Epidemiology, Equilibria, and Ebola: A Dynamical Analysis

Epidemiology, Equilibria, and Ebola: A Dynamical Analysis Graham Harper Department of Mathematics Colorado State University - Pueblo May 1, 2015 1 Abstract In order to understand the forces that drive outbreaks such as the recent Ebola outbreak in West Africa, the force of infection must be broken down into a system of differential equations. With the current understanding of how the equations and diseases interact, it s possible to build diseases from equations and build equations from diseases. Breaking down the differential equations farther leads to a deeper understanding of what stability looks like for a disease and ultimately enables the prediction of the behavior of diseases, including points of peak infection rates and the total number of people affected by the disease. This paper will break down the mathematics behind compartmental epidemic models and then use them to develop a model which affirms a hopeful future with the decline of Ebola in West Africa so that it may take its place in history as another disease scare that never fully developed. Acknowledgements Thanks to Dr. Frank Zizza for advising, supporting, and giving me muchappreciated assistance throughout the semester Thanks to Haven Hall and many other classmates for allowing me to talk with them and giving me inspirations that allowed me to move forward Thanks to the Mathematics Department at Colorado State University - Pueblo and the University Honors Program for supporting this undergraduate thesis 1

2 Introduction 2.1 Compartmental Epidemic Models In order to model how a disease behaves in a relevant population, the modeler must determine a population to study. This population may be as small as a town, or as large as a state, country, or planet. Then each person in the population must be labeled based on his or her status with regards to the disease. The healthy people are called susceptible. This category is denoted with S. The people infected with the disease are called infected (or infective). This category is denoted with I. These two categories are the most important to understanding the way a disease moves through a population; however, some diseases require other categories like R, for recovered (with immunity); E, for exposed (but not yet contagious); D, for dead (when death is significant); V, for vaccinated (and immune), which may not be significantly different from R, depending on the model. These categorical divisions of the population are by no means permanent, however, because at any time someone in S may fall ill and move into I. Thus in order to count the number of people in each category, counting functions S(t), I(t), etc. must be constructed. These are the compartments for the compartmental epidemic model. These compartments are partitions of the population because any given person must fall under one group and no person may belong to more than one group. In order to understand how these groups interact, a system of ordinary differential equations must be constructed. The basis for the equations lies in the Lotka-Volterra model for population dynamics in a predator-prey system. If X is the prey, and Y is the predator, then the Lotka-Volterra equations are Ẋ = αx βxy Ẏ = δxy γy where α, β, δ, and γ are constants. According to the model, in the absence of the predator (Y ), the prey (X) will reproduce exponentially, but in the absence of the prey, the predator will starve and die out; however, the most important part of this model is the XY factor. This factor, along with the coefficient β or δ represents the probability of an encounter between predator prey as well as the outcome. If the predator and prey meet, there is a chance that the prey will be eaten, which gives the predator a better chance of survival. Now consider a population of susceptible people and infected people. De Jong et al. point out that there are three factors which are relevant in the spread of the disease, but they have meaning in both models: 1. The number of significant contacts with other people per unit time 2. The probability of this contact taking place with an infected individual 3. The probability that the contact with an infected results in a transmission of disease 2

The first factor is a measure of the density of a population. A dense population is going to see more contacts per unit time than a sparse population, regardless of whether it s predator and prey or people and illness. The second factor is what the XY factor represents in the Lotka-Volterra model. Thus the probability of a susceptible and an infected coming into contact must be related to S(t)I(t). The third factor is a success factor. If a predator and prey meet, there is a chance that the prey will escape, but there is also a chance that the prey will not escape. This represents the contagiousness of the disease being studied. The disease transferring is a success, otherwise it is a failure. This is as good of a time as any to bring in the basic reproduction number, which is commonly denoted with R 0. R 0 is a factor which is described as the number of people 1 person will infect before losing contagiousness if he or she only comes in contact with others who are susceptible. R 0 is directly related to the third factor on the list. R 0 is based on several factors, including the medium through which the disease moves. If the disease is airborne, like measles, then R 0 is quite large, but if the disease if only transferred through bodily fluids, like the Ebola virus disease, then R 0 may be small. Also, if the disease has a longer contagious period, then R 0 will be larger because there is more time to infect others. The culmination of all of this information leads to the force of infection in compartmental disease models, which is written as βsi. From here forward, S and I are assumed to be functions of time. β is a constant that represents a combination of both societal and viral factors to decide the speed of the spread of the disease. Sometimes it s referred to as the contact rate for a disease, but since β is related to R 0, it may be more appropriate to refer to it as the infection factor. βsi is used when writing the differential equations for a model to describe what happens when a successful contact takes place. Here s what the most basic compartmental model, the SI model, looks like Ṡ = βsi I = βsi Every time there is a successful contact, 1 person leaves S and enters I. Notice that, unlike the Lotka-Volterra equations, there are no growth or death factors for the population in this model. That s because unless this model is considered over a significantly long period of time, total population changes may not have an impact on the behavior of the disease. The SI model the foundation for all compartmental epidemic models. 3

2.2 Pseudo Mass-Action vs. True Mass-Action Models While it was stated above that S and I are used to represent the number of individuals in a population who are susceptible or infected, it is also possible to use them to represent the ratio of individuals in a population who are susceptible or infected. The model where the compartments represent the number of individuals in a population who are susceptible or infected is called a True Mass-Action model. When writing the differential equations for an SI true mass-action model, one normally writes Ṡ = βsi N and I = βsi N where N is the total number of people in the population as a function of time. This model is the most accurate for modeling the way a population changes over time because it takes into account each and every person in the population instead of ratios. This also allows for factors that account for growth and death in the population to be taken into account, and those will influence the progression of the disease through changes in N. It is worth noting, however, that this model is much harder to manipulate algebraically and computationally because generally N looks like N = S + I with the possibility of additional terms being added in based on the number of compartments in the model. If there are no births or deaths in a population, then it may be more beneficial to write N(t) = k where k is the constant population size. The model where the compartments represent the ratio of individuals in a population who are susceptible or infected is called a Pseudo Mass-Action model. In an SI pseudo mass-action model, N(t) 1 so Ṡ simplifies to βsi and I simplifies to βsi. This kind of model is much simpler to manipulate algebraically and it will be the focus for the analysis in section 4. It s important to note that while these two models appear significantly different, de Jong et al. show evidence supporting the claim that there is no significant difference in the way these models behave when applied to real world data. This means that either model may be chosen for the user s convenience, so while the stability analysis may focus on the pseudo mass-action models, the parameter estimation section may focus on using a true mass-action model with a fixed population assumption. 4

2.3 Application to the Ebola Virus Disease With the current understanding of the Ebola Virus Disease (hereafter referred to as Ebola), the most common deterministic epidemic model that can be applied is the SEIR model, which assumes immunity after the patient recovers, which has not been proven by science yet. Letting β be the same as mentioned in section 2.1, the other coefficients ɛ and γ, are defined as the reciprocals of the exposed and infectious periods (in the same units of time as β), respectively. Assuming that S, E, I, and R are functions of time, a pseudo mass-action model for Ebola may look like Ṡ = βsi Ė = βsi ɛe I = ɛe γi Ṙ = γi Notice that since Ṡ+Ė+ I+Ṙ = 0, the population in this model is a conserved quantity (hence why the pseudo mass-action model was chosen). This model may be applied if the time period under consideration is small enough to see no significant change in the population due to births or death, or if birth and death rates are not important enough to be considered. While this model works well for modeling Ebola, there is one problem in applying this model with real data: making measurements to track movement in and out of the exposed group is near impossible. This means that when using real data, a simpler model is required. An SIR model may work just as well, and it looks like Ṡ = βsi I = βsi γi Ṙ = γi Since data is available through the WHO for the cumulative infected counts, and cumulative death tolls, it makes sense to utilize this model instead. One slightly difficult part of applying this model to real data is that if a country like Guinea has 12 million people, this model claims the entire population will fall ill and recover eventually. This is erroneous to assume, so S must be calculated by predicting the total number of cases and subtracting the cumulative infections. This introduces some error that will be accounted for in section 5. 5

2.4 Disease Building through Differential Equations Using compartmental epidemic models, it s possible to build a disease with a physical interpretation in mind. For example, assume there is a disease which is characterized by the following: 1. After the disease infects the victim, the victim spends an average of 5 days unaware that he is ill and without being contagious. 2. The victim then shows symptoms and is contagious for an average of 10 days. 3. The victim has a 40% chance of developing an immunity and a 60% chance of entering the cycle again immediately. In order to construct a pseudo mass-action model, the compartments must be defined first. Based on the description, there are S, E, I, and V compartments, where V represents permanent immunity, which is equivalent to a vaccination. Then the transfer speeds between the groups are based on the number of people who leave a specific group every day. This is calculated by the reciprocal of the time spent in the compartment times the number of people in the compartment. Thus the model should look like Ṡ = βsi + 0.06I Ė = βsi 0.2E I = 0.2E 0.1I V = 0.04I Once again, the 0.06I term in the first equation is because 60% of infected go straight to being susceptible again after a 10 day infectious period. The 0.2E term in the second and third equations is because 1 person per 5 days in the exposed category moves on. The 0.1I term in the third equation is because 1 person per 10 days in the infected category moves on. The 0.04I term in the last equation is because 40% of people who are infected move on to being permanently immune after a 10 day infectious period. β is not given for this model, but any value can likely be justified. Assuming β = 1 and initial conditions of S = 0.99, I = 0.01, and E = V = 0, here is a plot of the solution to this system from Mathematica. It s important to note that most of these systems cannot be solved analytically by hand because of the SI term that is always required to show up in the equations. This model is interesting because it looks like the number of vaccinated people in the population is increasing close to 95%, while the remaining 5% never stay immune to the disease. The peak infected growth period seems to occur around 20 days, and the infected category peaked at 30 days. 6

The point is without knowing how a disease fits into the framework of the model system, a model can be logically constructed relatively easily, and then it can be solved. This is beneficial for improvising models when presented with diseases that don t quite fit the most common models. Figure 1: Solutions to the SEIV Model 7

3 Common Compartmental Models and Representations Here is a list of commonly used compartmental models, their definitions, and some diseases that match the models well. These are not intended to be exact matchings to the diseases, but each model is given a line of reasoning as to why it is a good fit to help recognize the reasoning behind the model. Each of the models are pseudo mass-action models without accounting for population growth. The constant β represents the infection factor, ɛ represents the reciprocal of the length of the exposed period, γ represents the reciprocal of the length of the infectious period, and ρ represents the reciprocal of the recovery period before becoming susceptible again. Table 1: Compartmental Disease Models and Example Diseases Model Formulation Example Diseases SI Ṡ = βsi AIDS (Once the disease is caught, there is no I = βsi cure for it, and the victim is contagious forever) SIS SIR SIRS SEIR SEIRS Ṡ = βsi + γi I = βsi γi Ṡ = βsi I = βsi γi Ṙ = γi Ṡ = βsi + ρr I = βsi γi Ṙ = γi ρr Ṡ = βsi Ė = βsi ɛe I = ɛe γi Ṙ = γi Ṡ = βsi + ρr Ė = βsi ɛe I = ɛe γi Ṙ = γi ρr The Common Cold (Due to the many strains, it may be caught anytime, but this needs to assume a common γ for all colds) Chickenpox (It may only be caught once in a lifetime, but once it is caught and the victim recovers, it cannot be caught again) One strain of seasonal influenza (Once immunity is developed to an influenza strain, immunity lasts until the next season) Ebola (A victim may walk around exposed but not yet showing symptoms or being contagious, and after being contagious the victim may either die or recover) Ebola (Assuming that Ebola may be caught again, although there is no scientific evidence for this yet) Beyond this list of compartmental models, there is another way to represent the way a model looks. This is done with a flow diagram. They visually convey all of the same exact information that the equations do, but in a much less difficult way. The flow diagram is constructed by creating a box for each of the compartments, and then drawing flow arrows between the compartments to represent how people are moving between the compartments. Each of the arrows are labeled with the factor from the differential equation. For example, in the SIRS model, there are 3 compartments. This means a box should be drawn for S, I, and R. People flow from S to I, I to R, and R to S, so arrows 8

should be drawn appropriately. The rate that people leave S and enter I is βsi, the rate that people leave I and enter R is γi, and the rate that people leave R and re-enter S is ρr. This means the flow diagram for an SIRS model should look like this. Figure 2: Flow Diagram for SIRS Model Likewise, since everybody in an SEIR model ultimately progresses to the R stage, a flow diagram for an SEIR model needs 4 boxes, and it looks like Figure 3: Flow Diagram for SEIR Model This is a useful tool for conveying ideas about almost any epidemic model when the differential equations may not be appropriate. This is also a great intermediate step between describing the disease and writing down the differential equations. 9

4 Equilibria of Disease Models Many epidemic models can be solved for equilibrium points in order to better understand the system, and these equilibrium points can be taken and analyzed farther to see what behavior is being predicted by the differential equations around the points. For example, the SIS pseudo mass-action model is Ṡ = βsi + γi I = βsi γi where S and I are functions of time, and beta and gamma are constants. Setting Ṡ = I = 0 yields a trivial fixed point of S = I = 0, and in fact, any point of the form (S,0) is a fixed point since setting I = 0 solves all equations simultaneously. However, this only means that a population that starts with no sick people in it isn t going to have a spontaneous illness appear. Excluding the case where I = 0 yields S = γ β. Since this model is pseudo-mass action, there is the additional constraint of S + I = 1, so substitution yields I = β γ β (or 1 γ β may be written when convenient), so the two fixed points of the pseudo mass-action SIS system are (1, 0) and ( γ β, β γ β ). In order to assess what the system s behavior at these fixed points, it s important to linearize the differential equation around the points by forming the matrix of partials ( ) ṠS Ṡ I J = I S II In the case for the SIS model, this matrix is ( βi βs + γ βi βs γ One of the eigenvalues of this matrix must be 0 because the matrix has determinant 0, but the other eigenvalue requires much more work, and in this case it is βi + βs γ. At any point on the line S + I = 1, the eigenvectors for this system are ( ) ( 1 γ βi, S ) I 1 1 where the second vector corresponds to the 0 eigenvalue. These vectors aren t significant without considering them at an equilibrium point, so plugging in S = γ β γ β and I = β yields the eigenvalues γ β and 0 with eigenvectors ( ) ( ) 1 0, 1 1 ) 10

respectively, while plugging in S = 1 and I = 0 yields the eigenvalues γ and 0 with eigenvectors ( ) ( 1 γ β, 1 ) 1 1 respectively. This means that, as suspected, the eigenvector corresponding to the negative (attracting) eigenvalue is parallel to the line S + I = 1 but the eigenvector corresponding to the 0 eigenvalue is pointing off of S + I = 1, but this only holds in both cases if γ β 0. Considering the eigenvalues of γ β and 0 brings the concern that γ β may be positive, forcing ( γ β, β γ β ) to be a repelling fixed point. Letting γ β = 0 yields γ β = 1, which means that since S = γ β = 1, this crossover from attracting to repelling happens when the fixed point reaches the susceptible axis at the point (1,0). Figure 4: Phase Portrait of Solutions to an SIS model with β = 1 2 and γ = 1 12 The figure above is a phase portrait of multiple directed solutions to an SIS pseudo mass-action model. The fixed point of interest in this phase portrait is the point ( 1 6, 5 6 ), and it is attracting because the eigenvalue is β(1 γ β )+β( γ β ) γ = γ β = 1 12 1 2 = 5 12, so this is a stable fixed point. The phase portrait of this system also explains the eigenvalue of 0. Leaving the line S + I = 1 is impossible here because a crucial assumption was S + I = 1, so the eigenvector corresponding to the eigenvalue 0 should be a vector that is not parallel to S + I = 1. 11

Figure 5: Phase Portrait of Solutions to an SIS model with β = 1 20 and γ = 1 12 The figure above, in different from Figure 4, only in that β changed from 1 2 to 1 20. Since β γ now, the equilibrium point that was so obvious in Figure 4 of ( γ β, 1 γ β ) has shifted to the point ( 5 3, 2 3 ), and it is now an unstable fixed point because its eigenvalues are 0 and γ β = 1 30. This means that the fixed point at (1, 0) is the one drawing everything in now because it still has eigenvalues of 0 and γ = 1 12. Similar methods can be applied to any compartmental models to determine simple behavior, including the following SEIS model: Ṡ = βsi + γi Ė = βsi ɛe I = ɛe γi Setting Ṡ = Ė = I = 0 still provides an equilibrium point at (1,0,0) S R for this system, but excluding this case yields S = γ β again. Using S + E + I = 1 yields a more complicated solution, which is found with the substitution γi = ɛe from I = 0, so γ β + γ ɛ I + I = 1 and γ β + E + ɛ γ E = 1. This ultimately yields S = γ β, E = γ(β γ) β(γ+ɛ) ɛ(β γ), and I = β(γ+ɛ). It s important to note that unlike the SIS model, the SEIS model has 3 compartments and thus it may leave the line S + I = 1 in the SI phase diagram. When finding the equilibrium point in the SI phase diagram, the E term is dropped because and only ( γ β, ɛ(β γ) β(γ+ɛ) ) is plotted. 12

Figure 6: Phase Portrait of Solutions to an SEIS model with β = 1 2, ɛ = 1 20, and γ = 1 12 The above figure shows solutions to an SEIS model plotted in the SI plane using various initial conditions along the line S + I = 1 and E = 0, with β = 1 2, ɛ = 1 20, and γ = 1 12. Using ( the equations ) from the previous page, the equilibrium point in the SI plane is γ β, ɛ(β γ) β(γ+ɛ) = ( 1 6, 5 16 ), which is where all of the solutions are heading. While much more work can be done to calculate the eigenvalue of the point, the importance of this work on the SEIS model is that the procedures to find the equilibria can be broadly applied in order to understand a variety of epidemic models. In short, since the equilibrium points can be found regardless of the model, it makes more sense to compile all of the information into a table. From the table on the next page, it s much easier to see the equilibrium points, and this table is very useful for determining where a disease likes to settle in a population because it may move toward a state where the population has consistent proportions of susceptible, infected, exposed, or recovered. For example, if a disease has a long infectious period and a high infection factor, and people continuously revert back to being susceptible to it, it s likely that the disease will settle in the population with a high number of infected at any given point in time, and a low number of susceptible people. 13

Table 2: Compartmental Disease Models and Equilibrium Points Model Formulation Equilibria SI Ṡ = βsi I = βsi (S, I): (1, 0), (0, 1) SIS Ṡ = βsi + γi I = βsi γi (S, I): (1, 0), ( γ β, β γ β ) SIR Ṡ = βsi (S, I, R): Any point of the form (t, 0, 1 t), I = βsi γi where t [0, 1] Ṙ = γi SIRS SEI SEIS SEIR SEIRS Ṡ = βsi + ρr I = βsi γi Ṙ = γi ρr Ṡ = βsi Ė = βsi ɛe I = ɛe Ṡ = βsi + γi Ė = βsi ɛe I = ɛe γi Ṡ = βsi Ė = βsi ɛe I = ɛe γi Ṙ = γi Ṡ = βsi + ρr Ė = βsi ɛe I = ɛe γi Ṙ = γi ρr (S, I, R): (1, 0, 0), (S, E, I): (1, 0, 0), (0, 0, 1) (S, E, I): (1, 0, 0), ( ) γ β, ρ(β γ) β(γ+ρ), γ(β γ) β(γ+ρ) ( ) γ β, γ(β γ) β(γ+ɛ), ɛ(β γ) β(γ+ɛ) (S, E, I, R): Any point of the form (t, 0, 0, 1 t), where t [0, 1] (S, ( E, I, R): (1, 0, 0, 0), β γ, γρ(β γ) β(ɛρ+γɛ+ργ), ɛρ(β γ) β(ɛρ+γɛ+ργ, ) γɛ(β γ) β(ɛρ+γɛ+ργ) The reason that some of these equilibria have parametric definitions is because with the SIR model, for example, as long as I = 0, the entire system goes to 0, so this means that S can be a free variable in [0, 1], so setting S = t, with t [0, 1] forces R = 1 t from the requirement that S + I + R = 1. This condition may also be simply expressed as follows: a population without any illness is never going to develop illness. Thus there are an infinite number of equilibria for the SIR model. One way to show off the behavior of epidemics may also be to develop a program that switches between phase portraits and also allows for changes in parameters. Luckily Mathematica has a Manipulate function that allows for a lot of those features to be realized. The following image is a manipulator of an SEIRS model that allows for switching between different phase portraits while changing parameters and the number of solutions to plot. 14

Figure 7: Phase Portrait of Solutions to an SEIRS model The code for this is located in section 7.1, but the interactivity of this model makes it another excellent way to analyze or track equilibria. Notice that in this phase portrait, it looks like the S and R are settling to 45% and 40%. Utilizing the tabs can provide a totally different image of how the disease is settling. 15

5 Parameter Estimation of the Ebola Outbreak in West Africa It s possible to estimate the parameters of an outbreak given the data on the outbreak. Start by taking the collection of data (for Ebola, there is definitive data on infected and deaths with respect to time provided by the WHO) and form a hypothesized total effective population for the disease. Certainly for a country like Guinea it may not make sense to consider the entire population of 12 million people to be at risk for catching Ebola because the model would claim that all 12 million people will fall ill, so the data in this section is based on a population of only 5000 people in Guinea, and 10000 people in Sierra Leone, which may be unsanitary rural populations that spread Ebola and don t have much contact with the more hygienic part of the countries. WHO data is available for Guinea and Sierra Leone for cumulative infected and cumulative death counts, but the data available for Liberia is limited to cumulative infected. Having cumulative cases and deaths is beneficial because an average mortality rate may also be calculated for the disease in both Guinea and Sierra Leone. Here is a summary of the data available from the WHO Figure 8: West Africa Cumulative Cases (since 8/29/2014) 16

Figure 9: West Africa Deaths (since 8/29/2014) Notice that the cumulative cases data jumps by almost 2000 for Liberia at October 29, 2014, and the Liberia death data is scattered and scarce. This is why Liberia will not be studied in this section, although its data is present. The difficulty in constructing a model to model the West Africa Ebola outbreak is that the available data counts cumulative cases and deaths. This means there is a need for functions to count the total cases, and not just the present cases. In the style of Chowell et al., the most appropriate model for this is a SIJDR model. Here is the flow chart for it Figure 10: Flow Diagram for SIJDR Model The dashed line represents that J, the counting function, is counting the number of cases that pass through the solid line. Here δ represents the death rate in the population for the disease, which is about 40% in Sierra Leone and 60% in Guinea. The equations for this model look like 17

Ṡ = βsi I = βsi γi J = βsi Ḋ = δγi Ṙ = (1 δ)γi In order to calculate an estimation for β and γ, an error function needs to be defined. This function, G : R 2 R, is defined as G(β, γ) = i (S(t i ), J(t i ), D(t i )) (S ti, J ti, D ti ) Where S, J, and D are solutions of the SIJDR model utilising β and γ, and J ti, D ti are the data sets from the WHO. Recall that S ti is calculated with k J ti where k is the assumed total population size. Once this work is done, this function may be programmed into Mathematica, which can do a comprehensive search for the minimum error of this function and return the β and γ values. The code for this function is given in section 7.2. For Sierra Leone, the minimum is G(0.045153, 0.018449) = 26762.15, and for Guinea, the minimum is G(0.034935, 0.023391) = 7386.19. Notice that the error for the Guinea model is much lower than the error for the Sierra Leone model. This suggests that either the assumed populations of 5000 and 10000 caused variation, or that the Guinea data is much smoother. Both β and γ aren t separated by more than 0.03, which means it seems like the estimates respect each other well. Since the assumptions about the populations under consideration also seem like they could be erroneous, the minimum of G was found based on many different total population assumptions to see if there is are more appropriate population assumptions that could be made. 18

Table 3: Min(G(β, γ)) Based on Assumed Population Size Population (S + I) Guinea Sierra Leone 4000 10281.90 4900 7524.90 5000 7386.19 5500 6930.68 5600 6875.66 6000 6743.35 6100 6730.71 6200 6728.04 6300 6731.86 6500 6751.11 7000 6858.07 8000 7217.15 8600 58671.75 8700 54697.67 9000 44257.98 9200 38518.94 9500 31809.36 9800 27777.32 9900 27100.23 10000 26762.15 10100 26706.45 10200 26824.97 10500 27613.24 11000 29624.66 Looking at this table shows that the minimum error is minimized for Guinea when a model with a population size of about 6200 is assumed and for Sierra Leone when a population size of about 10100 is assumed. Looking back at the SIJDR model, this means that G(0.034016, 0.025537) = 6728.04 for Guinea and G(0.044932, 0.018703) = 26706.45 for Sierra Leone are better estimates of the parameter, but these also help predict the end behavior of the disease. 19

6 Results With the refined values of β = 0.034016, γ = 0.025537 for Guinea and β = 0.044932, γ = 0.018703 for Sierra Leone. Utilizing the β and γ values and solving the SIJDR model again yields the following solutions Figure 11: SIJD Model for Guinea Figure 12: SIJD Model for Sierra Leone For Guinea, the solution predicts a total of 3509.98 cases and a total of 2392.94 deaths by May 25, 2017, which corresponds to t = 1000. Assuming 20

this model is an accurate model of the real world situation, this means that the worst of the Ebola outbreak in Guinea is in the past, and there won t be much more progression of the disease in the future. Likewise for Sierra Leone, the solution predicts a total of 9030.22 cases and a total of 3992.08 deaths by May 25, 2017. This means that the worst is behind Sierra Leone, and Ebola is not going progress much more. This means that if this model is an accurate representation of the Ebola outbreak in West Africa, it is likely that the outbreak is coming to a close. Beyond that, these models are just some of the tools that can be used to predict the progression of Ebola if another outbreak occurs in another country. The β and γ factors, or even the death ratios, may be used as estimators for an outbreak in a nearby country like Côte d Ivoire, should one happen. Knowing the way these models behave can help determine critical time periods for fighting the disease in future events so that funding to fight the outbreak can be appropriately allocated to reduce the damage caused by the disease in its early stages, and then funding can be slowly removed as the disease slows down. Looking forward, taking derivatives of this model and finding the points where the derivative is maximized will help determine when future outbreaks may take the largest toll on countries and aid in the prediction and prevention of full-fledged outbreaks. 21

7 Code Index 7.1 Plotting Figures To plot Figure 1: sol = (NDSolve[ {s [t] == -s[t] i[t] + 0.06 i[t], e [t] == s[t] i[t] - 0.2 e[t], i [t] == 0.2 e[t] - 0.1 i[t], v [t] == 0.04 i[t], s[0] == 0.99, i[0] == 0.01, e[0] == v[0] == 0}, {s, e, i, v}, {t, 0, 100}][[1]]); {a, b, c, d} = {s, e, i, v} /. sol; Legend = SwatchLegend[{Blue, Red, RGBColor[0.8, 0.6, 0.], RGBColor[0.1, 0.6, 0.1]}, {"Susceptible", "Exposed", "Infected", "Vaccinated"}, LegendMarkers -> Graphics[{EdgeForm[Black], Opacity[0.5], Rectangle[]}], LegendLabel -> "Categories", LegendFunction -> (Framed[#, RoundingRadius -> 5] &), LegendMargins -> 5]; ParametricPlot[ {{t, a[t]}, {t, b[t]}, {t, c[t]}, {t, d[t]}}, {t, 0, 100}, AspectRatio -> 1, AxesLabel -> {"Time (days)", "Count"}, PlotLegends -> Legend] To plot Figures 4 and 5: Show[ Table[ParametricPlot[Evaluate[First[{s[t], i[t]} /. NDSolve[ {s [t] == (-\[Beta] s[t] i[t] + \[Gamma] i[t]), i [t] == \[Beta] s[t] i[t] - \[Gamma] i[t], Thread[{s[0], i[0]} == {j, 1 - j}]} /. {\[Beta] -> 1/2, \[Gamma] -> 1/12}, {s, i}, {t, 0, 5}]]], {t, 0, 5}, PlotRange -> {{0, 1}, {0, 1}}, AspectRatio -> 1, AxesLabel -> {Susceptibles, Infected}], {j, 0, 1,.05}] /. Line[x_] :> {Arrowheads[Table[.03, {1}]], Arrow[x]}, ParametricPlot[{t, 1 - t}, {t, 0, 1}]] To plot Figure 6: Show[ Table[ParametricPlot[Evaluate[First[{s[t], i[t]} /. NDSolve[ {s [t] == (-\[Beta] s[t] i[t] + \[Gamma] i[t]), e [t] == \[Beta] s[t] i[t] - \[Epsilon] e[t], 22

i [t] == \[Epsilon] e[t] - \[Gamma] i[t], Thread[{s[0], e[0], i[0]} == {j, 0, 1 - j}]} /. {\[Beta] -> 1/2, \[Epsilon] -> 1/20, \[Gamma] -> 1/12}, {s, i}, {t, 0, 50}]]], {t, 0, 25}, PlotRange -> {{0, 1}, {0, 1}}, AspectRatio -> 1, AxesLabel -> {Susceptibles, Infected}], {j, 0, 1,.05}] /. Line[x_] :> {Arrowheads[Table[.03, {1}]], Arrow[x]}, ParametricPlot[{t, 1 - t}, {t, 0, 1}]] To plot Figure 7: MakeFunction[l_] := Table[(l[[k]])[t], {k, 1, Length[l]}] MakeFunction[a_, b_, c_, d_] := Table[MakeFunction[x], {x, {a, b, c, d}}] bound = ParametricPlot[{t, 1 - t}, {t, 0, 1}]; Manipulate[ Sol = Table[ NDSolve[ {s [t] == -\[Beta] s[t]*i[t] + \[Rho] r[t], e [t] == \[Beta] s[t]*i[t] - \[Epsilon] e[t], i [t] == \[Epsilon] e[t] - \[Gamma] i[t], r [t] == \[Gamma] i[t] - \[Rho] r[t], Thread[{s[0], e[0], i[0], r[0]} == {j, 0, 1 - j, 0}]}, {s, e, i, r}, {t, 0, 100}][[1]], {j, 0, 1-1/Res, 1/Res}]; Show[ParametricPlot[ Transpose[ MakeFunction[s /. Sol, e /. Sol, i /. Sol, r /. Sol][[{x, y}]]], {t, 0, 100}, PlotRange -> {{0, 1}, {0, 1}}, AspectRatio -> 1, AxesLabel -> {{"Susceptible", "Exposed", "Infected", "Recovered"}[[ x]], {"Susceptible", "Exposed", "Infected", "Recovered"}[[y]]}], bound, ImageSize -> 400], {{x, 1}, {1 -> "S", 2 -> "E", 3 -> "I", 4 -> "R"}}, {{y, 2}, {1 -> "S", 2 -> "E", 3 -> "I", 4 -> "R"}}, {{Res, 5}, 1, 15, 1, Appearance -> "Labeled"}, {{\[Beta], 1./3}, 0, 1, Appearance -> "Labeled"}, {{\[Gamma], 1./8}, 0, 1, Appearance -> "Labeled"}, {{\[Epsilon], 1./9}, 0, 1, Appearance -> "Labeled"}, {{\[Rho], 1./20}, 0, 1, Appearance -> "Labeled"}] To plot Figure 8: ListPlot[{ICS[[All,{1,3}]],ICG[[All,{1,3}]],ICL[[All,{1,3}]]}, 23

PlotRange->{{0,250},{0,10000}, PlotLegends->{"Sierra Leone","Guinea","Liberia"}, AxesLabel->{"Time","Infected"}] To plot Figure 9: ListPlot[{ICS[[All,{1,4}]],ICG[[All,{1,4}]],ICL[[All,{1,4}]]}, PlotRange->{{0,250},{0,10000}, PlotLegends->{"Sierra Leone","Guinea","Liberia"}, AxesLabel->{"Time","Deaths"}] To plot Figures 11, 12: Below 7.2 Estimating Parameters To estimate parameters for the Guinea and Sierra Leone outbreaks and then build a plot for them: IC = Import[ "EbolaData_Sierra.xlsx", {"Data", 1, Table[n, {n, 2, 95(*95*)}], {2, 3, 4, 7}}]; n = IC[[1, 2]] + IC[[1, 3]]; G[\[Beta]_, \[Gamma]_] := Module[{sol, u, v, w, x, s, i, j, d}, sol = NDSolve[ {s [t] == -\[Beta] s[t] i[t]/n, i [t] == \[Beta] s[t] i[t]/n - \[Gamma] i[t], j [t] == \[Beta] s[t] i[t]/n, d [t] == \[Gamma] (\[Delta]) i[t], s[0] == IC[[1, 2]], i[0] == IC[[1, 3]], j[0] == i[0], d[0] == IC[[1, 4]]}, {s, j, d}, {t, 0, 500}]; {u, v, w} = {s, j, d} /. sol[[1]]; Sum[Norm[{u[IC[[k, 1]]], v[ic[[k, 1]]], w[ic[[k, 1]]]} - {IC[[k, 2]], IC[[k, 3]], IC[[k, 4]]}], {k, 2, 94}]] ans = FindMinimum[G, {0., 1.}, {0., 1.}, Method -> "PrincipalAxis", WorkingPrecision -> 10] \[Beta] = Last[ans][[1]]; \[Gamma] = Last[ans][[2]]; {SS, II, JJ, DD} = {s, i, j, d} /. NDSolve[ {s [t] == -\[Beta] s[t] i[t]/n, i [t] == \[Beta] s[t] i[t]/n - \[Gamma] i[t], j [t] == \[Beta] s[t] i[t]/n, d [t] == \[Gamma] (\[Delta]) i[t], 24

s[0] == IC[[1, 2]], i[0] == IC[[1, 3]], j[0] == i[0], d[0] == IC[[1, 4]]}, {s, i, j, d}, {t, 0, 1000}][[1]]; Legend = SwatchLegend[{Blue, Red, RGBColor[0.8, 0.6, 0.], RGBColor[0.1, 0.6, 0.1]}, {"Susceptible", "Infected", "Cumulative Infected", "Cumulative Dead"}, LegendMarkers -> Graphics[{EdgeForm[Black], Opacity[0.5], Rectangle[]}], LegendLabel -> "Categories", LegendFunction -> (Framed[#, RoundingRadius -> 5] &), LegendMargins -> 5]; Show[ParametricPlot[{{t, SS[t]}, {t, II[t]}, {t, JJ[t]}, {t, DD[t]}}, {t, 0, 1000}, PlotRange -> {{0, 1000}, {0, n}}, AspectRatio -> 1, AxesLabel -> {"Time (days)", "Count"}, PlotLegends -> Legend, PlotStyle -> Thick], ListPlot[IC[[All, {1, 2}]]], ListPlot[IC[[All, {1, 4}]], PlotStyle -> RGBColor[0.1, 0.4, 0.1]]] {SS[1000], II[1000], JJ[1000], DD[1000]} 25

8 WHO Data (for reference) Here is the data that was utilized in this paper from the WHO. Each of these data sets represents the date versus the number of total confirmed cases/deaths in a specific country. The data came from https://data.hdx.rwlabs.org/ebola where much more Ebola data is stored, but this specific data came from the csv file on https://data.hdx.rwlabs.org/dataset/ebola-cases-2014 The 4 data points marked with * are missing death counts that were set to an arbitrary close value instead of 0 to prevent large amounts of error entering the models. Guinea Cases 8/29/2014 482 9/5/2014 604 9/8/2014 664 9/12/2014 678 9/16/2014 743 9/18/2014 750 9/22/2014 818 9/24/2014 832 9/26/2014 876 10/1/2014 950 10/3/2014 977 10/8/2014 1044 10/10/2014 1097 10/15/2014 1184 10/17/2014 1217 10/22/2014 1289 10/25/2014 1312 10/29/2014 1391 10/31/2014 1409 11/5/2014 1457 11/7/2014 1479 11/12/2014 1612 11/14/2014 1647 11/19/2014 1698 11/21/2014 1745 11/26/2014 1850 12/1/2014 1921 12/3/2014 1929 12/10/2014 2051 12/15/2014 2115 12/17/2014 2127 12/22/2014 2259 12/23/2014 2284 12/24/2014 2284 12/26/2014 2342 12/29/2014 2384 12/30/2014 2397 12/31/2014 2397 1/2/2015 2435 1/5/2015 2465 1/6/2015 2471 1/7/2015 2471 1/8/2015 2477 1/9/2015 2493 1/12/2015 2508 1/13/2015 2508 1/14/2015 2514 1/15/2015 2522 1/16/2015 2525 1/19/2015 2539 1/20/2015 2539 1/21/2015 2539 1/22/2015 2542 1/23/2015 2545 1/26/2015 2559 1/27/2015 2569 1/28/2015 2569 1/29/2015 2571 1/30/2015 2575 2/2/2015 2593 2/3/2015 2608 2/4/2015 2608 2/5/2015 2621 2/6/2015 2628 2/10/2015 2674 2/11/2015 2674 2/12/2015 2685 2/13/2015 2693 2/16/2015 2720 2/17/2015 2727 2/18/2015 2727 2/19/2015 2732 2/20/2015 2734 2/23/2015 2758 2/25/2015 2762 2/26/2015 2781 2/27/2015 2790 3/2/2015 2808 3/4/2015 2808 3/5/2015 2833 3/6/2015 2840 3/10/2015 2871 3/11/2015 2871 3/12/2015 2901 3/13/2015 2911 3/16/2015 2957 3/17/2015 2966 3/18/2015 2966 3/20/2015 2988 3/23/2015 3007 3/24/2015 3011 3/25/2015 3011 3/26/2015 3032 3/27/2015 3042 26

Liberia Cases 8/29/2014 322 9/5/2014 614 9/8/2014 634 9/12/2014 654 9/16/2014 790 9/18/2014 812 9/22/2014 863 9/24/2014 890 9/26/2014 914 10/1/2014 927 10/3/2014 931 10/8/2014 941 10/10/2014 943 10/15/2014 950 10/22/2014 965 10/25/2014 965 10/29/2014 2515 10/31/2014 2515 11/5/2014 2451 11/7/2014 2514 11/12/2014 2553 11/14/2014 2562 11/19/2014 2643 11/21/2014 2669 11/26/2014 2727 12/1/2014 2801 12/3/2014 2801 12/10/2014 2830 12/15/2014 2946 12/17/2014 2946 12/22/2014 3085 Sierra Leone Cases 8/29/2014 935 9/5/2014 1146 9/8/2014 1234 9/12/2014 1287 9/16/2014 1464 9/18/2014 1513 9/22/2014 1640 9/24/2014 1745 9/26/2014 1816 10/1/2014 2076 10/3/2014 2179 10/8/2014 2455 12/23/2014 3085 12/24/2014 3085 12/26/2014 3085 12/29/2014 3108 12/30/2014 3108 12/31/2014 3110 1/2/2015 3110 1/5/2015 3116 1/6/2015 3118 1/7/2015 3118 1/8/2015 3118 1/9/2015 3123 1/12/2015 3123 1/13/2015 3123 1/14/2015 3127 1/15/2015 3127 1/16/2015 3127 1/19/2015 3131 1/20/2015 3135 1/21/2015 3135 1/22/2015 3136 1/23/2015 3136 1/26/2015 3136 1/27/2015 3138 1/28/2015 3138 1/29/2015 3138 1/30/2015 3138 2/2/2015 3138 2/3/2015 3143 2/4/2015 3143 2/5/2015 3143 2/6/2015 3143 10/10/2014 2593 10/15/2014 2849 10/17/2014 2977 10/22/2014 3223 10/25/2014 3389 10/29/2014 3700 10/31/2014 3778 11/5/2014 4057 11/7/2014 4149 11/12/2014 4523 11/14/2014 4683 11/19/2014 5056 11/21/2014 5152 2/10/2015 3146 2/11/2015 3146 2/12/2015 3146 2/13/2015 3147 2/16/2015 3149 2/17/2015 3149 2/18/2015 3149 2/19/2015 3152 2/20/2015 3152 2/23/2015 3153 2/25/2015 3153 2/26/2015 3153 2/27/2015 3153 3/2/2015 3153 3/4/2015 3153 3/5/2015 3150 3/6/2015 3150 3/10/2015 3150 3/11/2015 3150 3/12/2015 3150 3/13/2015 3150 3/16/2015 3150 3/17/2015 3150 3/18/2015 3150 3/20/2015 3150 3/23/2015 3151 3/24/2015 3151 3/25/2015 3151 3/26/2015 3151 3/27/2015 3151 11/26/2014 5441 12/1/2014 5831 12/3/2014 5978 12/10/2014 6375 12/15/2014 6638 12/17/2014 6702 12/22/2014 6975 12/23/2014 7017 12/24/2014 7017 12/26/2014 7160 12/29/2014 7326 12/30/2014 7354 12/31/2014 7354 27

1/2/2015 7476 1/5/2015 7570 1/6/2015 7602 1/7/2015 7602 1/8/2015 7759 1/9/2015 7759 1/12/2015 7766 1/13/2015 7766 1/14/2015 7786 1/15/2015 7802 1/16/2015 7825 1/19/2015 7883 1/20/2015 7903 1/21/2015 7903 1/22/2015 7909 1/23/2015 7921 1/26/2015 7963 1/27/2015 7968 1/28/2015 7968 Guinea Deaths 8/29/2014 287 9/5/2014 362 9/8/2014 400 9/12/2014 403 9/16/2014 429 9/18/2014 435 9/22/2014 465 9/24/2014 468 9/26/2014 481 10/1/2014 535 10/3/2014 562 10/8/2014 587 10/10/2014 598 10/15/2014 653 10/17/2014 671 10/22/2014 710 10/25/2014 732 10/29/2014 732* 10/31/2014 732* 11/5/2014 837 11/7/2014 850 11/12/2014 934 11/14/2014 958 11/19/2014 982 11/21/2014 998 1/29/2015 7977 1/30/2015 7989 2/2/2015 8042 2/3/2015 8059 2/4/2015 8059 2/5/2015 8063 2/6/2015 8084 2/10/2015 8135 2/11/2015 8135 2/12/2015 8138 2/13/2015 8155 2/16/2015 8199 2/17/2015 8212 2/18/2015 8212 2/19/2015 8223 2/20/2015 8223 2/23/2015 8223 2/25/2015 8289 2/26/2015 8308 11/26/2014 1050 12/1/2014 1102 12/3/2014 1117 12/10/2014 1207 12/15/2014 1255 12/17/2014 1262 12/22/2014 1323 12/23/2014 1344 12/24/2014 1344 12/26/2014 1385 12/29/2014 1422 12/30/2014 1433 12/31/2014 1433 1/2/2015 1463 1/5/2015 1488 1/6/2015 1499 1/7/2015 1499 1/8/2015 1504 1/9/2015 1515 1/12/2015 1523 1/13/2015 1523 1/14/2015 1530 1/15/2015 1537 1/16/2015 1541 1/19/2015 1556 1/20/2015 1557 2/27/2015 8320 3/2/2015 8353 3/4/2015 8353 3/5/2015 8383 3/6/2015 8389 3/10/2015 8428 3/11/2015 8428 3/12/2015 8463 3/13/2015 8469 3/16/2015 8484 3/17/2015 8487 3/18/2015 8487 3/20/2015 8508 3/23/2015 8518 3/24/2015 8520 3/25/2015 8520 3/26/2015 8529 3/27/2015 8532 1/21/2015 1557 1/22/2015 1560 1/23/2015 1561 1/26/2015 1574 1/27/2015 1578 1/28/2015 1578 1/29/2015 1579 1/30/2015 1581 2/2/2015 1595 2/3/2015 1597 2/4/2015 1597 2/5/2015 1600 2/6/2015 1608 2/10/2015 1643 2/11/2015 1643 2/12/2015 1651 2/13/2015 1659 2/16/2015 1678 2/17/2015 1683 2/18/2015 1683 2/19/2015 1686 2/20/2015 1688 2/23/2015 1699 2/25/2015 1704 2/26/2015 1714 2/27/2015 1721 28

3/2/2015 1735 3/4/2015 1735 3/5/2015 1749 3/6/2015 1755 3/10/2015 1778 3/11/2015 1778 Liberia Deaths 8/29/2014 225 9/5/2014 431 9/8/2014 508 9/12/2014 498 9/16/2014 563 9/18/2014 631 Sierra Leone Deaths 8/29/2014 380 9/5/2014 443 9/8/2014 461 9/12/2014 478 9/16/2014 514 9/18/2014 517 9/22/2014 545 9/24/2014 552 9/26/2014 557 10/1/2014 574 10/3/2014 575 10/8/2014 725 10/10/2014 753 10/15/2014 926 10/17/2014 932 10/22/2014 986 10/25/2014 1008 10/29/2014 1008* 10/31/2014 1008* 11/5/2014 893 11/7/2014 921 11/12/2014 960 11/14/2014 978 11/19/2014 1041 11/21/2014 1058 11/26/2014 1189 12/1/2014 1321 12/3/2014 1374 12/10/2014 1559 12/15/2014 1824 12/17/2014 1876 3/12/2015 1792 3/13/2015 1797 3/16/2015 1821 3/17/2015 1829 3/18/2015 1829 3/20/2015 1846 9/22/2014 670 9/24/2014 671 9/26/2014 792 10/1/2014 890 10/3/2014 934 10/8/2014 1018 10/10/2014 1072 12/22/2014 2190 12/23/2014 2216 12/24/2014 2216 12/26/2014 2289 12/29/2014 2366 12/30/2014 2392 12/31/2014 2392 1/2/2015 2461 1/5/2015 2549 1/6/2015 2577 1/7/2015 2577 1/8/2015 2611 1/9/2015 2611 1/12/2015 2683 1/13/2015 2683 1/14/2015 2696 1/15/2015 2701 1/16/2015 2717 1/19/2015 2766 1/20/2015 2779 1/21/2015 2779 1/22/2015 2787 1/23/2015 2793 1/26/2015 2829 1/27/2015 2833 1/28/2015 2833 1/29/2015 2841 1/30/2015 2850 2/2/2015 2908 2/3/2015 2910 2/4/2015 2910 2/5/2015 2920 29 3/23/2015 1863 3/24/2015 1865 3/25/2015 1865 3/26/2015 1875 3/27/2015 1878 10/22/2014 1241 10/25/2014 1241 2/10/2015 3826 2/11/2015 3826 2/12/2015 3826 2/13/2015 3858 2/6/2015 2935 2/10/2015 2975 2/11/2015 2975 2/12/2015 2984 2/13/2015 2997 2/16/2015 3035 2/17/2015 3042 2/18/2015 3042 2/19/2015 3057 2/20/2015 3057 2/23/2015 3057 2/25/2015 3095 2/26/2015 3113 2/27/2015 3124 3/2/2015 3164 3/4/2015 3164 3/5/2015 3199 3/6/2015 3210 3/10/2015 3263 3/11/2015 3263 3/12/2015 3289 3/13/2015 3297 3/16/2015 3321 3/17/2015 3325 3/18/2015 3325 3/20/2015 3346 3/23/2015 3376 3/24/2015 3381 3/25/2015 3381 3/26/2015 3398 3/27/2015 3407

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