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Mathematical Modelling of Some Diseases Related to Water

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Mathematical Modelling of Some Diseases Related to Water 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.
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