Understanding the Role of Health Care Workers in a Trade-Off Model

Understanding the Role of Health Care Workers in a Trade-oﬀ Model between Contact and Transmission for Ebola Virus Disease

Thesis

Presented in Partial Fulﬁllment of the Requirements for the Degree Master in Mathematical Sciences in the Graduate School of The Ohio State University

By Eduan E. Mart´ınez-Soto,B.S. Graduate Program in Mathematical Sciences

The Ohio State University 2016

Thesis Committee: Joseph H. Tien, Advisor Adriana T. Dawes c Copyright by Eduan E. Mart´ınez-Soto 2016 Abstract

During 2014 in West Africa, Ebola virus disease (EVD) was a serious public health concern and risk where the most deadly outbreak occurred since the initial epidemic in 1976 and where health care workers (HCWs) population suffered in a dramatic way. EVD transmission is possible through the movement of infected individuals where infection is classified as asymptomatic or severe. This could imply that as symptoms severity increase, individuals with severe infections may be unable to move as much and therefore contacts with other individuals will decrease. This involves a trade-off between contact and probability of transmission given contact in the general population (non-HCWs population). In contrast, HCWs population do not experience the same trade-off as the general population, because as symptom severity becomes greater, contacts with HCWs will increase. A mathematical model was developed in order to study this scenario and also to interpret the effect of the trade-off involving both populations by an-

alyzing the basic reproductive number R0 as a function of the probability of transmission given contact. It was found that R0 in the model represents the contribution of the general population and the contribution of the HCWs population regarding the new cases of EVD. Moreover, suﬃcient conditions were determined in order to acquire an increasing monotone function of the probability of transmission given contact including intermediate ranges for relative transmissibility parameters. Finally, performing parameter estimation of the model it was possible to estimate the initial growth rate of an epidemic r using the cumulative inci-

ii dence data from 2014 that corresponds to the outbreak. Additionally, R0 was estimated from r providing a match between the data, literature, and the deterministic model. This could help to interpret in a clean way how much the contribution in disease transmission from HCWs during the 2014 Ebola outbreak was, how fast the EVD outbreak grew, and how to be prepared for another catastrophic outbreak with the necessary intervention capacity and adequate measurements for control.

iii Dedicated to my uncle Rafael “Charito” Gonz´alezGarc´ıa (1970 − 2005)

iv Acknowledgements

I am indebted to my advisor, Dr. Joseph H. Tien, for his generous support, remarkable patience, consistent guidance, encouragement, and flexibility. This work would not have been possible without his passion and dedication; thank you for being an inspiration to me. The interest, transparency, commit- ment and positive attitude that you showed in my success and development is something for which I will feel very grateful for the rest of my life. I would also like to thank Dr. Adriana T. Dawes for her support since I came to The Ohio State University (OSU). Thank you for every advice you gave me (specially the poster presentation advice during your MOLGEN 5660 course). In summary, thank you for putting all your trust in me, for listening me and for always say “yes” to me, including for serving as a thesis committee member. Mr. Roman Nitze has been a fundamental part of my success. He was the first person I met at OSU. His good sense of humor and kindness was extremely helpful to make me feel comfortable at the Mathematics Department. Thank you! On the other hand, I would like to thank Evelyn Rodr´ıguez-Correafor her patience with me and for allowed me to interrupt her every time I wanted (even though she was very busy). We were a great team! In addition, I would like to thank Juan GuzmánRoca who helped me a lot in how to be better in MATLAB and for taking of his time to provide me uncountable help and therefore to become a survivor in scientific computing aspects.

v Finally, I would like to thank all my friends and family members. Specially to my mom, Luz Eneida, who taught to my little brother, Eduar, and to me good manners and also how to face life in a positive way no matter how difficult could be the circumstances, among other wonderful things. Thank you to my dad Edwin and grandmother Mayra for all your support. Also thank you to my girlfriend, Jennifer, for her kind ear and for always being there for me, for listening all my concerns and for all your love, including all your patience. I would like to thank my grandparents and cousin Arnaldo, Mar´ıa,and Rubénfor being always for me here at Ohio and for taking care of me. To conclude, thank you very much to my stepfather Iván“Coach Star”, grandmother Luz Mar´ıa“Abuelongas Lupa”, and to my uncle Woody. ¡Esto es para y por ustedes, mi familia; los amo mucho!

vi Vita

2014 - Present ...... Graduate Teaching Associate, The Ohio State University

2013 ...... B.S. in Biomathematics, AGMUS Institute of Mathematics, Universidad Metropolitana

1991 ...... Born in Santurce, San Juan, Puerto Rico, United States

Fields of Study

Major Field: Mathematical Sciences Specialization: Mathematical Biosciences

vii Table of Contents

Page

Abstract ...... ii Dedication ...... iv Acknowledgments ...... v Vita ...... vii List of Figures ...... x List of Tables ...... xii

1 Introduction ...... 1 1.1 Background ...... 1 1.2 Symptoms, Transmission, Control and Prevention ...... 3 1.3 2014 West Africa EVD Outbreak ...... 5 1.4 HCWs and Model Implementation...... 7

2 Stage Progression Model ...... 10 2.1 SIR Model with HCWs ...... 10 2.2 SIR Model with HCWs Equations ...... 11 2.3 General Assumptions from the Model ...... 16

viii 3 Model Analysis ...... 17 3.1 Disease Free Equilibrium (DFE) ...... 17

3.2 Basic Reproductive Number R0 ...... 18 3.4 Endemic Equilibrium ...... 23 3.4 Initial Outbreak...... 30 3.5 Trade-Oﬀ Study...... 32

G 3.5.1 Analysis for R0 ...... 35 H 3.5.2 Analysis for R0 ...... 36 G H 3.5.2 Analysis for R0 + R0 ...... 37

4 Model Parametrization ...... 47 4.1 2014 West Africa Ebola Outbreak Data...... 47 4.2 Parameter Estimation from Literature ...... 48 4.3 Estimating r from Data ...... 53

4.4 Estimating R0 from the Estimated Initial Growth Rate ...... 56

5 Conclusion ...... 59

Bibliography ...... 62

ix List of Figures

Figure Page

1 Ebola Virion Shape and its Components ...... 2

2 Africa Schematic Map...... 5

3 Flow Diagram for the Stage Progression SIR Model with HCWs..... 11

4 Four Possible Outcomes Regarding the System behavior...... 25

1 5 Increasing Monotone Behavior for R0 when ν > σ (green) and ν < σ (red)...... 40

1 6 Increasing Monotone Behavior for R0 when ν ≫ σ (green) 1 and Non-Monotone Behavior for R0 when ν ≪ σ (red)...... 42

2 7 Increasing Monotone Behavior for R0 when ν = σ (green) and 2 Non-Monotone Behavior for R0 when ν > σ (red)...... 44

1 2 8 Relative Contribution of R0 and R0 Case 1...... 45

x 1 2 9 Relative Contribution of R0 and R0 Case 2...... 45

1 2 10 Relative Contribution of R0 and R0 Case 3...... 46

11 Exponential Distribution Behavior of g(t) ...... 51

12 2014 West Africa Ebola Outbreak Incidence Cases by Week ...... 54

13 2014 West Africa Ebola Outbreak Disease Prevalence by Week...... 54

14 Fitting Cumulative Incidence Data for 45 Weeks with the Logistic Model...... 55

G H 15 Comparison Between R0 and R0 ...... 57

16 Comparison Between r and R0 ...... 57

xi List of Tables

Table Page

1 Variables of the SIR Model with HCWs ...... 14

2 Parameters of the SIR Model with HCWs...... 15

3 Summary of Set of Parameters...... 52

xii Chapter 1: Introduction

1.1 Background

Discovered barely forty years ago, Ebola Virus Disease (EVD) is one of the most notorious zoonotic diseases, which are diseases that can be passed between animals and humans. EVD is a viral hemorrhagic fever of humans and other mammals (e.g. macaques, chimpanzees, antelopes, rodents and other related species). The virus name is derived from the Ebola River that crosses the village of Yambuku in the Democratic Republic of the Congo (formerly Zaire), the place where the worst cases of hemorrhagic fever were recorded in 1976 [28]. The causative agent of EVD is an RNA virus of the order Monone- gavirales, family Filoviridae and genus Ebolavirus. Five different Ebolavirus subtypes or strains have been identified, namely Zaire ebolavirus (EBOV), Tai Forest ebolavirus (TAFV), Sudan ebolavirus (SUDV), Bundibugyo ebolavirus (BDBV) and Reston ebolavirus (RESTV) [2, 39]. Among the viruses of the Ebolavirus genus, EBOV is considered as the deadliest and the RESTV as non-pathogenic to humans [2, 40]. To describe its structure, the name of the family Filoviridae comes from the Latin word filum or thread, because the virion shape resembles a twisted thread when viewed under an electron microscope.

1 Figure 1: Ebola virion shape and its components [5].

Contact with corpses (dead bodies) can be signiﬁcant sources of infection during an epidemic, but not considered as the natural reservoir of EVD. However, it was found that the natural host are probably asymptomatic infected fruit bats of the Pteropodidae family [2, 6, 7, 9, 17, 28, 42]. Although non-human primates have been a source of infection for humans they are not thought to be the reservoir, but rather an accidental host like human beings. Currently, the National Institute of Allergy and Infectious Diseases (NI- AID) is supporting the development of multiple EVD vaccine candidates [26]. Speciﬁcally, the NIAID Vaccine Research Center (VRC) in collaboration with the U.S. Army Medical Research Institute of Infectious Diseases and Okairos, a Swiss-Italian biotechnology company acquired by GlaxoSmithKline (GSK) developed an Ebola vaccine candidate known as NIAID/GSK or cAd3-EBOZ. This candidate vaccine is based on a type of chimpanzee cold virus which is used as a vector to deliver Ebola genetic material. This gene expresses an Ebola virus protein designed to prompt the human body to make an im-

2 mune response. Findings presented in February 2016 indicate the vaccine was well-tolerated and induced an immune response at Phase 2 which 87% of the volunteers had measurable Ebola antibodies (from 500 volunteers). The next step will be to test cAd3-EBOZ at Phase 3 with more volunteers (among 28,000) to observe if this vaccine candidate will be enough to confer immunity. In addition, other vaccine candidates have been tested and developed such as rVSV-ZEBO, a vaccine trial launched in April 2015 by The Centers for Disease and Control and Prevention (CDC) and the U.S. Department of Health and Human Services (HHS) Office of the Assistant Secretary of Preparedness and Response, in partnership with the Sierra Leone College of Medicine and Al- lied Health Sciences and the Sierra Leone Ministry of Health and Sanitation. Other methods have been implemented for example in developing vaccines based on an existing rabies vaccine that would protect against Ebola, among other 30 different filovirus vaccine formulations including licensed drugs for anti-filovirus activity [26, 35].

1.2 Symptoms, Transmission, Control and Prevention

Early symptoms of EVD infection may be mild, but as the virus replicates, symptoms become more severe. According to the World Health Organization (WHO) and CDC, symptoms of EVD begin anywhere between 2 and 21 days (incubation period) after exposure to the Ebola virus involving low fever and fatigue. This period may vary, depending on the route of exposure and the amount of virus a patient has come in contact with [6, 7]. However, this is a very dangerous stage, because it could easily be confused with other illnesses

3 that cause fever such as flu or malaria. As the Ebola virus replicates in the person’s body, the effects become much more severe having advanced signs that include high fever, muscle aches, headache, abdominal pain, vomiting, diarrhea, bleeding from the eyes, ears, gums, nose, rectum, and internal organs, organ failure, low blood pressure, delirium, coma, among other fatal symptoms. In the worse case scenario of illness, patients can lose 6 to 10 quarts (more than 5700 milliliters) of fluid per day through diarrhea and vomiting [9]. The question is how this deadly disease is being introduced in the population? The answer has been studied for the past decades and the explanation is that this virus has different pathways of transmission. The first way for EVD to spread is through the animal-to-person transmission, for example, by having contact with non-human primates as mentioned earlier. However, non-human primates also get sick and die from EVD. This completely rejects the hypothesis that primates could be the natural reservoir of EVD. Another way to get infected is by having contact with several species of fruit bats. According to researchers, this is the most probable natural reservoir of EVD, because these species do not get infected by the disease. On the other hand, EVD is spread in the community through person-to-person transmission, with infection resulting from direct contact with the blood, secretions, organs or other bodily fluids (e.g. saliva, tears, vomit, feces) of infected people, and by having indirect contact with environments contaminated with such fluids [14, 28, 42]. Surprisingly, burial ceremonies in which mourners have direct contact with the body of the deceased person can also play a role in the transmission of EVD [2]. Fortunately, this plague is not airborne and people who survived

4 from it are no longer contagious. Men and women have to abstain from sex, because virus may be still present in breast milk, semen, and in the chambers of the eyes for several months. Finally, there is no cure for this disease, but it can be controlled and prevented. Some techniques can be applied including basic and respiratory hygiene, correct use of personal protective treatment and safe injection prac- tices. Also, the samples taken from suspected cases for laboratory diagnosis should be handled by trained staﬀ and processed in suitably equipped labo- ratories (e.g. biosafety level 4 or BSL-4). The community should help during this process, for example, by informing the population about the nature of the disease and about outbreak containment measures, including burial activities [6, 7].

1.3 2014 West Africa EVD Outbreak

Figure 2: Africa schematic map. Purple color represents West Africa region.

5 The epidemic of EVD observed in 2014 in West Africa is the largest outbreak which occurred since the first case of this disease in 1976 [28]. The number of cases and deaths has already exceeded the number of recorded cases in all previous epidemics together with 20,406 total number of suspected, probable and confirmed cases, and 8,036 reported deaths by December 31st, 2014 followed by 22,996 total number of cases of suspected, probable and confirmed cases and 9,477 reported deaths by February 9th, 2015. This epidemic was not identified until the end of March 2014 and WHO declared Ebola epidemic in West Africa a Public Health Emergency of International concern on Au- gust 2014 [2, 7]. The most affected countries were Guinea, Liberia and Sierra Leone. Unfortunately, there is much more information to collect in order to fully understand this lethal outbreak. The first step during the process of understanding this “chess game” is to comprehend the infectious diseases logistics, in this case Ebola. According to a successfully previous work [4], a disease can be spread through the movement of infected individuals. In order to transmit this disease, it requires contact between individuals that are infected and those who are not. This could imply that as symptoms severity increase, there is a tendency to increase the probability of disease transmission given contact with a susceptible individual. Thus, those individuals with severe infections may be unable to move as much and therefore contact rates will decrease. This involves an interesting trade-off between contact and transmission given contacts and it can be applied to EVD. According to the 2014 West Africa outbreak incidence reports, some secondary cases of EVD involved health care workers (HCWs) [12, 23, 37]. This is a huge concern that needs to be studied in order to understand the

6 epidemiological features such as intervention efficacy, disease control and preventive measurements. The motivation of this research starts with the question on how this trade-off can be implemented to analyze in depth EVD? Could this problem be minimized by assessing the role of HCWs in EVD transmission? Under what conditions HCWs are important for the disease spread in the population? These questions can be analyzed and answered using a mathematical model showing the interaction between the general population and HCWs population and simultaneously involving the trade-off between contact and transmission given contact as symptom severity goes up over time.

1.4 HCWs and Model Implementation

HCWs play a critical and high risk role in responding to the Ebola epidemic and in working to meet the health needs of their communities during the outbreak. For this thesis project, the term “health care worker” includes clinical staﬀ, drivers, cleaners, and volunteers. Preliminary analysis showed that, depending on their occupation in the health service, HCWs are between 21 and 32 times more likely to be infected with EVD than people in the general population [37]. The 2014 West Africa EVD outbreak includes a high number of doctors, nurses, and other HCWs who have been infected and this has had a devastating impact on the already fragile health workforces in Guinea, Liberia and Sierra Leone. If there is no HCWs, who will face this epidemic? Diﬀerent studies and analysis regarding the impact of EVD on HCWs remains unclear. Because common sense is not enough to handle this particular problem, we turn to mathematics.

7 A mathematical framework will be developed in order to explore the com- plex dynamics of EVD. The situation is formulated by a compartmental model relating the general population (general community or non-HCWs population) and HCWs population. In a compartmental model, the independent variable is the time t and the transfer rates between compartments are expressed as derivatives with respect to time of the sizes of the compartments, and as a result the model will be formulated as ordinary diﬀerential equations (ODEs). These “compartments” can stratify both populations in diﬀerent health states

labeled S, I1, I2, R, and SH . Let S(t) denote the number of individuals in the general community (not including HCWs individuals) who are susceptible to the disease, in other words, individuals who are not infected by the disease at time t. By nature, HCWs individuals belong to the general community as well (e.g. HCWs individuals could visit grocery stores, family members special activities, among other places), but for simplicity, they will not be part of the susceptible individuals in the general community compartment for this model. Next, I1 and I2 denote the number of infected individuals that are able to spread the disease by contact with susceptible individuals at the asymptomatic stage and severe stage, respectively. Then, R(t) represents the number of individuals who have been infected and then removed from the population. Finally, SH (t) represents the number of individuals in the HCWs population who are susceptible to the disease. To conclude, the model will have a focus on the trade-oﬀ between contact and transmission by using the basic reproductive number R0 in terms of symptom severity as the target reference. This mathematical tool provides the average number of secondary infections caused by a typical infected individ-

8 ual in a totally susceptible population. Unfortunately, the HCWs population do not experience the same trade-oﬀ as the general community, because as symptom severity goes up, contacts with HCWs will increase. Our main goal is to study this interesting twist and of course, the contribution of HCWs in spreading the disease and assess under what conditions HCWs become important drivers of transmission.

9 Chapter 2: Stage Progression Model

2.1 SIR Model with HCWs

EVD infection can be summarized into two stages: asymptomatic or mild stage (stage 1) and severe stage (stage 2). The infection intensity can be assessed by the amount of pathogen an individual is shedding, with higher shedding rates corresponding to more severe infections [4, 21]. At stage 1, asymptomatic infected individuals have the ability to move in the population. This will increase the contacts between individuals, but the probability of transmission given contact is low due to low pathogen shedding rate. At stage 2, the disease is getting worse, and as a consequence contact rates in the general population will decrease, because infected individuals will not have the ability to move. On the other hand, the probability of transmission given contact will be higher in stage 2. Due to the severe symptoms, stage 2 individuals will seek HCWs and now the latter will be at risk to be infected, so the dynamics of EVD will be more difficult to understand. A flow diagram of the model is given in Figure 3. What is novel here is the consideration of how symptom severity affects the trade-off between contacts and transmission given contacts by adding the HCWs population in the scenario.

10 Figure 3: Flow diagram for the stage progression SIR model with HCWs. S represents the susceptible individuals from the general community (non-health care workers), I1 represents the infected individuals at asymptomatic or mild stage (stage 1), I2 represents the infected individuals at severe stage (stage 2), and SH represents the susceptible individuals from the HCWs population.

2.2 SIR Model with HCWs Equations

The following ODE system is studied and it follows a frequency dependent transmission of the EVD. This means the per capita contact rate between susceptible and infected individuals does not depend on the population density, thus the transmission rates do not change with density. One of the conse- quences of making this assumption is that the per capita transmission rates β SI β SI 1 1 and 2 2 decline with increasing population size N. In addition, it N N will be clear to mention when the population size is not constant, frequency dependent transmission is not the same as density dependent transmission (transmission rates changes with density). However, if we consider a closed

11 population, then N is constant and frequency dependent and density dependent transmission will be the same [30].

 ˙ S  S = Λ − (β1I1 + β2I2) − µS,  N     S f1I1pH1 + f2I2pH2  I˙ = (β I + β I ) − γ I − α (S )I − µI + S φ ,  1 N 1 1 2 2 1 1 1 H 1 1 H ξ + f I + f I  1 1 2 2

  I˙ = γ I − γ I − α (S )I − µI ,  2 1 1 2 2 2 H 2 2      ˙ f1I1pH1 + f2I2pH2  SH = ΛH − SH φ − µSH ξ + f1I1 + f2I2 (1)

This model presents two new features that play a key role in understanding Ebola infectious disease epidemiology. The ﬁrst one is a Michaelis-Menten type ˙ function for the HCWs incidence (last term in I1 equation and second term in ˙ SH equation). Speciﬁcally, this term summarizes the event where individuals in general population seek HCWs when they feel sick with the assumption that an individual will visit the hospital with a certain fraction f at any stage of infection. Then, this implies that there is a probability pH that HCWs will be infected by having contacts with an individual coming from I1 or I2. In summary, this term represents the overall contact rate of Ebola patients at any stage and this type of function is chosen, because there is a limited capacity of beds where HCWs treat patients.

The second feature is the disease-induced mortality term α1 and α2 at

12 stage 1 and stage 2, respectively. This term represents the rate at which people who is sick from EVD die as a consequence of it. In order to link the interaction with HCWs, this term will be expressed as a function of susceptible individuals in the health care population. This will help to understand the contribution of HCWs regarding secondary cases and deaths of EVD. Tables 1 and 2 summarize the mathematical model variables and parameters.

13 Table 1. Variables for the SIR Model with HCWs (1)

S general (non-health care workers) susceptible population

14 I1 infectious population at stage 1

I2 infectious population at stage 2

SH health care workers susceptible population N total population Table 2. Parameters for the SIR Model with HCWs (1)

α1 disease-induced mortality rate at stage 1

α2 disease-induced mortality rate at stage 2

β1 transmission rate from S to I1

β2 transmission rate from S to I2

γ1 recovery rate at stage 1

γ2 recovery rate at stage 2 µ birth/death rate

c1 contact rate for infected individuals at stage 1

15 c2 contact rate for infected individuals at stage 2 φ maximum contact rate of HCWs with patients

p1 probability of transmission given contacts for general population at stage 1

p2 probability of transmission given contacts for general population at stage 2

pH1 probability of transmission given contacts for HCWs at stage 1

pH2 probability of transmission given contacts for HCWs at stage 2

f1 fraction of individuals in I1 that seek HCW

f2 fraction of individuals in I2 that seek HCW ξ level of infection in the population at which the half maximum capacity is reached Λ recruitment rate for susceptible individuals in general population

ΛH recruitment rate for susceptible individuals in HCWs population 2.3 General Assumptions from the Model

This stage progression model is based on several assumptions. The ﬁrst one is that the epidemic process is deterministic, this means the behavior of both populations are determined by its history and by the rules which describe the model. Also, both populations have a constant rate of susceptible recruitment ˙ ˙ ˙ ˙ Λ and ΛH , respectively. Because the equations for S, I1, I2, and SH do not depend of the variable R, the dynamics of R˙ can be ignored. On the other hand, the transmission parameter β1 and β2 are deﬁned as the product between contact rate for infected individuals at stage 1 and 2, respectively, times the probability of transmission given contact of an individual from the general population at stage 1 and 2, respectively. Generally speaking, β1 = c1p1 and

β2 = c2p2. Finally, recall that the contacts at stage 1 are greater than the contacts at stage 2 and the probability of transmission at stage 1 is less than the probability of transmission at stage 2. As a result we have: c1 ≥ c2, p1 ≤ p2, pH1 ≤ pH2 and f1 ≤ f2, where p1, p2, pH1, pH2, f1, f2 [0, 1].

16 Chapter 3: Model Analysis

In this chapter, some fundamental quantities for the resulting SIR model with the introduction of HCWs population, such as the disease free equilibrium, basic reproductive number, endemic equilibrium, initial growth rate, and trade-oﬀ analysis are calculated.

3.1 Disease Free Equilibrium (DFE)

The disease free equilibrium (DFE) is denoted as E0 and it is deﬁned as the point at which the disease is not present in the population [2, 4, 31]. This is represented in the model as I1 = 0 and I2 = 0. The system of equation simpliﬁes to

 ˙  S = Λ − µS,   ˙  I1 = 0, (2) ˙  I2 = 0,   ˙  SH = ΛH − µSH

∗ ∗ ∗ ∗ ∗ ∗ ˙ with E0 = (S ,I1 ,I2 ,SH ) = (S , 0, 0,SH ). Solving for S in the S equation ˙ and solving for SH in the SH , the DFE of this system lies at the point

Λ Λ E = , 0, 0, H . 0 µ µ

17 3.2 Basic Reproductive Number R0

The basic reproductive number R0 is one of the most important quantities in mathematical epidemiology. It is defined as the average number of secondary infections from a single infected individual in a totally susceptible population. It will determine if the disease (or diseases) will invade the population (pathogen explosion) or if it will die out (pathogen extinction). If R0 < 1, the disease will not be a concern in the population. Now, if R0 > 1, then the disease could be a potential harm to the population if safety measures are not taken into account. Please note that this is not always guarantee since it could change for different diseases and other deterministic models. Mathematically, the basic reproductive number is defined as the spectral radius of the next

−1 generation matrix in the epidemic model or generally speaking R0 = ρ(FV ) [2, 21, 31]. The meaning of the word “generation” in epidemic models are the waves of secondary infections that ﬂow from previous infection. Thus, the ﬁrst generation of an epidemic is all the secondary infections that results from an infectious contact with the index case (primary case), who is of generation zero. If Ri denotes the reproductive number of the ith generation, then R0 is the number of infections generated by generation zero.

18 Now, in order to ﬁnd the basic reproductive number for this model consider ˙ ˙ I1 and I2, because they are the infected compartments. Also, assume that our compartmental ODE model is on the form:

x˙ = b(x) where x is a vector of state variables in Rn and b : Rn → Rm the right hand side of system (1), where n = 4 and m = 2. According to [30] let’s split the right hand side of b(x) into two parts: F and V. Let F be the rate at which new infections occur and let V be the transfers of existing infections among the diﬀerent infected compartments. Finally, the compartmental ODE model form turns to:

x˙ = F(x) − V(x).

where

    S SH (f1I1pH1φ + f2I2pH2φ) I (β1I1 + β2I2) + 1  N ξ + f I + f I  F =   =  1 1 2 2  , I2 0 and

    I1 γ1I1 + α1(SH )I1 + µI1 V =   =   . I2 −γ1I1 + γ2I2 + α2(SH )I2 + µI2

19 Now, let F and V denote the linearization of F and V at the DFE by calcu- lating the Jacobian. After linearizing the system, it turns to the following:

 ∗ ∗ ∗ ∗  S SH φf1ph1 S SH φf2ph2 β1 + β2 +  N ξ N ξ  F = JF =   , 0 0

  γ1 + α1(SH ) + µ 0 V = JV =   . −γ1 γ2 + α2(SH ) + µ

φf p φf p Let β = 1 H1 and β = 2 H2 , giving H1 ξ H2 ξ

∗ ∗  S ∗ S ∗  β1 + SH βH1 β2 + SH βH2 F = JF =  N N  , 0 0

  γ1 + α1(SH ) + µ 0 V = JV =   . −γ1 γ2 + α2(SH ) + µ

20 The inverse matrix of V , is given by

 1  0  γ1 + α1(SH ) + µ  −1   V =   .    γ1 1 

(γ1 + α1(SH ) + µ)(γ2 + α2(SH ) + µ) γ2 + α2(SH ) + µ

Let a1 = γ1 + α1(SH ) + µ and a2 = γ2 + α2(SH ) + µ, now:

 1  0  a1  −1   V =   .    γ1 1 

a1a2 a2

The basic reproductive number R0 will be the largest eigenvalue of the next generation matrix

  1   S∗ S∗  0  β + S∗ β β + S∗ β  a1  −1  1 H H1 2 H H2   R0 = ρ(FV ) = ρ  N N        0 0  γ1 1 

a1a2 a2

21  ∗  ∗  ∗  S ∗ S ∗ S ∗ β1 + SH βH1 β2 + SH βH2 β2 + SH βH2  N  N  N   + γ1     a1 a1a2 a2  −1   R0 = ρ(FV ) = ρ   .       0 0

The largest eigenvalue of this next generation matrix is the ﬁrst entry in the diagonal, giving the following expression for R0:

∗ S β1 γ1β2 ∗ βH1 γ1βH2 R0 = + + SH + . (3) N a1 a1a2 a1 a1a2

This expression for the basic reproductive number represents the contribution

G of the general population R0 and the contribution of the health care workers H population R0

∗ G S β1 γ1β2 H ∗ βH1 γ1βH2 R0 = + , R0 = SH + . (4) N a1 a1a2 a1 a1a2

A more general version of the basic reproductive number has the following

G H form R0 = R0 + R0 .

22 3.3 Endemic Equilibrium

The endemic equilibrium (EE) is denoted as Ee and it is deﬁned as the equilibrium point where the disease is present at a constant level without any re-introduction. This point is biologically feasible only if R0 > 1, which agrees with earlier deﬁnitions about when an epidemic is possible. Before deriving the EE it is necessary to understand DFE is locally asymptotically stable when

R0 < 1 and unstable when R0 > 1 [34]. In the opposite sense, EE is locally asymptotically stable when R0 > 1 and unstable when R0 < 1. This is not guarantee in general, this could change for other diseases and deterministic models. Traditional methods suggests that S∗ at EE is the reciprocal of the basic reproductive number and the procedure to show the expression for the other variables is very straight forward [16, 30, 31, 34]. Unfortunately, for this stage progression model this approach does not work, and thus another approach was taken. The system of four variables in section 2.2 can be expressed at equilibrium into one equation involving a single variable. For convenience, the system of equations will be expressed in terms of I1 and the aim will con- sist to show geometrically that the terms involving this single equation have a unique intersection at a point when R0 > 1, so this can assure the existence of an unique EE for R0 > 1. After expressing the entire system in terms of I1 ˙ at equilibrium, the I1 equation becomes

23 ˙ ΛI1 (a2β1 + β2γ1) I1 = + I1a2β1 + I1β2γ1 + Na2 µ

φI1ΛH (pH1a2f1 + pH2f2γ1) − a1I1 = 0. I1pH1a2f1φ + I1pH2f2φγ1 + I1a2f1µ + I1f2µγ1 + a2µξ

ΛI1 (a2β1 + β2γ1) Let f(I1) = and I1a2β1 + I1β2γ1 + Na2 µ

φI1ΛH (pH1a2f1 + pH2f2γ1) let g(I1) = I1pH1a2f1φ + I1pH2f2φγ1 + I1a2f1µ + I1f2µγ1 + a2µξ the equation becomes

f(I1) + g(I1) − a1I1 = 0.

What is the intersection or intersections between f(I1) + g(I1) and the line a1I1? In order to answer this question it is necessary to investigate the shape of f(I1) + g(I1). Intuitively, it looks like a quadratic function, but there is a possibility this function crosses the line a1I1 several times, because at some point it could change its concavity. To avoid this situation, we will show that in fact f(I1) + g(I1) does not change concavity. Four diﬀerent pictures are then possible, as shown in Figure 4.

24 f (I1) + g (I1) f (I1) + g (I1) a1 I1 a1 I1

4A 4B

I1 I1

f (I1) + g (I1) f (I1) + g (I1) a1 I1 a1 I1

4C 4D

I1 I1

Figure 4: Four possible outcomes regarding the system behavior.

The process of choosing which one represents the correct behavior under biological meaning is not diﬃcult at all. First, the cases at the bottom, Figure 4C and Figure 4D, can be ignored because in order to make possible those behaviors most of the parameters in the SIR model with HCWs should be negative, and by nature, parameters for disease dynamics are non-negative. Figures 4A and 4B represent the curves with positive parameters values. The question is how we can get a concise conclusion that the image selected represents the curve where the EE can be found for this system? To demonstrate this, it will be necessary to show that if f(I1) + g(I1) is an increasing function, 0 0 concave down and limI1→0 [f (I1) + g (II )] is greater than the slope m of a1I1, then it can be concluded that Figure 4B will never happen and for sure there is an EE and hence its unique. The proof will be supported by the following theorem.

25 Theorem 1. For R0 > 1, system (1) has a unique endemic equilibrium.

0 0 (i) f (I1) + g (II ) > 0

00 00 (ii) f (I1) + g (II ) < 0

0 0 (iii) limI1→0 [f (I1) + g (II )] > a1

Proof. In order to show (i), it turns that

2 0 Λ(a2β1 + β2γ1) ΛI1 (a2β1 + β2γ1) f (I1) = − 2 , I1a2β1 + I1β2γ1 + Na2µ (I1a2β1 + I1β2γ1 + Na2µ)

0 φΛH (pH1a2f1 + pH2f2γ1) g (I1) = I1pH1a2f1φ + I1pH2f2φγ1 + I1a2f1µ + a2µξ

φΛH I1 (pH1a2f1 + pH2f2γ1)(pH1a2f1φ + pH2f2φγ1 + a2f1µ + f2µγ1) − 2 . (I1pH1a2f1φ + I1pH2f2φγ1 + I1a2f1µ + a2µξ)

0 For f (I1) > 0:

2 Λ(a2β1 + β2γ1) ΛI1 (a2β1 + β2γ1) > 2 , I1a2β1 + I1β2γ1 + Na2µ (I1a2β1 + I1β2γ1 + Na2µ)

I a β + I β γ + Na µ multiplying by 1 2 1 1 2 1 2 on both sides the inequality Λ(a2β1 + β2γ1) becomes

I1 (a2β1 + β2γ1) 1 > ⇒ Na2µ > 0. I1a2β1 + I1β2γ1 + Na2µ

26 0 For g (I1) > 0:

φΛ (p a f + p f γ ) H H1 2 1 H2 2 1 > I1pH1a2f1φ + I1pH2f2φγ1 + I1a2f1µ + a2µξ

φΛH I1 (pH1a2f1 + pH2f2γ1)(pH1a2f1φ + pH2f2φγ1 + a2f1µ + f2µγ1) 2 (I1pH1a2f1φ + I1pH2f2φγ1 + I1a2f1µ + a2µξ)

I p a f φ + I p f φγ + I a f µ + a µξ multiplying by 1 H1 2 1 1 H2 2 1 1 2 1 2 on both sides φΛH (pH1a2f1 + pH2f2γ1) the inequality becomes

I1 (pH1a2f1φ + pH2f2φγ1 + a2f1µ + f2µγ1) 1 > ⇒ a2µξ > 0. I1pH1a2f1φ + I1pH2f2φγ1 + I1a2f1µ + a2µξ

0 0 Therefore, f (I1)+g (I1) > 0 when Na2µ > 0 and a2µξ > 0.

Now, to show (ii), we ﬁnd

2 3 00 −2Λ (a2β1 + β2γ1) 2ΛI1 (a2β1 + β2γ1) f (I1) = 2 + 3 , (I1a2β1 + I1β2γ1 + Na2µ) (I1a2β1 + I1β2γ1 + Na2µ)

00 −2φΛH (pH1a2f1 + pH2f2γ1)(pH1a2f1φ + pH2f2φγ1 + a2f1µ + f2µγ1) g (I1) = 2 (I1pH1a2f1φ + I1pH2f2φγ1 + I1a2f1µ + I1f2µγ1a2µξ)

2 2φΛH I1 (pH1a2f1 + pH2f2γ1)(pH1a2f1φ + pH2f2φγ1 + a2f1µ + f2µγ1) + 3 (I1pH1a2f1φ + I1pH2f2φγ1 + I1a2f1µ + I1f2µγ1a2µξ)

27 00 For f (I1) < 0:

3 2 2ΛI1 (a2β1 + β2γ1) 2Λ (a2β1 + β2γ1) 3 < 2 (I1a2β1 + I1β2γ1 + Na2µ) (I1a2β1 + I1β2γ1 + Na2µ)

2 (I1a2β1 + I1β2γ1 + Na2µ) multiplying by 2 on both sides the inequality 2Λ (a2β1 + β2γ1) becomes

I (a β + β γ ) 1 2 1 2 1 < 1 I1a2β1 + I1β2γ1 + Na2µ

0 < Na2µ ⇒ Na2µ > 0

00 For g (I2) < 0:

2 2φΛH I1 (pH1a2f1 + pH2f2γ1)(pH1a2f1φ + pH2f2φγ1 + a2f1µ + f2µγ1) 3 < (I1pH1a2f1φ + I1pH2f2φγ1 + I1a2f1µ + I1f2µγ1a2µξ)

2φΛH (pH1a2f1 + pH2f2γ1)(pH1a2f1φ + pH2f2φγ1 + a2f1µ + f2µγ1) 2 (I1pH1a2f1φ + I1pH2f2φγ1 + I1a2f1µ + I1f2µγ1a2µξ)

(I p a f φ + I p f φγ + I a f µ + I f µγ a µξ)2 multiplying by 1 H1 2 1 1 H2 2 1 1 2 1 1 2 1 2 2φΛH (pH1a2f1 + pH2f2γ1)(pH1a2f1φ + pH2f2φγ1 + a2f1µ + f2µγ1) on both sides the inequality becomes

28 I (p a f φ + p f φγ + a f µ + f µγ ) 1 H1 2 1 H2 2 1 2 1 2 1 < 1 I1pH1a2f1φ + I1pH2f2φγ1 + I1a2f1µ + I1f2µγ1a2µξ

0 < a2µξ ⇒ a2µξ > 0.

Finally, to show (iii)

0 0 lim [f (I1) + g (II )] > a1, I1→0

we divide by a1 on both sides giving

lim [f 0(I ) + g0(I )] I1→0 1 I > 1. (5) a1

0 0 limI1→0 [f (I1) + g (II )] In order to satisfy (iii), R0 needs to be equal to . a1

The above expression becomes to

lim [f 0(I ) + g0(I )] Λ(a β + β γ ) φΛ (p a f + p f γ ) I1→0 1 I = 2 1 2 1 + H H1 2 1 H2 2 1 , a1 Na1a2µ a1a2µξ

Λ Λ φf p φf p recall S = , S = H , β = 1 H1 and β = 2 H2 consequently, µ H µ H1 ξ H2 ξ

0 0 limI1→0 [f (I1) + g (II )] S β1 β2γ1 βH1 γ1βH2 = + + SH + . a1 N a1 a1a2 a1 a1a2

Note that the right hand side expression is equal to equation (3), the basic reproductive number,

29 0 0 limI1→0 [f (I1) + g (II )] = R0. a1

Replacing R0 on the left hand side of inequality (5) we can conclude that a unique EE exist when the basic reproductive number R0 > 1. In summary,

R0 > 1 guarantees that (i), (ii) and, (iii) hold and therefore Figure 4A corresponds to the system behavior at EE.

3.4 Initial Outbreak

The initial outbreak or initial growth rate of an epidemic is a measure of disease spread, and is mostly used to derive the basic reproductive number R0 [22]. Mathematically, is known as the dominant eigenvalue of the Jacobian matrix at the DFE. This approach will be very helpful later on in order to analyze and understand the relative contributions of general population and HCWs population during the 2014 West Africa Ebola outbreak. Also, this measurement will allow us to ﬁt our deterministic model to real incidence data from this largest outbreak. In this section, the Jacobian matrix from system (1) will be used in order to give an expression for the initial model growth φf p rate. Recall that a = γ + α (S ) + µ, a = γ + α (S ) + µ, β = 1 H1 , 1 1 1 H 2 2 2 H H1 ξ φf p and β = 2 H2 , the Jacobian matrix J at DFE is: H2 ξ

30  Λβ Λβ  −µ − 1 − 2 0  µN µN         Λβ Λ β Λβ Λ β   1 H H1 2 H H2   0 − a1 + + 0   µN µ µN µ    J =        0 γ1 −a2 0           ΛH βH1 ΛH βH2  0 − − −µ µ µ

The eigenvalues of J are: r1,2 = −µ √ w + w + w r = w ± 2 3 4 (6) 3,4 1 2µN

where:

w1 = −Na1µ − Na2µ + NΛH βH1 + Λβ1 2 2 2 2 2 2 2 2 2 w2 = N a1µ − 2N a1a2µ − 2N a1µΛH βH1 + N a2µ 2 2 2 2 2 w3 = 2N a2µΛH βH1 + 4N µΛH βH2γ1 + N ΛH βH1 − 2ΛNa1µβ1 2 2 w4 = 2ΛNa2µβ1 + 4ΛNµβ2γ1 + 2ΛNΛH β1βH1 + Λ β1 .

To conclude this section, r1,2 will be ignored, because negative parameter values makes no biological sense. Therefore, we determine that the possible

dominant eigenvalues could be r3,4. We will see in chapter 4 the exact dominant eigenvalue of J corresponding to the initial growth rate of an epidemic.

31 3.5 Trade-oﬀ Study

In order to analyze the eﬀectiveness of the trade-oﬀ between contact and

transmission for general and HCWs population we express R0 in terms of the probability of transmission given contact, generally speaking, R0 will be a function in terms of p2. This will help to understand the contribution of HCWs and how the spectrum of symptom severity will impact the shape of

R0, for example, by ﬁnding the suﬃcient conditions where R0 is a monotone increasing function of the probability of transmission.

Recall from last sections that:

∗ S β1 γ1β2 ∗ βH1 γ1βH2 R0 = + +SH + . N a1 a1a2 a1 a1a2 | {z } | {z } G H R0 R0

In order to relate the contact rate and probability of transmission given

contact, the basic reproductive number can be expressed in terms of p2 by G making some assumptions. Doing this step by step, for R0 be in terms of p2

it will be necessary to use previous deﬁnitions for β1 = c1p1 and β2 = c2p2. Next, contact rate c will be assumed to be a monotone decreasing function with respect to symptom severity in the community population. Because the probability of transmission at stage 1 is diﬀerent from the probability of trans-

mission at stage 2, let p1 = σp2, where σ [0, 1] is the relative transmissibility

32 of the general community infections. This parameter will ﬁx the contact rate at stage 1 to be greater than the contact rate at stage 2 and at the same time will ﬁx the probability of transmission at stage 1 to be less than the prob-

G ability of transmission at stage 2. Now, the transmission terms for R0 will

be β1 = c(p1)p1 and β2 = c(p2)p2 and by using the ﬁxed parameter σ they

become β1 = c(σp2)σp2 and β2 = c(p2)p2. On the other hand, we use the same H approach to express R0 in terms of p2. This time the transmission parameters φf for HCWs population will be β = h p and β = h p , where h = 1 , H1 1 H1 H2 2 H2 1 ξ φf h = 2 , respectively and contact rate h is assumed to be a monotone in- 2 ξ creasing function with respect of symptom severity in the HCW population. In order to ﬁx the probability of HCWs transmission at stage 2 to be greater than the probability of HCWs transmission at stage 1, let pH1 = σνp2, where ν [0, 1], is the relative transmissibility of HCW infections. Now, the transmis-

H sion terms for R0 will be βH1 = h(σνp2)σνp2 and βH2 = h(νp2)νp2. Finally,

the entire R0 can be expressed in terms of p2 in the following form

∗ S c(σp2)σp2 c(p2)p2γ1 ∗ h(σνp2)σνp2 γ1h(νp2)νp2 R0 = + + SH + . N a1 a1a2 a1 a1a2 | {z } | {z } G H R0 R0

33 Because we want to study its monotonicity, we need

dR S∗ c0(σp )σ2p S∗ c(σp )σ S∗ c0(p )p γ S∗ c(p )γ 0 = 2 2 + 2 + 2 2 1 + 2 1 dp2 N a1 N a1 N a1a2 N a1a2 | {z } G dR0 dp2

S∗ h0(σνp )σ2ν2p S∗ h(σνp )σν S∗ h0(νp )ν2p γ S∗ h(νp )νγ + H 2 2 + H 2 + H 2 2 1 + H 2 1 a1 a1 a1a2 a1a2 | {z } H dR0 dp2 (7)

In order to fully understand this approach, it will be convenient to study the dRG dRH contributions for 0 and 0 from the general community and HCWs popu- dp2 dp2 lation separately. This will help to clarify the dynamics when both populations are interacting.

34 G 3.5.1 Analysis for R0

G The general population basic reproductive number is denoted as R0 . The G following analysis for R0 was done in [4].

A. For small p2 (p2 ≈ 0)

G When p2 is small, the derivative of R0 becomes:

dRG S∗ c(σp )σ c(p )γ 0 = 2 + 2 1 dp2 N a1 a1a2

G Thus, for small p2, R0 is always monotone increasing as symptoms severity goes up.

B. For bigger values of p2

G dR0 When p2 is not small, the entire expression of is used. In [4], the dp2 −τσp2 speciﬁc contact rate functions c are exponential functions, c(σp2) = ke

−τp2 and c(p2) = ke , where τ denotes the rate at which contact scales with transmissibility and k is a positive and real constant. According to the results

G 2 1 1 in [4] suﬃcient conditions for R0 to be monotone are σ < and τ < . τp2 p2

35 H 3.5.2 Analysis for R0

H The basic reproductive number for HCWs population is denoted as R0 .

A. For small p2 (p2 ≈ 0) and large p2

G The derivative of R0 in terms of p2 is expressed as:

dRH S∗ h0(σνp )σ2ν2p S∗ h(σνp )σν 0 = H 2 2 + H 2 dp2 a1 a1

S∗ h0(νp )ν2p γ S∗ h(νp )νγ + H 2 2 1 + H 2 1 a1a2 a1a2

The speciﬁc contact rate functions h are Michaelis-Menten type functions, σνp2 νp2 h(σνp2) = and h(νp2) = , where δ is a positive and real δ + σνp2 δ + νp2 H 0 dR0 constant. Please note that h ≥ 0 and h ≥ 0, ∀ p2 ≥ 0 making > 0 as dp2 symptoms severity goes up.

36 G H 3.5.3 Analysis for R0 + R0 (entire R0)

A. For small p2 (p2 ≈ 0)

When p2 is small, the derivative becomes:

dR S∗ c(σp )σ S∗ c(p )γ S∗ h(σνp )σν S∗ h(νp )νγ 0 = 2 + 2 1 + H 2 + H 2 1 dp2 N a1 N a1a2 a1 a1a2

For small p2, R0 is always monotonically increasing as symptoms severity goes up.

B. For bigger values of p2

In order to complete this analysis, it will be convenient to divide equation dR1 dR2 (7) into two diﬀerent groups where 0 is represented in blue color and 0 dp2 dp2 is represented in red color

dR S∗ c0(σp )σ2p S∗ c(σp )σ S∗ c0(p )p γ S∗ c(p )γ 0 = 2 2 + 2 + 2 2 1 + 2 1 dp2 N a1 N a1 N a1a2 N a1a2 | {z } G dR0 dp2 S∗ h0(σνp )σ2ν2p S∗ h(σνp )σν S∗ h0(νp )ν2p γ S∗ h(νp )νγ + H 2 2 + H 2 + H 2 2 1 + H 2 1 . a1 a1 a1a2 a1a2 | {z } H dR0 dp2

37 This can be summarized as

dR1 S∗ c0(σp )σ2p S∗ c(σp )σ S∗ h0(νp )ν2p γ S∗ h(νp )νγ 0 = 2 2 + 2 + H 2 2 1 + H 2 1 (8) dp2 N a1 N a1 a1a2 a1a2

dR2 S∗ c0(p )p γ S∗ c(p )γ S∗ h0(σνp )σ2ν2p S∗ h(σνp )σν 0 = 2 2 1 + 2 1 + H 2 2 + H 2 . dp2 N a1a2 N a1a2 a1 a1 (9)

1 1 dR0 2 If R0 is monotone increasing > 0 and if R0 is monotone increasing dp2 2 dR0 1 2 > 0 , then this will guarantee the entire R0, R0 + R0, is a monotone dp2 increasing function. The main focus will be on the comparison between the rate of change of general community contacts (negative function) and the rate of change of HCWs population contacts (positive function) as disease symptoms severity increase. In order to make this possible the equations (8) and (9) become:

dR1 S∗ c0(σp )σ2p S∗ h0(νp )ν2p γ 0 = 2 2 + H 2 2 1 dp2 N a1 a1a2

dR2 S∗ c0(p )p γ S∗ h0(σνp )σ2ν2p γ 0 = 2 2 1 + H 2 2 1 . dp2 N a1a2 a1

dR1 dR2 The other terms of 0 and 0 were ignored because first of all, they are dp2 dp2 positive terms and also because we want to find sufficient conditions for when

38 dR 0 > 0. By analyzing the rate of change at which infected individuals have dp2 contacts with general community and HCWs population individuals we can observe how will be the shape of R0 and therefore understand how much the HCWs population drives to transmission and thus to the spread of Ebola disease.

dR1 i) Suﬃcient Condition for 0 > 0 dp2

S∗ c0(σp )σ2p S∗ h0(νp )ν2p γ 2 2 + H 2 2 1 > 0. N a1 a1a2

Noting that c(σp2) < 0 we have:

S∗ h0(νp )ν2p γ S∗ |c0(σp )| σ2p H 2 2 1 > 2 2 a1a2 N a1

S∗ 0 2 γ1 |c (σp2)| σ N > 0 2 ∗ a2 h (νp2) ν SH

S∗ 0 2 γ1 |c (σp2)| σ N > 0 2 ∗ γ2 + α2 + µ h (νp2) ν SH

S∗ 0 2 infectious period at stage 2 |c (σp2)| σ N > 0 2 ∗ . (10) infectious period at stage 1 h (νp2) ν SH

39 a) LHS is ﬁxed with a large number

Let us consider when inequality (10) will be satisﬁed. If the infectious period at stage 1 lasts for very short time relative to infectious period at stage

1 2, then the LHS of the inequality is big. The suﬃcient condition for R0 to be a monotone increasing function of the probability of transmission should be that the rate of change at which contacts with HCWs increases needs to be suﬃciently faster than the rate of change at which contacts with general

G community decreases. As a consequence, the shape of R0 will increase (as shown in Figure 5) as symptoms severity increases.

1.4

1.2

0.8 R01 0.6

0.4

0.2

0 0 0.2 0.4 0.6 0.8 1 p2

1 Figure 5: Increasing monotone behavior for R0 when ν > σ (green) and ν < σ

(red). Parameters values for green were ﬁxed as γ1 = 0.9, γ2 = 0.2, ν = 0.35,

σ = 0.02, α1 = α2 = µ = 0. Parameters values for red were ﬁxed as γ1 = 0.9,

γ2 = 0.2, ν = 0.02, σ = 0.35, α1 = α2 = µ = 0.

40 b) LHS is ﬁxed with a small number

Considering again when inequality (10) will be satisﬁed, if the infectious period at stage 1 lasts for very long time relative to the infectious period at stage 2, then the LHS of the inequality is small. Now, this will make it more diﬃcult to satisfy this criterion. Looking at the rate of change between the 0 |c (σp2)| contact rate of general population and contact of HCWs population, 0 , h (νp2) in order to satisfy inequality (10), the rate at which HCWs contacts are increasing with symptoms needs to be extremely bigger than the magnitude of the rate at which general community contacts are decreasing. Also, parameters

1 σ and ν may impact the shape of R0. If ν is suﬃciently large in comparison 1 with σ we may have a monotone increasing function for R0, because ν can tell us how infectious the contacts with HCWs could be. On the other hand, if σ is suﬃciently large in comparison with ν we may have a non-monotone

1 shape of R0, because σ can tell us how infectious the contacts with general

community individuals could be. If R0 is non-monotone, this means that contacts in HCWs population are not suﬃciently infectious, because preventive and control measurements are being strictly followed. In Figure 6 we will see

1 both cases when R0 is an increasing monotone function and non-monotone function with respect to symptom severity.

41 1.2

0.8

0.6 R01

0.4

0.2

0 0 0.2 0.4 0.6 0.8 1 p2

1 Figure 6: Increasing monotone behavior for R0 when ν ≫ σ (green) and non- 1 monotone behavior for R0 when ν ≪ σ (red). Parameters values for green were ﬁxed as γ1 = 0.2, γ2 = 0.9, ν = 0.75, σ = 0.02, α1 = α2 = µ = 0.

Parameters values for red were ﬁxed as γ1 = 0.2, γ2 = 0.9, ν = 0.02, σ = 0.75,

α1 = α2 = µ = 0.

42 dR2 ii) Suﬃcient Condition for 0 > 0 dp2

S∗ c0(p )p γ S∗ h0(σνp )σ2ν2p γ 2 2 1 + H 2 2 1 > 0. N a1a2 a1

Noting that c(p2) < 0 we have:

S∗ h0(σνp )σ2ν2p γ S∗ |c0(p )| p γ H 2 2 1 > 2 2 1 a1 N a1a2

0 ∗ h (νσp2) 2 2 SH γ1 0 σ ν ∗ > |c (p2)| S a2 N 0 ∗ γ1 h (νσp2) 2 2 SH < 0 σ ν ∗ a2 |c (p2)| S N 0 ∗ γ1 h (νσp2) 2 2 SH < 0 σ ν ∗ γ2 + α2 + µ |c (p2)| S N S∗ 0 γ2 + α2 + µ |c (p2)| 1 N > 0 2 2 ∗ γ1 h (νσp2) σ ν SH

S∗ 0 infectious period at stage 1 |c (p2)| 1 N > 0 2 2 ∗ . (11) infectious period at stage 2 h (νσp2) σ ν SH

2 In order to R0 be a monotone increasing function of the probability of 0 |c (p2)| transmission we need in 0 that the rate at which HCWs contacts are h (σνp2)

43 increasing with symptoms needs to be much bigger than the magnitude of the rate at which general community contacts are decreasing. Surprisingly, 1 there is a new term in this inequality. According to the criterion for σ2ν2 1 2 2 R0, the expression ν is at the bottom of the fraction and σ is at the top. But for this speciﬁc case, σ2 is at the bottom of the fraction together with ν2. This could have diﬀerent implications, one of them is that σ must be in an

2 intermediate range to satisfy R0 in order to be a monotone increasing function of the probability of transmission. This is shown in Figure 7.

1.2

0.8

0.6 R02

0.4

0.2

0 0 0.2 0.4 0.6 0.8 1 p2

2 Figure 7: Increasing monotone behavior for R0 when ν = σ (green) and non- 2 monotone behavior for R0 when ν > σ (red), also the same behavior occurs if

ν < σ. Parameters values for green were ﬁxed as γ1 = 0.9, γ2 = 0.2, ν = 0.35,

σ = 0.35, α1 = α2 = µ = 0. Parameters values for red were ﬁxed as γ1 = 0.9,

γ2 = 0.2, ν = 0.75, σ = 0.35, α1 = α2 = µ = 0.

1 2 Now, observing the contribution of both R0 and R0 we are able to see the entire behavior of R0 for diﬀerent cases in Figures 8, 9 and 10.

44 2 R01 R02 1

0 0 0.2 0.4 0.6 0.8 1 p2 3 R0 2

0 0 0.2 0.4 0.6 0.8 1 p2

1 2 Figure 8: Relative contribution of R0 and R0 case 1. Both contributions make

the entire R0 non-monotone function. Parameter values were ﬁxed as γ1 = 0.2,

γ2 = 0.9, ν = 0.02, σ = 0.75, α1 = α2 = µ = 0..

1 R01 R02 0.5

0 0 0.2 0.4 0.6 0.8 1 p2 1 R0

0.5

0 0 0.2 0.4 0.6 0.8 1 p2

1 2 Figure 9: Relative contribution of R0 and R0 case 2. Both contributions make

R0 a monotone increasing function. Parameter values were ﬁxed as γ1 = 0.9,

γ2 = 0.2, ν = 0.75, σ = 0.35, α1 = α2 = µ = 0..

45 1.5 R01 1 R02

0.5

0 0 0.2 0.4 0.6 0.8 1 p2 2 R0

0 0 0.2 0.4 0.6 0.8 1 p2

1 2 1 Figure 10: Relative contribution of R0 and R0 case 3. R0 is a monotone 2 increasing function and R0 is not. The entire R0 is a monotone increasing function. Parameter values were ﬁxed as γ1 = 0.9, γ2 = 0.2, ν = 0.35,

σ = 0.02, α1 = α2 = µ = 0..

In summary, the longer that one stage lasts relative to the other, the easier it is to satisfy the inequalities (8) and (9). These suﬃcient conditions allows

R0 to be increasing or decreasing with respect to p2. In the next chapter we will see how EVD spread in the population during the 2014 West Africa Ebola outbreak keeping in mind this trade-oﬀ.

46 Chapter 4: Model Parametrization

4.1 2014 West Africa Ebola Outbreak Data

In section 3.4, the initial model growth rate r is equation (6). The dominant eigenvalue is

√ w + w + w r = r = w + 2 3 4 (12) 3 1 2µN where:

The epidemic early stage of an infectious disease, for this case EVD, is approximated by exponential growth in the number of reported cases [22, 34]. As mentioned before, r can be used to estimate the basic reproductive number. Fortunately, this measurement has many advantages in comparison with the

information that R0 could provide. First, r measures the speed of epidemic growth, bringing useful information about the time scale of disease spread. In

contrast, R0 is a single number with no speciﬁc information regarding time scale [22]. The data used for completing this chapter will be based on Liberia, because according to databases this was the most overwhelmed country with

47 Ebola cases (suspected, probable, and conﬁrmed) and deaths among general community and HCWs population. According to WHO, the Liberian Ministry of Health, and the Social Welfare and Humanitarian Data Exchange reports, 8,905 cases and 3,858 deaths were recorded in Liberia by February 2015 (since April 2014), with 378 cases and 172 deaths in HCWs.

4.2 Parameter Estimation from Literature

In this section, derivation of parameters will be presented in detail. The rest of the parameters will be computed numerically in section 4.4. The Liberian life expectancy is around 58.21 years [13, 41]. Considering table 1 in [24], it was possible to estimate parameters for our model. An important observation is that our model lacks an exposed compartment, thus it will be necessary to ﬁt infectious period at stage 1 and stage 2 carefully. In order to complete this task the infectious period at stage 1 will be deﬁned as the contribution of the exposed period and the contribution of some small portion of the symptomatic phase. Using table 1 in [24], the mean exposed period of EVD in Liberia is 11.4 days and the mean time from symptom onset to hospital admission is 5 days. According to this information we decided for our model that the exposed period of EVD in Liberia is 9.8 days, the small portion of the symptomatic phase as 3.7 days, and therefore we concluded that the infectious period at 1 stage 1, is 13.5 days. On the other hand, for infectious period at stage 2 we γ1 took a guess for the information provided in [24] regarding the mean time from hospital admission to death (4.2 days) and mean time from hospital admission

48 to recovery of survivors (4.6 days). The decision was that the infectious period 1 at stage 2 for our model, is 4 days. Another parameter estimated from γ2 literature was the disease-induced fatality rate at stage 1 and 2, α1 and α2,

respectively. For α1 the assumption was that no one will die at stage 1 due EVD, because mild-infected individuals are not suﬃciently infected to develop deadly symptoms, then we stated that α1 = 0. The overall case fatality ratio

(OCFR) provided in [24] is 54% and it was used in order to relate α1 and α2:

α α α OCFR = 1 + 1 − 1 2 . (13) γ1 + α1 + µ γ1 + α1 + µ γ2 + α2 + µ

Solving for α2 we get α2 = 0.293534 per day. Additionally, recruitment rate for susceptible individuals in the general community and the HCWs pop-

ulation Λ and ΛH , respectively were calculated in the following way. The total population for Liberia in 2014 was approximately 4,092,310, but this huge amount will not be used to estimate these two parameters, because not all individuals in Liberia are at risk to contract EVD. We assume that only 100,000 individuals have a possibility to be infected by EVD. The relationship to get this estimation is based on the DFE point, S∗ = 100, 000. In section Λ 3.1 the definition of S∗ is = 100, 000. Therefore Λ = 1, 718 individuals per µ 1 year, where µ = per year. For Λ this is not obvious, because there 58.21 H is not enough information that could tells us the specific number of HCWs that were working during the outbreak in Liberia. By December 2009, the number of HCWs in Liberia was around 8,553 individuals [38] and the birth rate in Liberia was 35.07/1000 population [13]. Therefore, to find the number

49 of HCWs in 2014 we assumed that the proportion of HCWs is ﬁxed, so the number of HCWs could be approximated by solving the exponential growth diﬀerential equation:

N˙ = (b − d)N, where r = intrinsic growth rate = b − d

35.07 1 r = − = 0.017891 year−1. 1000 58.21

Let H be an exponential function for population growth that will tell us

tr the population at time t. This H function is deﬁned as H(t) = H0e , where

H0 is the initial population. The initial time t = 0 will be the year 2009 and the time at year 2014 will be t = 5. As a result:

HCWs in 2009 = H(0) = 8, 553

⇒ HCWs in 2014 = 8553e5(0.017891) = 9, 353.

We use the DFE point SH from section 3.1, thus we have SH = 9, 353 ⇒ Λ H = 9, 353. Therefore Λ = 1, 718 per year. Moreover, in order to ﬁnd f µ H 1 and f2 the following approach was used. For f1 we assume that the expected time infectious at stage 1 η is 3.7 days. We also assume that individuals who are infectious for more than 5 days in stage 1 will seek health care, therefore the behavior will be exponentially distributed (as shown in Figure 11). Thus, 1 g(t) = ηe−ηt, where η = per day. 3.7

50 0.3

0.25

0.2

0.15 g(t)

0.1

0.05

0 0 2 4 6 8 10 t (days)

Figure 11: Exponential Distribution Behavior of g(t).

Integrating the region under the curve starting at t = 5 up to inﬁnity we have:

Z ∞ f1 = g(t)dt = 0.25889017. 5

For f2 we used the same approach as equation (9). Using the overall fraction of individuals that seek HCWs (OFI = 80%) from [24]:

OFI = f1 + (1 − f1)f2,

and solving for the desired parameter it turns that f2 = 0.7301133447. Finally to estimate ξ, ﬁrst, we need a better interpretation of this parameter. This is the level of infection in the population at which the half maximum capacity is reached. The level of infection in the population in our model is measured by the contacts an individual could have. Most Ebola patients are removed from

51 the general population and delivered to Ebola Treatment Centers (ETC). An ETC is an isolated health center where HCWs treat patients with EVD. The capacity to isolate patients is dependent on the number of ETC beds and the number of new EVD cases. In Liberia, 660 from 1,990 beds were operational in 16 ETCs [40]. A bed is considered operational when it is staﬀed and ready to receive patients. WHO reported that the average was 13.9 beds per reported patient and median about 17.4. Therefore a proper estimation for ξ is 1,000 beds. The following table shows a summary of the estimated parameters from literature.

Table 3. Summary of Set of Parameters

Parameter Value Reference

f1 0.25889017 [11, 24, 39]

f2 0.73013447 [11, 15, 24] −1 α1 0.00 day assumption −1 α2 0.293534 day [24, 36]

1/γ1 13.5 days [11, 18, 24, 39]

1/γ2 4.0 days [24, 39] Λ 1, 718 year−1 [DFE, 13]

−1 ΛH 160 year [13, 36, 37, 38] µ 1/58.21 year−1 [3] ξ 1, 000 beds [40]

52 4.3 Estimating r from Data

The goal of this section will be to ﬁt the logistic model with the 2014 West Africa Ebola outbreak incidence data in order to estimate the initial

growth rate of an epidemic rest. Typical approaches suggest that rest can be estimated by fitting an exponential curve to the initial growth phase by using least squares approximation of a straight line to the logarithm of cumulative incidence [22]. The logistic model is used as a point of departure for imitating the shape of the cumulative incidence curve, which first grow exponentially, and then starts to level off. During an outbreak, cumulative incidence initially grows exponentially, but as time goes, it approaches to an equilibrium. The logistic model is defined for the following differential equation

y(t) y˙(t) = ry(t) 1 − , K

where y(t) stands for the expected cumulative number of cases, K for the carrying capacity, which y(t) approaches. This equation has a solution in the following form

K y(t) = −rt 1 + [(K/y0) − 1] e

where y0 is the total number of cases observed at time t = 0. This classical logistic model will allow us to use longer sequences of incidence data from the beginning of the outbreak. This model is chosen, because is likely to give a

53 more robust estimate [22]. The data for this ﬁtting process was acquired from the Humanitarian Data Exchange. The initial guess for the carrying capacity and growth rate were

K0 = 45, 000 and r0 = 0.01, respectively. Please, note that the final estimate is not sensitive to the initial values, K0 and r0. In addition, the data collected is from 45 weeks (from April 8, 2014 to February 9, 2015). In the following three figures it will be shown the incidence cases, the cumulative incidence cases, and the cumulative incidence data fitted with the logistic model.

1800 1600 1400 1200 1000 800 600 Incidence 400 200 0 -200

0 5 10 15 20 25 30 35 40 45 t (weeks)

Figure 12: 2014 West Africa Ebola outbreak incidence cases by week.

9000

8000

7000

6000

5000

4000

3000 Cumulative Incidence 2000

1000

0 0 5 10 15 20 25 30 35 40 45 t (weeks)

Figure 13: 2014 West Africa Ebola outbreak cumulative incidence cases by week.

54 9000 Cumulative Incidence 8000 Logistic Model

7000

6000

5000

4000

3000

2000

1000

0 0 5 10 15 20 25 30 35 40 45 t (weeks)

Figure 14: Fitting cumulative incidence data for 45 weeks with the logistic model.

Our estimate for the initial growth rate rest was 0.26258 per week. In the following section we will estimate R0 by using rest and also see how fast HCWs spread the disease in the population. However, we have a special case!

In system (1) if f1 = 0, f2 = 0, p1 = 0, ΛH = 0 and φ = 0, then contribu- ˙ tion of HCWs population is absent and SH is removed from the model. As a consequence, system (1) becomes a Susceptible-Exposed-Infected (SEI) model ˙ ˙ ˙ ˙ where E = I1 and I = I2

 S  S˙ = Λ − β I − µS,  2 2  N    ˙ S I1 = β2I2 − γ1I1 − −µI1, (14)  N     ˙  I2 = γ1I1 − γ2I2 − µI2.

55 4.4 Estimating R0 from the estimated initial growth rate

of an epidemic rest

A typical problem in epidemiology is to estimate the basic reproductive number R0 from the growth rate r [22, 34]. Before doing this, we need to have all the parameter values for our deterministic model. Unfortunately, we do not

have all the parameters values, (e.g. c1, c2, p1, p2, pH1, pH2, φ). The strategy

is to ﬁx p1 = 0.1, p2 = 0.5, pH1 = 0.1, pH2 = 0.4 and explore what happens

when c1, c2 and φ are changing in order to know what will be the value of R0 when both populations are interacting. This method is not unique, there are

diﬀerent ways to estimate R0 from rest, because of the unknown parameters.

The goal is to match the growth rate of our model r, equation (12), with rest

and therefore ﬁnd the value of R0. This will help to observe directly from

R0 and r the contribution of HCWs with respect to the disease spread. In

addition, we want to see if the R0 value for Liberia lies in the range that [24] found, (1.60 − 2.13). Next step is to ﬁnd zeros of the following function d(x)

involving two unknown variables, c1, c2 by relating the initial model growth

rate r and the observed empirical growth rate rest

d(x) = r − rest

where x is a vector containing solutions for c1 and c2. The parameter φ will be

ﬁxed from φ = 0.05 to φ = 0.46 in each step in order to get c1 and c2 such that

d(x) = 0. This method will decipher the value for r and the corresponding R0 value. The results are presented in Figure 15 and Figure 16.

56 1.6 R0G 1.4 R0H

1.2

0.8

0.6

0.4

0.2

0 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 phi

G H Figure 15: Comparison between R0 and R0 . This is showing the relative contribution of general community and HCWs population. The x−axis represents the maximum contact rate of HCWs individuals with patients φ.

1.6 r 1.4 R0

1.2

0.8

0.6

0.4

0.2 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 phi

Figure 16: Comparison between r and R0. This is showing the value of R0 according to the observed growth rate from the data. The x−axis represents the maximum contact rate of HCWs individuals with patients φ.

57 To conclude this chapter let’s discuss some points. First, it was possible to match the initial growth rate of the model r with the initial growth rate of

the data rest. Our model predicted that R0 is near 1.60 as reported during the Liberia outbreak in 2014. According to the data, there is an upper constraint at 1.60, roughly speaking, the Liberian data matched with our model up to the value 1.60 and after that point the initial growth rate of our model is not the same value as the initial growth rate of the data. To clarify, the constraint is not in R0, it seems to be in the maximum contact rate between HCWs with patients φ. What is the interpretation and why is this happening? Results can be interpreted due the presence of parameter φ. When this parameter is large, the transmission rate in the population will be higher and therefore the growth rate will grow faster. To answer the second question, this is happening because after R0 = 1.60 the contacts parameters values, c1 and c2, become negative numbers meaning that r could not be matched. Regarding HCWs, we can observe that at some point in Figure 15 they start driving transmission as contacts and symptoms severity increase. In contrast, Figure 16 do not give

us a clear interpretation regarding the contribution of HCWs: the entire R0 do not provide enough information about how much HCWs spread EVD. Contact tracing is the investigation and diagnosis of people who may have contact with

infected patients. In order to observe directly from R0 how much HCWs drive to transmission we need access to contact tracing data. In addition, by ﬁtting the model with more data points (e.g. by ﬁtting the model with the incidence curve, Figure 13) we will be able to have unique parameter values and therefore to observe how much HCWs induce to new cases of EVD.

58 Chapter 5: Conclusion

This model was able to study the trade-off between contact and probability of transmission given contact involving two different populations: general community and health care workers (HCWs). The study was possible thanks to the interpretation of the basic reproductive number as a function of the probability of transmission as symptom severity increase. It was found that the basic reproductive number is constructed in the following form ∗ S β1 γ1β2 ∗ βH1 γ1βH2 R0 = + +SH + , representing the relative contri- N a1 a1a2 a1 a1a2 G H butions of the general community, R0 and HCWs population, R0 . Performing several analysis of the trade-off we found sufficient conditions that makes R0 an increasing function in terms of the probability of transmission. This is possible by looking at the rate of change in contacts between the general pop- 0 |c (σp2)| ulation and HCWs population, 0 . Some of the results were that in h (νp2) order to have an increasing shape the rate at which HCWs contacts are increasing with symptoms needs to be extremely bigger than the magnitude of the rate at which general community contacts are decreasing. Another interesting result was regarding the relative transmissibility pa-

rameters σ and ν. Surprisingly, this could impact the shape of R0, specially as a consequence of the parameter σ that represents how infectious the contacts

1 2 with general community could be. It was found in R0 criterion that ν is at 2 2 2 the bottom of the fraction and σ is at the top. But for R0 criterion, σ is at the bottom of the fraction together with ν2 giving an interpretation that σ

must be in an intermediate range to make R0 a monotone increasing function.

59 We considered a parameter estimation in order to ﬁt our deterministic model with the 2014 West Africa Ebola Outbreak incidence data. We were able to successfully estimate model parameters from literature and also using a numerical approach. The cumulative cases of Ebola were ﬁtted with the logistic model using a MATLAB nonlinear least squares solver called lsqnonlin.A result from this observation was an estimate for the initial growth rate of an

epidemic rest = 0.26258 per week. This epidemiological tool was used in order to study the early phase of the outbreak and also to understand how fast secondary cases of Ebola arrive. On the other hand the basic reproductive

number was estimated from the data using rest. This was possible by ﬁxing the maximum contact rates of HCWs with patients φ from 0.05 to 0.46 and

then solving for c1 and c2. As a result, the relative contribution of general community and the relative contribution of HCWs with respect to φ were shown in Figure 15. Unfortunately, the relative contribution of HCWs could not be determined from initial growth rate alone and from the entire R0, so we need more data points such as the rest of the incidence curve and contact tracing data. A future work could be an extension of this study for the other two aﬀected countries in West Africa: Sierra Leone and Guinea. It will be interesting to

dive in depth and try to estimate R0 from the generation time interval. In order to make this possible, the distribution of the generation time interval function must be determined. Other future considerations should be ﬁrst, make an extension of this model regarding ceremony burials and vaccinations trials; second, by mixing the HCWs individuals in the susceptible general community compartment; and third, add an extra compartment representing

60 the infected individuals isolated in Ebola Treatment Centers and Community Treatment Centers where HCWs are having interaction with them. The latter will be very useful for future studies, because it could tell us the eﬀectiveness of the isolation technique in order to make a positive impact in the number of EVD cases and therefore to eradicate the disease from the population. In conclusion, do not forget this disease is intermittent. It may disappear for a couple of years, but there is a probability it will return. As mathe- maticians, epidemiologists and science researchers we need to be prepared and provide tools for stopping its development once the ﬁrst cases are reported.

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