Epidemiology, Equilibria, and Ebola: A Dynamical Analysis

Size: px
Start display at page:

Download "Epidemiology, Equilibria, and Ebola: A Dynamical Analysis"

Transcription

1 Epidemiology, Equilibria, and Ebola: A Dynamical Analysis Graham Harper Department of Mathematics Colorado State University - Pueblo May 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 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

3 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

4 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

5 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

6 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 I 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

7 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

8 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

9 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

10 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

11 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 = = 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

12 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 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 = 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 = 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

13 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 = 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

14 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

15 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

16 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 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

17 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

18 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( , ) = , and for Guinea, the minimum is G( , ) = 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 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

19 Table 3: Min(G(β, γ)) Based on Assumed Population Size Population (S + I) Guinea Sierra Leone 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 is assumed. Looking back at the SIJDR model, this means that G( , ) = for Guinea and G( , ) = for Sierra Leone are better estimates of the parameter, but these also help predict the end behavior of the disease. 19

20 6 Results With the refined values of β = , γ = for Guinea and β = , γ = 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 cases and a total of deaths by May 25, 2017, which corresponds to t = Assuming 20

21 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 cases and a total of deaths by May 25, 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

22 7 Code Index 7.1 Plotting Figures To plot Figure 1: sol = (NDSolve[ {s [t] == -s[t] i[t] i[t], e [t] == s[t] i[t] e[t], i [t] == 0.2 e[t] 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

23 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

24 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

25 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

26 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 where much more Ebola data is stored, but this specific data came from the csv file on 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/ /5/ /8/ /12/ /16/ /18/ /22/ /24/ /26/ /1/ /3/ /8/ /10/ /15/ /17/ /22/ /25/ /29/ /31/ /5/ /7/ /12/ /14/ /19/ /21/ /26/ /1/ /3/ /10/ /15/ /17/ /22/ /23/ /24/ /26/ /29/ /30/ /31/ /2/ /5/ /6/ /7/ /8/ /9/ /12/ /13/ /14/ /15/ /16/ /19/ /20/ /21/ /22/ /23/ /26/ /27/ /28/ /29/ /30/ /2/ /3/ /4/ /5/ /6/ /10/ /11/ /12/ /13/ /16/ /17/ /18/ /19/ /20/ /23/ /25/ /26/ /27/ /2/ /4/ /5/ /6/ /10/ /11/ /12/ /13/ /16/ /17/ /18/ /20/ /23/ /24/ /25/ /26/ /27/

27 Liberia Cases 8/29/ /5/ /8/ /12/ /16/ /18/ /22/ /24/ /26/ /1/ /3/ /8/ /10/ /15/ /22/ /25/ /29/ /31/ /5/ /7/ /12/ /14/ /19/ /21/ /26/ /1/ /3/ /10/ /15/ /17/ /22/ Sierra Leone Cases 8/29/ /5/ /8/ /12/ /16/ /18/ /22/ /24/ /26/ /1/ /3/ /8/ /23/ /24/ /26/ /29/ /30/ /31/ /2/ /5/ /6/ /7/ /8/ /9/ /12/ /13/ /14/ /15/ /16/ /19/ /20/ /21/ /22/ /23/ /26/ /27/ /28/ /29/ /30/ /2/ /3/ /4/ /5/ /6/ /10/ /15/ /17/ /22/ /25/ /29/ /31/ /5/ /7/ /12/ /14/ /19/ /21/ /10/ /11/ /12/ /13/ /16/ /17/ /18/ /19/ /20/ /23/ /25/ /26/ /27/ /2/ /4/ /5/ /6/ /10/ /11/ /12/ /13/ /16/ /17/ /18/ /20/ /23/ /24/ /25/ /26/ /27/ /26/ /1/ /3/ /10/ /15/ /17/ /22/ /23/ /24/ /26/ /29/ /30/ /31/

28 1/2/ /5/ /6/ /7/ /8/ /9/ /12/ /13/ /14/ /15/ /16/ /19/ /20/ /21/ /22/ /23/ /26/ /27/ /28/ Guinea Deaths 8/29/ /5/ /8/ /12/ /16/ /18/ /22/ /24/ /26/ /1/ /3/ /8/ /10/ /15/ /17/ /22/ /25/ /29/ * 10/31/ * 11/5/ /7/ /12/ /14/ /19/ /21/ /29/ /30/ /2/ /3/ /4/ /5/ /6/ /10/ /11/ /12/ /13/ /16/ /17/ /18/ /19/ /20/ /23/ /25/ /26/ /26/ /1/ /3/ /10/ /15/ /17/ /22/ /23/ /24/ /26/ /29/ /30/ /31/ /2/ /5/ /6/ /7/ /8/ /9/ /12/ /13/ /14/ /15/ /16/ /19/ /20/ /27/ /2/ /4/ /5/ /6/ /10/ /11/ /12/ /13/ /16/ /17/ /18/ /20/ /23/ /24/ /25/ /26/ /27/ /21/ /22/ /23/ /26/ /27/ /28/ /29/ /30/ /2/ /3/ /4/ /5/ /6/ /10/ /11/ /12/ /13/ /16/ /17/ /18/ /19/ /20/ /23/ /25/ /26/ /27/

29 3/2/ /4/ /5/ /6/ /10/ /11/ Liberia Deaths 8/29/ /5/ /8/ /12/ /16/ /18/ Sierra Leone Deaths 8/29/ /5/ /8/ /12/ /16/ /18/ /22/ /24/ /26/ /1/ /3/ /8/ /10/ /15/ /17/ /22/ /25/ /29/ * 10/31/ * 11/5/ /7/ /12/ /14/ /19/ /21/ /26/ /1/ /3/ /10/ /15/ /17/ /12/ /13/ /16/ /17/ /18/ /20/ /22/ /24/ /26/ /1/ /3/ /8/ /10/ /22/ /23/ /24/ /26/ /29/ /30/ /31/ /2/ /5/ /6/ /7/ /8/ /9/ /12/ /13/ /14/ /15/ /16/ /19/ /20/ /21/ /22/ /23/ /26/ /27/ /28/ /29/ /30/ /2/ /3/ /4/ /5/ /23/ /24/ /25/ /26/ /27/ /22/ /25/ /10/ /11/ /12/ /13/ /6/ /10/ /11/ /12/ /13/ /16/ /17/ /18/ /19/ /20/ /23/ /25/ /26/ /27/ /2/ /4/ /5/ /6/ /10/ /11/ /12/ /13/ /16/ /17/ /18/ /20/ /23/ /24/ /25/ /26/ /27/

30 References [1] Roy M Anderson and Robert M May. Population biology of infectious diseases: Part I. In: Nature 280 (1979), pp [2] Gerardo Chowell et al. SARS outbreaks in Ontario, Hong Kong and Singapore: the role of diagnosis and isolation as a control mechanism. In: Journal of Theoretical Biology (2003), pp [3] Herbert W Hethcote. The mathematics of infectious diseases. In: SIAM review 42.4 (2000), pp [4] John A Jacquez and Philip O Neill. Reproduction numbers and thresholds in stochastic epidemic models I. Homogeneous populations. In: Mathematical Biosciences (1991), pp [5] Mart CM de Jong, Odo Diekmann, and Hans Heesterbeek. How does transmission of infection depend on population size. In: Epidemic models: their structure and relation to data 5.2 (1995), pp [6] Denis Mollison. Epidemic models: their structure and relation to data. Vol. 5. Cambridge University Press, [7] S Towers and Z Feng. Pandemic H1N1 influenza: Predicting the course of vaccination programme in the united states [8] Sherry Towers, Oscar Patterson-Lomba, and Carlos Castillo-Chavez. Temporal variations in the effective reproduction number of the 2014 West Africa Ebola outbreak. In: PLoS currents 6 (2014). 30

Introduction to SEIR Models

Introduction to SEIR Models Department of Epidemiology and Public Health Health Systems Research and Dynamical Modelling Unit Introduction to SEIR Models Nakul Chitnis Workshop on Mathematical Models of Climate Variability, Environmental

More information

Mathematical modelling and controlling the dynamics of infectious diseases

Mathematical modelling and controlling the dynamics of infectious diseases Mathematical modelling and controlling the dynamics of infectious diseases Musa Mammadov Centre for Informatics and Applied Optimisation Federation University Australia 25 August 2017, School of Science,

More information

Thursday. Threshold and Sensitivity Analysis

Thursday. Threshold and Sensitivity Analysis Thursday Threshold and Sensitivity Analysis SIR Model without Demography ds dt di dt dr dt = βsi (2.1) = βsi γi (2.2) = γi (2.3) With initial conditions S(0) > 0, I(0) > 0, and R(0) = 0. This model can

More information

Markov Chains and Pandemics

Markov Chains and Pandemics Markov Chains and Pandemics Caleb Dedmore and Brad Smith December 8, 2016 Page 1 of 16 Abstract Markov Chain Theory is a powerful tool used in statistical analysis to make predictions about future events

More information

Non-Linear Models Cont d: Infectious Diseases. Non-Linear Models Cont d: Infectious Diseases

Non-Linear Models Cont d: Infectious Diseases. Non-Linear Models Cont d: Infectious Diseases Cont d: Infectious Diseases Infectious Diseases Can be classified into 2 broad categories: 1 those caused by viruses & bacteria (microparasitic diseases e.g. smallpox, measles), 2 those due to vectors

More information

Epidemics in Networks Part 2 Compartmental Disease Models

Epidemics in Networks Part 2 Compartmental Disease Models Epidemics in Networks Part 2 Compartmental Disease Models Joel C. Miller & Tom Hladish 18 20 July 2018 1 / 35 Introduction to Compartmental Models Dynamics R 0 Epidemic Probability Epidemic size Review

More information

Mathematical Analysis of Epidemiological Models: Introduction

Mathematical Analysis of Epidemiological Models: Introduction Mathematical Analysis of Epidemiological Models: Introduction Jan Medlock Clemson University Department of Mathematical Sciences 8 February 2010 1. Introduction. The effectiveness of improved sanitation,

More information

CS224W: Analysis of Networks Jure Leskovec, Stanford University

CS224W: Analysis of Networks Jure Leskovec, Stanford University Announcements: Please fill HW Survey Weekend Office Hours starting this weekend (Hangout only) Proposal: Can use 1 late period CS224W: Analysis of Networks Jure Leskovec, Stanford University http://cs224w.stanford.edu

More information

Australian Journal of Basic and Applied Sciences

Australian Journal of Basic and Applied Sciences AENSI Journals Australian Journal of Basic and Applied Sciences ISSN:1991-8178 Journal home page: www.ajbasweb.com A SIR Transmission Model of Political Figure Fever 1 Benny Yong and 2 Nor Azah Samat 1

More information

Mathematical Modeling and Analysis of Infectious Disease Dynamics

Mathematical Modeling and Analysis of Infectious Disease Dynamics Mathematical Modeling and Analysis of Infectious Disease Dynamics V. A. Bokil Department of Mathematics Oregon State University Corvallis, OR MTH 323: Mathematical Modeling May 22, 2017 V. A. Bokil (OSU-Math)

More information

Stochastic modelling of epidemic spread

Stochastic modelling of epidemic spread Stochastic modelling of epidemic spread Julien Arino Centre for Research on Inner City Health St Michael s Hospital Toronto On leave from Department of Mathematics University of Manitoba Julien Arino@umanitoba.ca

More information

A Note on the Spread of Infectious Diseases. in a Large Susceptible Population

A Note on the Spread of Infectious Diseases. in a Large Susceptible Population International Mathematical Forum, Vol. 7, 2012, no. 50, 2481-2492 A Note on the Spread of Infectious Diseases in a Large Susceptible Population B. Barnes Department of Mathematics Kwame Nkrumah University

More information

Approximation of epidemic models by diffusion processes and their statistical inferencedes

Approximation of epidemic models by diffusion processes and their statistical inferencedes Approximation of epidemic models by diffusion processes and their statistical inferencedes Catherine Larédo 1,2 1 UR 341, MaIAGE, INRA, Jouy-en-Josas 2 UMR 7599, LPMA, Université Paris Diderot December

More information

University of Minnesota Duluth Department of Mathematics and Statistics. Modeling of Ebola Control Strategies

University of Minnesota Duluth Department of Mathematics and Statistics. Modeling of Ebola Control Strategies University of Minnesota Duluth Department of Mathematics and Statistics Modeling of Ebola Control Strategies Duluth, May 216 Václav Hasenöhrl Acknowledgments I would like to express my appreciation to

More information

Models of Infectious Disease Formal Demography Stanford Summer Short Course James Holland Jones, Instructor. August 15, 2005

Models of Infectious Disease Formal Demography Stanford Summer Short Course James Holland Jones, Instructor. August 15, 2005 Models of Infectious Disease Formal Demography Stanford Summer Short Course James Holland Jones, Instructor August 15, 2005 1 Outline 1. Compartmental Thinking 2. Simple Epidemic (a) Epidemic Curve 1:

More information

Control of Epidemics by Vaccination

Control of Epidemics by Vaccination Control of Epidemics by Vaccination Erik Verriest, Florent Delmotte, and Magnus Egerstedt {erik.verriest,florent,magnus}@ece.gatech.edu School of Electrical and Computer Engineering Georgia Institute of

More information

arxiv: v2 [q-bio.pe] 7 Nov 2015

arxiv: v2 [q-bio.pe] 7 Nov 2015 Modeling Contact Tracing in Outbreaks with Application to Ebola Cameron Browne a,, Hayriye Gulbudak b,c, Glenn Webb a a Department of Mathematics, Vanderbilt University b School of Biology, Georgia Institute

More information

The SIRS Model Approach to Host/Parasite Relationships

The SIRS Model Approach to Host/Parasite Relationships = B I + y (N I ) 1 8 6 4 2 I = B I v I N = 5 v = 25 The IR Model Approach to Host/Parasite Relationships Brianne Gill May 16, 28 5 1 15 2 The IR Model... Abstract In this paper, we shall explore examples

More information

Electronic appendices are refereed with the text. However, no attempt has been made to impose a uniform editorial style on the electronic appendices.

Electronic appendices are refereed with the text. However, no attempt has been made to impose a uniform editorial style on the electronic appendices. This is an electronic appendix to the paper by Alun L. Lloyd 2001 Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods. Proc. R. Soc. Lond. B 268, 985-993.

More information

Global Analysis of an SEIRS Model with Saturating Contact Rate 1

Global Analysis of an SEIRS Model with Saturating Contact Rate 1 Applied Mathematical Sciences, Vol. 6, 2012, no. 80, 3991-4003 Global Analysis of an SEIRS Model with Saturating Contact Rate 1 Shulin Sun a, Cuihua Guo b, and Chengmin Li a a School of Mathematics and

More information

Name Student ID. Good luck and impress us with your toolkit of ecological knowledge and concepts!

Name Student ID. Good luck and impress us with your toolkit of ecological knowledge and concepts! Page 1 BIOLOGY 150 Final Exam Winter Quarter 2000 Before starting be sure to put your name and student number on the top of each page. MINUS 3 POINTS IF YOU DO NOT WRITE YOUR NAME ON EACH PAGE! You have

More information

Systems of Ordinary Differential Equations

Systems of Ordinary Differential Equations Systems of Ordinary Differential Equations Scott A. McKinley October 22, 2013 In these notes, which replace the material in your textbook, we will learn a modern view of analyzing systems of differential

More information

Global Stability of a Computer Virus Model with Cure and Vertical Transmission

Global Stability of a Computer Virus Model with Cure and Vertical Transmission International Journal of Research Studies in Computer Science and Engineering (IJRSCSE) Volume 3, Issue 1, January 016, PP 16-4 ISSN 349-4840 (Print) & ISSN 349-4859 (Online) www.arcjournals.org Global

More information

Kasetsart University Workshop. Mathematical modeling using calculus & differential equations concepts

Kasetsart University Workshop. Mathematical modeling using calculus & differential equations concepts Kasetsart University Workshop Mathematical modeling using calculus & differential equations concepts Dr. Anand Pardhanani Mathematics Department Earlham College Richmond, Indiana USA pardhan@earlham.edu

More information

Network Models in Epidemiology

Network Models in Epidemiology Network Models in Epidemiology Considering Discrete and Continuous Dynamics arxiv:1511.01062v1 [q-bio.pe] 19 Oct 2015 Edward Rusu University of Washington email epr24@uw.edu 24 April 2014 Abstract Discrete

More information

Applications in Biology

Applications in Biology 11 Applications in Biology In this chapter we make use of the techniques developed in the previous few chapters to examine some nonlinear systems that have been used as mathematical models for a variety

More information

Models of Infectious Disease Formal Demography Stanford Spring Workshop in Formal Demography May 2008

Models of Infectious Disease Formal Demography Stanford Spring Workshop in Formal Demography May 2008 Models of Infectious Disease Formal Demography Stanford Spring Workshop in Formal Demography May 2008 James Holland Jones Department of Anthropology Stanford University May 3, 2008 1 Outline 1. Compartmental

More information

Mathematical Epidemiology Lecture 1. Matylda Jabłońska-Sabuka

Mathematical Epidemiology Lecture 1. Matylda Jabłońska-Sabuka Lecture 1 Lappeenranta University of Technology Wrocław, Fall 2013 What is? Basic terminology Epidemiology is the subject that studies the spread of diseases in populations, and primarily the human populations.

More information

Predator-Prey Population Dynamics

Predator-Prey Population Dynamics Predator-Prey Population Dynamics Gonzalo Mateos Dept. of ECE and Goergen Institute for Data Science University of Rochester gmateosb@ece.rochester.edu http://www.ece.rochester.edu/~gmateosb/ October 2,

More information

Stochastic modelling of epidemic spread

Stochastic modelling of epidemic spread Stochastic modelling of epidemic spread Julien Arino Department of Mathematics University of Manitoba Winnipeg Julien Arino@umanitoba.ca 19 May 2012 1 Introduction 2 Stochastic processes 3 The SIS model

More information

The dynamics of disease transmission in a Prey Predator System with harvesting of prey

The dynamics of disease transmission in a Prey Predator System with harvesting of prey ISSN: 78 Volume, Issue, April The dynamics of disease transmission in a Prey Predator System with harvesting of prey, Kul Bhushan Agnihotri* Department of Applied Sciences and Humanties Shaheed Bhagat

More information

LAW OF LARGE NUMBERS FOR THE SIRS EPIDEMIC

LAW OF LARGE NUMBERS FOR THE SIRS EPIDEMIC LAW OF LARGE NUMBERS FOR THE SIRS EPIDEMIC R. G. DOLGOARSHINNYKH Abstract. We establish law of large numbers for SIRS stochastic epidemic processes: as the population size increases the paths of SIRS epidemic

More information

On the Spread of Epidemics in a Closed Heterogeneous Population

On the Spread of Epidemics in a Closed Heterogeneous Population On the Spread of Epidemics in a Closed Heterogeneous Population Artem Novozhilov Applied Mathematics 1 Moscow State University of Railway Engineering (MIIT) the 3d Workshop on Mathematical Models and Numerical

More information

MODELING THE SPREAD OF DENGUE FEVER BY USING SIR MODEL. Hor Ming An, PM. Dr. Yudariah Mohammad Yusof

MODELING THE SPREAD OF DENGUE FEVER BY USING SIR MODEL. Hor Ming An, PM. Dr. Yudariah Mohammad Yusof MODELING THE SPREAD OF DENGUE FEVER BY USING SIR MODEL Hor Ming An, PM. Dr. Yudariah Mohammad Yusof Abstract The establishment and spread of dengue fever is a complex phenomenon with many factors that

More information

HETEROGENEOUS MIXING IN EPIDEMIC MODELS

HETEROGENEOUS MIXING IN EPIDEMIC MODELS CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 2, Number 1, Spring 212 HETEROGENEOUS MIXING IN EPIDEMIC MODELS FRED BRAUER ABSTRACT. We extend the relation between the basic reproduction number and the

More information

Analysis of Numerical and Exact solutions of certain SIR and SIS Epidemic models

Analysis of Numerical and Exact solutions of certain SIR and SIS Epidemic models Journal of Mathematical Modelling and Application 2011, Vol. 1, No. 4, 51-56 ISSN: 2178-2423 Analysis of Numerical and Exact solutions of certain SIR and SIS Epidemic models S O Maliki Department of Industrial

More information

MATHEMATICAL MODELS Vol. III - Mathematical Models in Epidemiology - M. G. Roberts, J. A. P. Heesterbeek

MATHEMATICAL MODELS Vol. III - Mathematical Models in Epidemiology - M. G. Roberts, J. A. P. Heesterbeek MATHEMATICAL MODELS I EPIDEMIOLOGY M. G. Roberts Institute of Information and Mathematical Sciences, Massey University, Auckland, ew Zealand J. A. P. Heesterbeek Faculty of Veterinary Medicine, Utrecht

More information

Project 1 Modeling of Epidemics

Project 1 Modeling of Epidemics 532 Chapter 7 Nonlinear Differential Equations and tability ection 7.5 Nonlinear systems, unlike linear systems, sometimes have periodic solutions, or limit cycles, that attract other nearby solutions.

More information

Modelling of the Hand-Foot-Mouth-Disease with the Carrier Population

Modelling of the Hand-Foot-Mouth-Disease with the Carrier Population Modelling of the Hand-Foot-Mouth-Disease with the Carrier Population Ruzhang Zhao, Lijun Yang Department of Mathematical Science, Tsinghua University, China. Corresponding author. Email: lyang@math.tsinghua.edu.cn,

More information

Figure The Threshold Theorem of epidemiology

Figure The Threshold Theorem of epidemiology K/a Figure 3 6. Assurne that K 1/ a < K 2 and K 2 / ß < K 1 (a) Show that the equilibrium solution N 1 =0, N 2 =0 of (*) is unstable. (b) Show that the equilibrium solutions N 2 =0 and N 1 =0, N 2 =K 2

More information

Stability of SEIR Model of Infectious Diseases with Human Immunity

Stability of SEIR Model of Infectious Diseases with Human Immunity Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 1811 1819 Research India Publications http://www.ripublication.com/gjpam.htm Stability of SEIR Model of Infectious

More information

Age-dependent branching processes with incubation

Age-dependent branching processes with incubation Age-dependent branching processes with incubation I. RAHIMOV Department of Mathematical Sciences, KFUPM, Box. 1339, Dhahran, 3161, Saudi Arabia e-mail: rahimov @kfupm.edu.sa We study a modification of

More information

M469, Fall 2010, Practice Problems for the Final

M469, Fall 2010, Practice Problems for the Final M469 Fall 00 Practice Problems for the Final The final exam for M469 will be Friday December 0 3:00-5:00 pm in the usual classroom Blocker 60 The final will cover the following topics from nonlinear systems

More information

Three Disguises of 1 x = e λx

Three Disguises of 1 x = e λx Three Disguises of 1 x = e λx Chathuri Karunarathna Mudiyanselage Rabi K.C. Winfried Just Department of Mathematics, Ohio University Mathematical Biology and Dynamical Systems Seminar Ohio University November

More information

Spotlight on Modeling: The Possum Plague

Spotlight on Modeling: The Possum Plague 70 Spotlight on Modeling: The Possum Plague Reference: Sections 2.6, 7.2 and 7.3. The ecological balance in New Zealand has been disturbed by the introduction of the Australian possum, a marsupial the

More information

EPIDEMIC MODELS I REPRODUCTION NUMBERS AND FINAL SIZE RELATIONS FRED BRAUER

EPIDEMIC MODELS I REPRODUCTION NUMBERS AND FINAL SIZE RELATIONS FRED BRAUER EPIDEMIC MODELS I REPRODUCTION NUMBERS AND FINAL SIZE RELATIONS FRED BRAUER 1 THE SIR MODEL Start with simple SIR epidemic model with initial conditions S = βsi I = (βs α)i, S() = S, I() = I, S + I = N

More information

Princeton University Press, all rights reserved. Chapter 10: Dynamics of Class-Structured Populations

Princeton University Press, all rights reserved. Chapter 10: Dynamics of Class-Structured Populations Supplementary material to: Princeton University Press, all rights reserved From: Chapter 10: Dynamics of Class-Structured Populations A Biologist s Guide to Mathematical Modeling in Ecology and Evolution

More information

Modeling the Immune System W9. Ordinary Differential Equations as Macroscopic Modeling Tool

Modeling the Immune System W9. Ordinary Differential Equations as Macroscopic Modeling Tool Modeling the Immune System W9 Ordinary Differential Equations as Macroscopic Modeling Tool 1 Lecture Notes for ODE Models We use the lecture notes Theoretical Fysiology 2006 by Rob de Boer, U. Utrecht

More information

PARAMETER ESTIMATION IN EPIDEMIC MODELS: SIMPLIFIED FORMULAS

PARAMETER ESTIMATION IN EPIDEMIC MODELS: SIMPLIFIED FORMULAS CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 19, Number 4, Winter 211 PARAMETER ESTIMATION IN EPIDEMIC MODELS: SIMPLIFIED FORMULAS Dedicated to Herb Freedman on the occasion of his seventieth birthday

More information

Fixed Point Analysis of Kermack Mckendrick SIR Model

Fixed Point Analysis of Kermack Mckendrick SIR Model Kalpa Publications in Computing Volume, 17, Pages 13 19 ICRISET17. International Conference on Research and Innovations in Science, Engineering &Technology. Selected Papers in Computing Fixed Point Analysis

More information

Epidemics in Complex Networks and Phase Transitions

Epidemics in Complex Networks and Phase Transitions Master M2 Sciences de la Matière ENS de Lyon 2015-2016 Phase Transitions and Critical Phenomena Epidemics in Complex Networks and Phase Transitions Jordan Cambe January 13, 2016 Abstract Spreading phenomena

More information

SIR model. (Susceptible-Infected-Resistant/Removed) Outlook. Introduction into SIR model. Janusz Szwabiński

SIR model. (Susceptible-Infected-Resistant/Removed) Outlook. Introduction into SIR model. Janusz Szwabiński SIR model (Susceptible-Infected-Resistant/Removed) Janusz Szwabiński Outlook Introduction into SIR model Analytical approximation Numerical solution Numerical solution on a grid Simulation on networks

More information

GLOBAL DYNAMICS OF A MATHEMATICAL MODEL OF TUBERCULOSIS

GLOBAL DYNAMICS OF A MATHEMATICAL MODEL OF TUBERCULOSIS CANADIAN APPIED MATHEMATICS QUARTERY Volume 13, Number 4, Winter 2005 GOBA DYNAMICS OF A MATHEMATICA MODE OF TUBERCUOSIS HONGBIN GUO ABSTRACT. Mathematical analysis is carried out for a mathematical model

More information

(e) Use Newton s method to find the x coordinate that satisfies this equation, and your graph in part (b) to show that this is an inflection point.

(e) Use Newton s method to find the x coordinate that satisfies this equation, and your graph in part (b) to show that this is an inflection point. Chapter 6 Review problems 6.1 A strange function Consider the function (x 2 ) x. (a) Show that this function can be expressed as f(x) = e x ln(x2). (b) Use the spreadsheet, and a fine subdivision of the

More information

MA 138 Calculus 2 for the Life Sciences Spring 2016 Final Exam May 4, Exam Scores. Question Score Total

MA 138 Calculus 2 for the Life Sciences Spring 2016 Final Exam May 4, Exam Scores. Question Score Total MA 138 Calculus 2 for the Life Sciences Spring 2016 Final Exam May 4, 2016 Exam Scores Question Score Total 1 10 Name: Section: Last 4 digits of student ID #: No books or notes may be used. Turn off all

More information

Transmission in finite populations

Transmission in finite populations Transmission in finite populations Juliet Pulliam, PhD Department of Biology and Emerging Pathogens Institute University of Florida and RAPIDD Program, DIEPS Fogarty International Center US National Institutes

More information

ME 406 S-I-R Model of Epidemics Part 2 Vital Dynamics Included

ME 406 S-I-R Model of Epidemics Part 2 Vital Dynamics Included ME 406 S-I-R Model of Epidemics Part 2 Vital Dynamics Included sysid Mathematica 6.0.3, DynPac 11.01, 1ê13ê9 1. Introduction Description of the Model In this notebook, we include births and deaths in the

More information

Modeling the Spread of Epidemic Cholera: an Age-Structured Model

Modeling the Spread of Epidemic Cholera: an Age-Structured Model Modeling the Spread of Epidemic Cholera: an Age-Structured Model Alen Agheksanterian Matthias K. Gobbert November 20, 2007 Abstract Occasional outbreaks of cholera epidemics across the world demonstrate

More information

Estimating the Exponential Growth Rate and R 0

Estimating the Exponential Growth Rate and R 0 Junling Ma Department of Mathematics and Statistics, University of Victoria May 23, 2012 Introduction Daily pneumonia and influenza (P&I) deaths of 1918 pandemic influenza in Philadelphia. 900 800 700

More information

A NEW SOLUTION OF SIR MODEL BY USING THE DIFFERENTIAL FRACTIONAL TRANSFORMATION METHOD

A NEW SOLUTION OF SIR MODEL BY USING THE DIFFERENTIAL FRACTIONAL TRANSFORMATION METHOD April, 4. Vol. 4, No. - 4 EAAS & ARF. All rights reserved ISSN35-869 A NEW SOLUTION OF SIR MODEL BY USING THE DIFFERENTIAL FRACTIONAL TRANSFORMATION METHOD Ahmed A. M. Hassan, S. H. Hoda Ibrahim, Amr M.

More information

Dynamical models of HIV-AIDS e ect on population growth

Dynamical models of HIV-AIDS e ect on population growth Dynamical models of HV-ADS e ect on population growth David Gurarie May 11, 2005 Abstract We review some known dynamical models of epidemics, given by coupled systems of di erential equations, and propose

More information

The death of an epidemic

The death of an epidemic LECTURE 2 Equilibrium Stability Analysis & Next Generation Method The death of an epidemic In SIR equations, let s divide equation for dx/dt by dz/ dt:!! dx/dz = - (β X Y/N)/(γY)!!! = - R 0 X/N Integrate

More information

Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission

Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission P. van den Driessche a,1 and James Watmough b,2, a Department of Mathematics and Statistics, University

More information

An intrinsic connection between Richards model and SIR model

An intrinsic connection between Richards model and SIR model An intrinsic connection between Richards model and SIR model validation by and application to pandemic influenza data Xiang-Sheng Wang Mprime Centre for Disease Modelling York University, Toronto (joint

More information

Species 1 isocline. Species 2 isocline

Species 1 isocline. Species 2 isocline 1 Name BIOLOGY 150 Final Exam Winter Quarter 2002 Before starting please write your name on each page! Last name, then first name. You have tons of time. Take your time and read each question carefully

More information

Epidemic model for influenza A (H1N1)

Epidemic model for influenza A (H1N1) Epidemic model for influenza A (H1N1) Modeling the outbreak of the pandemic in Kolkata, West Bengal, India, 2010 Sandipan Dey India July 8, 2017 2 Summary In this report, the spread of the pandemic influenza

More information

7 Planar systems of linear ODE

7 Planar systems of linear ODE 7 Planar systems of linear ODE Here I restrict my attention to a very special class of autonomous ODE: linear ODE with constant coefficients This is arguably the only class of ODE for which explicit solution

More information

STABILITY ANALYSIS OF A GENERAL SIR EPIDEMIC MODEL

STABILITY ANALYSIS OF A GENERAL SIR EPIDEMIC MODEL VFAST Transactions on Mathematics http://vfast.org/index.php/vtm@ 2013 ISSN: 2309-0022 Volume 1, Number 1, May-June, 2013 pp. 16 20 STABILITY ANALYSIS OF A GENERAL SIR EPIDEMIC MODEL Roman Ullah 1, Gul

More information

University of Leeds. Project in Statistics. Math5004M. Epidemic Modelling. Supervisor: Dr Andrew J Baczkowski. Student: Ross Breckon

University of Leeds. Project in Statistics. Math5004M. Epidemic Modelling. Supervisor: Dr Andrew J Baczkowski. Student: Ross Breckon University of Leeds Project in Statistics Math5004M Epidemic Modelling Student: Ross Breckon 200602800 Supervisor: Dr Andrew J Baczkowski May 6, 2015 Abstract The outbreak of an infectious disease can

More information

SIR Epidemic Model with total Population size

SIR Epidemic Model with total Population size Advances in Applied Mathematical Biosciences. ISSN 2248-9983 Volume 7, Number 1 (2016), pp. 33-39 International Research Publication House http://www.irphouse.com SIR Epidemic Model with total Population

More information

APPM 2360 Lab 3: Zombies! Due April 27, 2017 by 11:59 pm

APPM 2360 Lab 3: Zombies! Due April 27, 2017 by 11:59 pm APPM 2360 Lab 3: Zombies! Due April 27, 2017 by 11:59 pm 1 Introduction As you already know, in the past month, zombies have overrun much of North America, including all major cities on both the East and

More information

ECS 289 / MAE 298, Lecture 7 April 22, Percolation and Epidemiology on Networks, Part 2 Searching on networks

ECS 289 / MAE 298, Lecture 7 April 22, Percolation and Epidemiology on Networks, Part 2 Searching on networks ECS 289 / MAE 298, Lecture 7 April 22, 2014 Percolation and Epidemiology on Networks, Part 2 Searching on networks 28 project pitches turned in Announcements We are compiling them into one file to share

More information

Available online at Commun. Math. Biol. Neurosci. 2015, 2015:29 ISSN:

Available online at   Commun. Math. Biol. Neurosci. 2015, 2015:29 ISSN: Available online at http://scik.org Commun. Math. Biol. Neurosci. 215, 215:29 ISSN: 252-2541 AGE-STRUCTURED MATHEMATICAL MODEL FOR HIV/AIDS IN A TWO-DIMENSIONAL HETEROGENEOUS POPULATION PRATIBHA RANI 1,

More information

The Estimation of the Effective Reproductive Number from Disease. Outbreak Data

The Estimation of the Effective Reproductive Number from Disease. Outbreak Data The Estimation of the Effective Reproductive Number from Disease Outbreak Data Ariel Cintrón-Arias 1,5 Carlos Castillo-Chávez 2, Luís M. A. Bettencourt 3, Alun L. Lloyd 4,5, and H. T. Banks 4,5 1 Statistical

More information

Introduction to Modeling

Introduction to Modeling Introduction to Modeling CURM Background Material, Fall 2014 Dr. Doreen De Leon 1 What Is a Mathematical Model? A mathematical model is an idealization of a real-world phenomenon, which we use to help

More information

Mathematical Model of Tuberculosis Spread within Two Groups of Infected Population

Mathematical Model of Tuberculosis Spread within Two Groups of Infected Population Applied Mathematical Sciences, Vol. 10, 2016, no. 43, 2131-2140 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2016.63130 Mathematical Model of Tuberculosis Spread within Two Groups of Infected

More information

Gerardo Zavala. Math 388. Predator-Prey Models

Gerardo Zavala. Math 388. Predator-Prey Models Gerardo Zavala Math 388 Predator-Prey Models Spring 2013 1 History In the 1920s A. J. Lotka developed a mathematical model for the interaction between two species. The mathematician Vito Volterra worked

More information

Mathematical Analysis of Epidemiological Models III

Mathematical Analysis of Epidemiological Models III Intro Computing R Complex models Mathematical Analysis of Epidemiological Models III Jan Medlock Clemson University Department of Mathematical Sciences 27 July 29 Intro Computing R Complex models What

More information

Introduction to Epidemic Modeling

Introduction to Epidemic Modeling Chapter 2 Introduction to Epidemic Modeling 2.1 Kermack McKendrick SIR Epidemic Model Introduction to epidemic modeling is usually made through one of the first epidemic models proposed by Kermack and

More information

Dynamics of Disease Spread. in a Predator-Prey System

Dynamics of Disease Spread. in a Predator-Prey System Advanced Studies in Biology, vol. 6, 2014, no. 4, 169-179 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/asb.2014.4845 Dynamics of Disease Spread in a Predator-Prey System Asrul Sani 1, Edi Cahyono

More information

Mathematical models on Malaria with multiple strains of pathogens

Mathematical models on Malaria with multiple strains of pathogens Mathematical models on Malaria with multiple strains of pathogens Yanyu Xiao Department of Mathematics University of Miami CTW: From Within Host Dynamics to the Epidemiology of Infectious Disease MBI,

More information

BIO S380T Page 1 Summer 2005: Exam 2

BIO S380T Page 1 Summer 2005: Exam 2 BIO S380T Page 1 Part I: Definitions. [5 points for each term] For each term, provide a brief definition that also indicates why the term is important in ecology or evolutionary biology. Where I ve provided

More information

1 Disease Spread Model

1 Disease Spread Model Technical Appendix for The Impact of Mass Gatherings and Holiday Traveling on the Course of an Influenza Pandemic: A Computational Model Pengyi Shi, Pinar Keskinocak, Julie L Swann, Bruce Y Lee December

More information

Multistate Modelling Vertical Transmission and Determination of R 0 Using Transition Intensities

Multistate Modelling Vertical Transmission and Determination of R 0 Using Transition Intensities Applied Mathematical Sciences, Vol. 9, 2015, no. 79, 3941-3956 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.52130 Multistate Modelling Vertical Transmission and Determination of R 0

More information

Food Chains. energy: what is needed to do work or cause change

Food Chains. energy: what is needed to do work or cause change Have you ever seen a picture that shows a little fish about to be eaten by a big fish? Sometimes the big fish has an even bigger fish behind it. This is a simple food chain. A food chain is the path of

More information

Epidemiology and Chaos Jonathan Wills Math 053: Chaos! Prof. Alex Barnett Fall Abstract:

Epidemiology and Chaos Jonathan Wills Math 053: Chaos! Prof. Alex Barnett Fall Abstract: Abstract: Epidemiology and Chaos Jonathan Wills Math 053: Chaos! Prof. Alex Barnett Fall 2009 Mathematical epidemiological models for the spread of disease through a population are used to predict the

More information

Delay SIR Model with Nonlinear Incident Rate and Varying Total Population

Delay SIR Model with Nonlinear Incident Rate and Varying Total Population Delay SIR Model with Nonlinear Incident Rate Varying Total Population Rujira Ouncharoen, Salinthip Daengkongkho, Thongchai Dumrongpokaphan, Yongwimon Lenbury Abstract Recently, models describing the behavior

More information

A Time Since Recovery Model with Varying Rates of Loss of Immunity

A Time Since Recovery Model with Varying Rates of Loss of Immunity Bull Math Biol (212) 74:281 2819 DOI 1.17/s11538-12-978-7 ORIGINAL ARTICLE A Time Since Recovery Model with Varying Rates of Loss of Immunity Subhra Bhattacharya Frederick R. Adler Received: 7 May 212

More information

STUDY OF THE DYNAMICAL MODEL OF HIV

STUDY OF THE DYNAMICAL MODEL OF HIV STUDY OF THE DYNAMICAL MODEL OF HIV M.A. Lapshova, E.A. Shchepakina Samara National Research University, Samara, Russia Abstract. The paper is devoted to the study of the dynamical model of HIV. An application

More information

Smoking as Epidemic: Modeling and Simulation Study

Smoking as Epidemic: Modeling and Simulation Study American Journal of Applied Mathematics 2017; 5(1): 31-38 http://www.sciencepublishinggroup.com/j/ajam doi: 10.11648/j.ajam.20170501.14 ISSN: 2330-0043 (Print); ISSN: 2330-006X (Online) Smoking as Epidemic:

More information

Australian Journal of Basic and Applied Sciences. Effect of Education Campaign on Transmission Model of Conjunctivitis

Australian Journal of Basic and Applied Sciences. Effect of Education Campaign on Transmission Model of Conjunctivitis ISSN:99-878 Australian Journal of Basic and Applied Sciences Journal home page: www.ajbasweb.com ffect of ducation Campaign on Transmission Model of Conjunctivitis Suratchata Sangthongjeen, Anake Sudchumnong

More information

Walking across a snowy field or mountain, you might not notice many living things. But if you dig into the snow, you ll find a lot of life!

Walking across a snowy field or mountain, you might not notice many living things. But if you dig into the snow, you ll find a lot of life! Beetle, it s cold outside! Featured scientists: Caroline Williams & Andre Szejner Sigal, University of California, Berkeley, & Nikki Chambers, Biology Teacher, West High School, Torrance, CA Research Background:

More information

MAS1302 Computational Probability and Statistics

MAS1302 Computational Probability and Statistics MAS1302 Computational Probability and Statistics April 23, 2008 3. Simulating continuous random behaviour 3.1 The Continuous Uniform U(0,1) Distribution We have already used this random variable a great

More information

Epidemic Modeling with Contact Heterogeneity and Multiple Routes of Transmission

Epidemic Modeling with Contact Heterogeneity and Multiple Routes of Transmission Epidemic Modeling with Contact Heterogeneity and Multiple Routes of Transmission An Undergraduate Study of the model first published by Kiss et al. in 2006, "The effect of contact heterogeneity and multiple

More information

Behavior Stability in two SIR-Style. Models for HIV

Behavior Stability in two SIR-Style. Models for HIV Int. Journal of Math. Analysis, Vol. 4, 2010, no. 9, 427-434 Behavior Stability in two SIR-Style Models for HIV S. Seddighi Chaharborj 2,1, M. R. Abu Bakar 2, I. Fudziah 2 I. Noor Akma 2, A. H. Malik 2,

More information

Supplemental Information Population Dynamics of Epidemic and Endemic States of Drug-Resistance Emergence in Infectious Diseases

Supplemental Information Population Dynamics of Epidemic and Endemic States of Drug-Resistance Emergence in Infectious Diseases 1 2 3 4 Supplemental Information Population Dynamics of Epidemic and Endemic States of Drug-Resistance Emergence in Infectious Diseases 5 Diána Knipl, 1 Gergely Röst, 2 Seyed M. Moghadas 3, 6 7 8 9 1 1

More information

Section 8.1 Def. and Examp. Systems

Section 8.1 Def. and Examp. Systems Section 8.1 Def. and Examp. Systems Key Terms: SIR Model of an epidemic o Nonlinear o Autonomous Vector functions o Derivative of vector functions Order of a DE system Planar systems Dimension of a system

More information

A Producer-Consumer Model With Stoichiometry

A Producer-Consumer Model With Stoichiometry A Producer-Consumer Model With Stoichiometry Plan B project toward the completion of the Master of Science degree in Mathematics at University of Minnesota Duluth Respectfully submitted by Laura Joan Zimmermann

More information

Copyright is owned by the Author of the thesis. Permission is given for a copy to be downloaded by an individual for the purpose of research and

Copyright is owned by the Author of the thesis. Permission is given for a copy to be downloaded by an individual for the purpose of research and Copyright is owned by the Author of the thesis. Permission is given for a copy to be downloaded by an individual for the purpose of research and private study only. The thesis may not be reproduced elsewhere

More information

Systems of Differential Equations

Systems of Differential Equations WWW Problems and Solutions 5.1 Chapter 5 Sstems of Differential Equations Section 5.1 First-Order Sstems www Problem 1. (From Scalar ODEs to Sstems). Solve each linear ODE; construct and solve an equivalent

More information