(t) 9(t) x˜1(t) u˜(t)=min 1, max 0, . (9) 8 8 2 ⇣ !2 ⌘ 99 < < == The optimal controlu ˜:given by: (9) is unique due to the boundedness;; of the state and adjoint functions and the Lipschitz property of systems (2) and (8). We would like to note that if we consider the optimal control problem without any restriction on the number of available vaccines, that is, to find the optimal solution (˜x, u˜), withu ˜ ⌦, which minimizes the cost functional (3) subject to the control 2 system (2), initial conditions (5), and free final conditions (x1(tf ),...,x9(tf )), then the adjoint functions (1,...,9) must satisfy transversality conditions i(tf ) = 0, i =1,...,9, and, since ˜9 = 0, the optimal control is given by

˜ ˜ 1 1(t) 8(t) x˜1(t) u˜(t)=min 1, max 0, . 8 8 2 ⇣ !2 ⌘ 99 < < == In a concrete situation, the: number: of available vaccines is always;; limited. There- fore, it is also important to study the optimal control problem with such kind of constraints. This is done in Section 5. Both problems, with and without constraints, are numerically solved in Section 6.

5. Optimal control with a mixed state control constraint. A particularly challenging situation in vaccination programs happens when there is a limited sup- ply of vaccines at each instant of time for a fixed interval of time [0,tf ]. In order to study this health public problem, from the optimal point of view, we formulate an optimal control problem with a mixed state control constraint (see, e.g., [6]). The cost functional (3) remains (3), the one considered in previous section, as well as the set of admissible controls ⌦ (6). The end point state constraint (4) is replaced by the following mixed state control constraint:

S(t)u(t) #, # 0 , for almost all t [0,t ],  2 f which should be satisfied at almost every instant of time during the whole vaccina- tion program. Analogously to [6], we observe that in our optimal control problem the di↵erential equation dW (t)=S(t)u(t) dt does not appear neither in the cost and in any other di↵erential equation, nor in the mixed state control constraint. Thus, in this section, the control system does not include the last equation and x is used to denote

8 x(t)=(x1(t),...,x8(t)) = (S(t),E(t),I(t),R(t),D(t),H(t),B(t),C(t)) R . 2 Let us consider the initial conditions (5). The control system can be rewritten in the following way: dx(t) = f(x(t)) + Ax(t)+Bx(t)u(t), dt 10 I. AREA, F. NDA¨IROU, J. J. NIETO, C. J. SILVA AND D. F. M. TORRES with µ 00 0 0 0 00 0 µ 00 0 0 00 0 0 ⇤ 00 0001 B 00 ( + µ)0 00C A = B 1 3 2 C , B 00✏ 0 ( + ⇠)0 00C B 1 C B 00⌧ 00( + + µ)0 0C B 2 2 C B 000 0 ⇠ 0 C B 1 2 C B 000 000µC B 3 C @ A ⇤ = ( + ✏ + ⌧ + µ), 1 B =(bZ) , where b =( 1 0 0 0 0 0 0 1)T and Z =0with0the7 8 null matrix, and ⇥ f =(f1 f2 0 0 0 0 0 0) , with f (x)=µN i SI h SH d SD r SR 1 N N N N and f (x)= i SI + h SH + d SD + r SR. 2 N N N N It follows from Theorem 23.11 in [11] that problem (2)–(6) has a solution (see also [6]). Let (˜x, u˜) denote such solution. To determine it, we apply the Pontryagin Maximum Principle (see, e.g., Theorem 7.1 in [12]): there exists multipliers 0 0, 8 1  AC([0,tf ]; R ), and L ([0,tf ]; R), such that 2 2 min (t) : t [0,tf ] >0 (nontriviality condition); • d(t){| |@ 2 } = H2 (˜x(t), u˜(t), ,(t), (t)) (adjoint system); • dt @x 0 (t)Bx˜(t)+ (t)˜x (t)+ w u˜2(t) (˜u(t)) a.e. and • 1 0 2 2N[0,1] (˜x(t), u˜(t), ,(t), (t)) (˜x(t),v, ,(t), (t)), v [0, 1] :x ˜ (t)v # H2 0 H2 0 8 2 1  (minimality condition); (t)(˜x1(t)˜u(t) #) = 0 and (t) 0 a.e.; • (t )=(0,...,0) (transversality conditions); • f where the Hamiltonian for problem (2)–(6)isdefinedby H2 (x, u, ,, )= w x + w u2 + (f(x)+Ax + Bxu)+ (Su #), H2 0 0 1 3 2 and [0,1](˜u(t)) stands for the normal cone from convex analysis to [0, 1] at the optimalN controlu ˜(t) (see, e.g., [11]). The optimal solution (˜x, u˜) is normal (see [6] for details), so we can choose 0 = 1. Analogously to previous section, we obtain a closed form of the unique optimal controlu ˜: ˜ ˜ 1 1(t) 8(t) (t) x˜1(t) u˜(t)=min 1, max 0, . 8 8 2 ⇣ !2 ⌘ 99 < < == The theoretical results: obtained: in Sections 4 and 5 are illustrated;; numerically in the next section. EBOLA MODEL AND OPTIMAL CONTROL 11

6. Numerical simulations. We start the numerical simulations by considering an intervention of 90 days, initial conditions given in (5), and the evolution of cumulative confirmed cases based on the data from the World Health Organization (WHO), following all the reports of the disease in the three main a↵ected countries of Western Africa of the 2014 Ebola outbreak, namely, Liberia, Guinea and Sierra Leone. The model (1) with the parameter values from Table 1 fits the real data from WHO, see Fig. 2a. We would like to emphasize that we are just considering the initial period of spreading of the disease, in which the vaccination should be introduced. Looking to a longer period of time (as considered in [4]), then the model fits quite well the real data: in [4, Figure 2] the `2 norm of the di↵erence between the real data and the prediction is 3181, which gives an error of less than 7.3 cases per day, as compared with about 15,000 cases at the end of the outbreak.

900 16

800 14

700 12 600

500 10

400 8

300 6 Cumulative confirmed cases Cumulative confirmed cases 200

4 100

0 2 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 time (days) time (days)

(a) Real data and solution of (1) (b) Unlimited supply of vaccines

Figure 2. (a) Cumulative confirmed cases: in dashed circle line the real data from WHO and in continuous line the values of I(t)+ R(t)+D(t)+H(t)+B(t)+C(t) µ(N S(t) E(t)) from (1) with the parameter values from Table 1. (b) Cumulative confirmed cases given in (2), when available an unlimited supply of vaccines, also with the parameter values from Table 1.

Assuming that, in the near future, an e↵ective vaccine against the Ebola virus will be available, as expected by WHO by the end of 2015, we study three di↵erent scenarios, which illustrate limitations on the number of vaccines available and on the capacity of administration of the vaccines by the health care services and hu- manitarian teams working in the a↵ected countries. The three vaccination scenarios are the following: unlimited supply of vaccines; limited total number of vaccines to be used; and limited supply of vaccines at each instant of time.

6.1. Unlimited supply of vaccines. Assume w1 = w2 = 1. If we administer from an unlimited supply of vaccines, then the number of total individuals who have an active infection I(t)+R(t)+D(t)+H(t)+B(t) µ(N S(t) E(t) C(t)) during the 90 days is a decreasing function in time, and is equal to 3.56 individuals at the final time (see Fig. 2b). If we compare the case where there is no vaccination with the opposite case of unlimited supply of vaccines, we observe that at the end of 90 days the class of completely recovered individuals has approximately 86.5 individuals in the case of no vaccination and 13468 in the case of unlimited vaccination, which represents 74.82 12 I. AREA, F. NDA¨IROU, J. J. NIETO, C. J. SILVA AND D. F. M. TORRES per cent of the total population (see Fig. 3–4). If vaccines are available, then the number of individuals that develop active disease is less than one at the end of 7.5 days and less that 0.1 at the end of 32.4 days. In the case of no vaccination, the class of active infected individuals has 61.7 individuals at the end of 90 days (see Fig. 3). If no vaccination is provided, then the number of deaths, hospitalizations and burials increases from 1.2 to 262.6, when compared to the case of unlimited supply of vaccines (see Fig. 4). The optimal vaccination policy suggested by the solution

4 Vaccination x 10 Vaccination 2 No vaccination 400 No vaccination

1.5 300

S 1 E 200

0.5 100

0 0 0 50 100 0 50 100 time (days) time (days)

80 Vaccination 150 Vaccination No vaccination No vaccination 60 100 I 40 R 50 20

0 0 0 50 100 0 50 100 time (days) time (days)

Figure 3. Individuals S(t), E(t), I(t) and R(t). In dashed line, the case of vaccination without limit on the supply of vaccines; in continuous line, the case with no vaccination with the parameter values from Table 1. of the optimal problem, implies a vaccination of 100 per cent of the susceptible population for approximately 1.62 days followed by a fast reduction of the fraction of susceptible population that is vaccinated. This is based on the fact that the vaccine is e↵ective and once all the susceptible population is vaccinated in a short period of time, then the number of susceptible individuals immediately decreases, since they are transferred to the class of completely recovered individuals, as well as the need of vaccination (see Fig. 5a). The previous results show the importance of an e↵ective vaccine for Ebola virus and the very good results that can be attained if the number of available vaccines satisfies the needs of the population. 6.2. Limited total number of vaccines. In Fig. 5b, we observe that at the end of 90 days, 33786 vaccines were used, if the supply of vaccines is unlimited. In this section, we consider the case where the total number of vaccines used in the 90 days period is limited. We consider the case where the total number of vaccines available is lower or equal than the initial number of susceptible individuals (W (90) 10000, W (90) 11000, W (90) 13000, W (90) 15000, W (90) 16000, W (90)  18000) and the case where the total number of vaccines available is bigger than the initial number of susceptible individuals (W (90) 20000). We first consider w1 = w2 = 1. The cumulative confirmed cases (see Fig. 6) increases in time in the case W (90)  EBOLA MODEL AND OPTIMAL CONTROL 13

15 Vaccination 30 Vaccination No vaccination No vaccination

10 20 D H 5 10

0 0 0 50 100 0 50 100 time (days) time (days) 4 x 10 300 Vaccination 2 No vaccination 1.5 200 B C 1 Vaccination 100 No vaccination 0.5

0 0 0 50 100 0 50 100 time (days) time (days)

Figure 4. Individuals D(t), H(t), B(t) and C(t), with the param- eter values from Table 1. In dashed line, the case of vaccination without limit on the supply of vaccines; in continuous line, the case of no vaccination.

4 x 10 3.5 1

0.9 3

0.8 2.5 0.7

0.6 2 u 0.5 W 1.5 0.4

0.3 1

0.2 0.5 0.1

0 0 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 time (days) time (days)

(a) Optimal control u(t) (b) Number of vaccines W (t)

Figure 5. Optimal control and number of vaccines with the pa- rameter values from Table 1, when an unlimited supply of vaccines is available.

10000 and decreases in the case W (90) 20000, t [0, 90]. At the end of the 90 days period, the total number of individuals who got2 active infection is approximately 76 and 9.5 individuals in the case W (90) 10000 and W (90) 20000, respectively. In the case W (90) 10000, the optimal control takes the maximum value for less than one day (approximately 0.72 days) with a cost equal to 322.74, and in the case W (90) 20000, the optimal control takes the maximum value for approximately 2.2 days with a cost equal to 72.35. The cost associated to the case W (90) 20000 is lower than the one in the case W (90) 10000, although more individuals are vaccinated, since the number of individuals in the class I is lower. Namely, in the 14 I. AREA, F. NDA¨IROU, J. J. NIETO, C. J. SILVA AND D. F. M. TORRES

80 ≤ W(90)≤10000 W(90) 10000 1 W(90)≤20000 70 W(90)≤20000

60 0.8

50 0.6

40 u

0.4 30

Cumulative confirmed cases 20 0.2

10 0

0 0 20 40 60 80 100 0 10 20 30 40 50 60 70 80 90 time (days) time (days)

(a) Cumulative confirmed cases (b) Optimal control

Figure 6. (a) Cumulative confirmed cases. (b) Optimal control for the case of limited total number of vaccines. Dashed line for W (90) 10000 and continuous line for W (90) 20000.  

4 x 10 2 W(90)≤10000 60 W(90)≤10000 W(90)≤20000 W(90)≤20000 1.5 40

S 1 E 20 0.5

0 0 0 50 100 0 50 100 time (days) time (days)

15 30 W(90)≤10000 ≤ W(90) 10000 W(90)≤20000 W(90)≤20000 10 20 I R

5 10

0 0 0 50 100 0 50 100 time (days) time (days)

Figure 7. Individuals S(t), E(t), I(t) and R(t). The dashed line represents the case where W (90) 10000 and the continuous line represents the case where W (90)  20000.  case W (90) 10000, the number of individuals with active infection at the end of 90 days is equal to I(90) = 8.4 and in the case W (90) 20000 the respective number is equal to I(90) = 0.67. This means that in the case W (90) 10000, in a epidemiological scenario corresponding to a basic reproduction number greater than one, 10000 vaccines will not be enough to eradicate the disease. Additionally, if we consider the maximum value for the total number of vaccines used during EBOLA MODEL AND OPTIMAL CONTROL 15

2 W(90)≤ 10000 4 W(90)≤10000 W(90)≤20000 W(90)≤20000 1.5 3

D 1 H 2

0.5 1

0 0 0 50 100 0 50 100 time (days) time (days) 4 x 10 60 2 ≤ W(90)≤10000 W(90) 10000 W(90)≤20000 W(90)≤20000 1.5 40 B C 1 20 0.5

0 0 0 50 100 0 50 100 time (days) time (days)

Figure 8. Individuals D(t), H(t), B(t) and C(t). The dashed line represents the case where W (90) 10000 and the continuous line represents the case where W (90)  20000.  the period of 90 days to be equal to 11000, 13000, 15000, 16000 and 18000, then we observe that the optimal control u remains more time at the maximum value 1 when the supply of vaccines is bigger, which means that when the total number of available vaccines is increased there will be resources to vaccinate all susceptible individuals for a longer period of time, which implies a bigger reduction of the number of individuals who get infected by the virus (see Fig. 9a and 9b for the optimal control strategy and respective zoom in the period of vaccination).

4 x 10 2.5 W(90)≤10000 1 W(90)≤10000 1 W(90)≤11000 W(90)≤11000 0.9 W(90)≤13000 W(90)≤13000 2 W(90)≤15000 0.8 W(90)≤15000 0.8 W(90)≤16000 W(90)≤16000 W(90)≤18000 0.7 W(90)≤18000 W(90)≤20000 W(90)≤20000 0.6 0.6 1.5 u u

0.5 W

0.4 0.4 1 W(90)≤10000 W(90)≤11000 0.3 W(90)≤13000 0.2 0.2 W(90)≤15000 0.5 W(90)≤16000 0.1 W(90)≤18000 0 ≤ 0 W(90) 20000 0 0 10 20 30 40 50 60 70 80 90 2 4 6 8 10 12 14 16 18 20 0 10 20 30 40 50 60 70 80 90 time (days) time (days) time (days) (a) Optimal control (b) Optimal control (c) Number of vaccines

Figure 9. Optimal control u(t) and number of vaccines W (t) for W (90) 10000, W (90) 11000, W (90) 13000, W (90) 15000, W (90)  16000, W (90)  18000 and W (90) 20000.     16 I. AREA, F. NDA¨IROU, J. J. NIETO, C. J. SILVA AND D. F. M. TORRES

Consider now the case where the weight constant associated with the cost of implementation of the vaccination strategy, designated by the optimal control u, is bigger than one, for example, consider w1 = 1 and w2 = 50, and w1 = 1 and w = 500. To simplify, consider in both cases W (90) 10000 and W (90) 20000. 2   When we increase the weight constant w2, the maximum value attained by the optimal control becomes lower than one (see Fig. 10). In the case W (90) 10000 for  w2 = 50, the optimal control starts with the value u(0) = 0.54 and is a decreasing function with a cost function 344.3. At the end of approximately 3.7 days, the control remains equal to zero. For w2 = 500, the optimal control starts with the value u(0) = 0.16 and is also a decreasing function, with a cost 399.62. At the end of 13.5 days, it remains equal to zero. The behavior of the optimal state variables S, E, I, R, D, H, B and C are similar.

12000 w =50 2 w =500 0.6 2 10000 w =50 0.5 2 w =500 2 8000 0.4 u

0.3 W 6000

0.2 4000

0.1

2000 0

−0.1 0 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 time (days) time (days)

(a) Control (b) Number of vaccines

Figure 10. Optimal control u(t) and number of vaccines W (t) for W (90) 10000. In dashed line the case w = 50 and in continuous  2 line the case w2 = 500.

6.3. Limited supply of vaccines at each instant of time. From previous re- sults, we observe that when there is no limit on the supply of vaccines, or when the total number of used vaccines is limited, the optimal vaccination strategy implies a vaccination of 100 per cent of the susceptible population in a very short period of time, sometimes smaller than one day. But we know that in practice this is a very dicult task, since there are limitations in the number of vaccines available and also in the number of health care workers or humanitarian teams in the regions a↵ected by Ebola virus with capacity to vaccinate such a big number of individuals almost simultaneously. From this point of view, it is important to study the case where there is a limited supply of vaccines at each instant of time. In this section, we consider w1 = w2 = 1, a shorter interval of time [0,tf ], with tf = 10, 15, 16, and we assume that at each instant of time there exist only 1000, 1200 and 900 available vaccines, respectively. From our point of view, these numbers of available vaccines at each instant of time and the number of days considered, correspond to possible real scenarios, which are possible to implement in a concrete endemic region and at the same time characterize lack of human and material resources to vaccinate the susceptible population in a short period of time. From the numerical simulations, for such mixed constraints, the number of cumulative confirmed cases EBOLA MODEL AND OPTIMAL CONTROL 17 increases with time (see Figure 11a). The cost associated with the vaccination cam- paign, associated with the solution of the optimal control problem with the mixed constraint S(t)u(t) 1000, t [0, 10], is equal to 45.8879. Such solution is the less costly of the three considered,2 followed by the constraint S(t)u(t) 1200 for t [0, 15] with a cost of 55.079. The most expensive vaccination strategy is the one2 associated with the mixed constraint S(t)u(t) 900, t [0, 16], with a cost of 59.109. The strategy associated with the constraint S(t)u(2t) 1000 is the one where a lowest number of susceptible individuals completely recover through vacci- nation, with 7540.9 individuals in the class C at the end of 10 days. If we consider that at each instant of time there are 1200 vaccines available during a period of 15 days, then 12438 completely recover. This is the strategy with more individuals in the class C. If we consider 16 days, but only 900 vaccines available for each instant of time, then only 10839 individuals completely recover (see Fig. 11b). For all three mixed constraint situations, the number of individuals in the classes E, I, R, D, H and B does not change significantly (therefore, the figures with these classes are omitted). As the number of available vaccines represent a small percentage of the susceptible population, in the three cases the optimal vaccination strategies for the constraints S(t)u(t) 1200 and S(t)u(t) 900 suggest that the percentage of the susceptible population that is vaccinated is always inferior than 18 percent. In the case of the constraint S(t)u(t) 1000, this percentage is always inferior to 8 percent (see Fig. 11c). 

14000 21 0.2 ≤ S*u ≤ 1000 S*u 1000 S*u ≤ 1000 ≤ S*u ≤ 1200 0.18 S*u 1200 12000 S*u ≤ 1200 ≤ 20 S*u ≤ 900 S*u 900 ≤ 0.16 S*u 900 10000 0.14 19 8000 0.12 C

18 u 0.1 6000 0.08 17 4000 0.06 Cumulative confirmed cases

0.04 16 2000 0.02

15 0 0 0 5 10 15 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 time (days) time (days) time (days)

(a) Total of (b) Completely recovered (c) Optimal control active infected

Figure 11. (a) Cumulative confirmed cases, (b) completely recov- ered, (c) optimal control. In (a), (b) and (c) the following mixed constraints are considered: S(t)u(t) 1000 for all t [0, 10], S(t)u(t) 1200 for all t [0, 15], and S(t)u(t) 9002 for all t [0, 16]. 2  2

7. Discussion. We assume that, in a near future, an e↵ective vaccine against the Ebola virus will be available. Under this assumption, three di↵erent scenarios have been studied: unlimited supply of vaccines; limited total number of vaccines to be used; and limited supply of vaccines at each instant of time. We have solved the optimal control problems analytically and we have performed a number of numerical simulations in the three aforementioned vaccination scenarios. Some authors have already considered the optimal control problem with vacci- nation for Ebola disease, but always with unlimited supply of vaccines [1, 32]. It turns out that the solution to this mathematical problem is obvious: the solution 18 I. AREA, F. NDA¨IROU, J. J. NIETO, C. J. SILVA AND D. F. M. TORRES consists to vaccinate all susceptible individuals in the beginning of the outbreak. This is a very particular case of our work, investigated in Section 6.1 (see Figure 5). If vaccines are available without any restriction, then one could completely eradi- cate Ebola in a very short period of time. These results show the importance of an e↵ective vaccine for Ebola virus and the very good results that can be attained if the number of available vaccines satisfy the needs of the population. Unfortunately, such situation is not realistic: in case an e↵ective vaccine for Ebola virus will ap- pear, there always will be restrictions on the number of available vaccines as well as constraints on how to inoculate them in a proper way and in a short period of time; economic problems might also exist. In our work, for first time in the literature of Ebola, an optimal control problem with state and control constraints has been considered. Mathematically, it rep- resents a health public problem of limited total number of vaccines. The results obtained in Section 6.2 provide useful information on the number of vaccines to be bought, in order to reduce the number of new infections with minimum cost. For example, the results between 10000 and 20000 vaccines (in 90 days) are completely di↵erent. With 10000 vaccines, the number of cumulative infected cases continues to increase, while with 20000 vaccines it is already possible to decrease the new infections. The optimal solution, in this case, is similar to the case of unlimited supply of vaccines, that is, it implies a vaccination of 100 per cent of the susceptible population in a very short period of time. In practice, this is an unrealistic task, due to the necessary number of vaccines and humanitarian teams in the regions a↵ected by Ebola. Therefore, we conclude that it is important to study the case where there is a limited supply of vaccines at each instant of time. This was investi- gated in Section 6.3. This situation is much richer and the optimal control solution is not obvious. For a given number of available vaccines at each instant of time, we have a di↵erent solution, which is the optimal rate of susceptible individuals that should be vaccinated. In this case, the optimal control implies the vaccination of a small subset of the susceptible population. It remains the ethical problem of how to choose the individuals to be vaccinated.

Acknowledgments. Three reviewers deserve special thanks for helpful and con- structive comments. The work of Area was partially supported by the Ministerio de Econom´ıa y Competitividad of Spain, under grants MTM2012–38794–C02–01 and MTM2016–75140–P, co-financed by the European Community fund FEDER. Nda¨ırou acknowledges the AIMS-Cameroon 2014–2015 fellowship, Nieto the partial financial support by the Ministerio de Econom´ıa y Competitividad of Spain under grants MTM2010–15314, MTM2013–43014–P and MTM2016–75140–P, and Xunta de Galicia under grants R2014/002 and GRC 2015/004, co-financed by the European Community fund FEDER. Silva was supported through the Portuguese Foundation for Science and Technology (FCT) post-doc fellowship SFRH/BPD/72061/2010. The work of Silva and Torres was partially supported by FCT through CIDMA and project UID/MAT/04106/2013, and by project TOCCATA, reference PTDC/EEI- AUT/2933/2014, funded by Project 3599 – Promover a Produ¸c˜aoCient´ıfica e De- senvolvimento Tecnol´ogico e a Constitui¸c˜aode Redes Tem´aticas (3599-PPCDT) and FEDER funds through COMPETE 2020, Programa Operacional Competitividade e Internacionaliza¸c˜ao(POCI), and by national funds through FCT. EBOLA MODEL AND OPTIMAL CONTROL 19

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