MODELING EPIDEMICS FROM ZOMBIES TO EBOLA A Thesis Presented to the Faculty of California State Polytechnic University, Pomona In Partial Fulfillment Of the Requirements for the Degree Master of Science In Mathematics By Chal Tomlinson 2017 SIGNATURE PAGE THESIS: MODELING EPIDEMICS FROM ZOMBIES TO EBOLA AUTHOR: Chal Tomlinson DATE SUBMITTED: Spring 2017 Department of Mathematics and Statistics Dr. Randall J. Swift Thesis Committee Chair Mathematics & Statistics Dr. Hubertus Von Bremen Mathematics & Statistics Dr. Ryan Szypowski Mathematics & Statistics ii ACKNOWLEDGMENTS I’d like to thank my friend and advisor, Randy Swift, for a whole lot of stuff. Thank you to Hubertus Von Bremen for putting in way more elbow grease than is required, and a special shout out to Hubertus’s wife for making him stay in Southern California. Thank you Ryan Szypowski for putting up with me. Thank you Dr. Rosin for helping Cal Poly Pomona to become a home for me. Thank you John Rock for surfing the “bureaucracy.” Thank you Berit Givens for knowing how to knit and reminding me of my mother. Thank you Robin Wilson for the friendship, guidance, and stressing me out in Topology. Thank you Jenny Switkes for all the fantastic courses; you truly have a gift! Thank you Adam King for showing me a new side of statistics and the insight into graduate school. Thank you to Diana, Moriya, Brian, and to all of the beautiful people I’ve had the pleasure of crossing paths with during my time at CPP. Venturing off campus, thank you to all the family and friends who’ve tolerated and endured my struggle to balance life. Finally, thank you to the spirit of peace, love, patience and freedom for allowing each day to be lived in a new and meaningful way. iii ABSTRACT 2014 saw by far the largest Ebola outbreak in history. In this thesis, we explain the ba­ sics of epidemic modeling. The classic SI, SIR, and SEIR models are detailed with a hypothetical zombie apocalypse. The importance of the basic reproduction number, R0, is shown. For Ebola, data is obtained from the World Health Organization on Guinea, Liberia, and Sierra Leone of West Africa (the most affected regions), and the Ebola out­ break is fit to the SEIR disease model. R0 is calculated as a function of time for all three countries: Guinea Liberia Sierra Leone (1:762)e−(0:00196)t (5: 183)(e−(0:0167)t 3: 121)e−(0:00679)t To introduce a hypothetical cure, the infected group of the SEIR model is split up into a modified S-E-I1-I2-R model to differentiate between patients whose disease is advanced versus those whose is not. More complicated models make calculating R0 increasingly difficult. The next-generation matrix (NGM) is introduced to show an effective way of obtaining R0. iv Contents Introduction 1 1 Modeling an Epidemic 2 1.1 An SI Model . 3 1.2 An SIR Model . 7 1.2.1 The Basic Reproduction Number, R0 . 11 1.2.2 Solving for R0 in the SIR Model . 15 1.3 An SEIR Model . 16 1.3.1 Solving for R0 in the SEIR Model . 19 2 Implementing the Model 20 2.1 Fitting the SEIR Model for Ebola . 21 2.1.1 Guinea Results . 23 2.1.2 Liberia Results . 25 2.1.3 Sierra Leone Results . 27 2.1.4 Discussion of Results . 29 2.2 Introducing a Cure . 32 2.2.1 The SEI1I2R Model . 32 2.2.2 Discussion of the Results . 37 v 3 R0 as the Spectral Radius of the Next Generation Matrix 39 3.1 The Next-Generation Matrix . 40 3.2 Examples of Calculating R0 . 41 3.2.1 SIR . 41 3.2.2 SEIR . 42 3.2.3 SEI1I2R . 43 3.2.4 SEI1I2R with a Few New Twists . 45 A Ebola Data 47 B MATLAB Code 53 Bibiliography 59 vi List of Figures 1.1 flow chart for the SI model . 3 1.2 sample SI models with one intitial zombie and a total population of 1,000 6 1.3 flow chart for the SIR model . 7 1.4 sample SIR models with one intitial zombie and a total population of 1,000 8 1.5 a generational look at R0 = 2 (R0 > 1) . 12 1.6 a generational look at R0 = 1 . 13 -1 1.7 a generational look at R0 = 3 (R0 < 1) . 14 1.8 flow chart for the SEIR model . 17 2.1 Guinea: modeling the cumulative number of cases compared to the data 23 2.2 Guinea: the number of infections from our model compared to the data . 23 2.3 Guinea: the numerical solution for the (S)EIR model . 24 2.4 Liberia: modeling the cumulative number of cases compared to the data 25 2.5 Liberia: the number of infections from our model compared to the data . 25 2.6 Liberia: the numerical solution for the (S)EIR model . 26 2.7 Sierra Leone: modeling the cumulative number of cases compared to the data . 27 2.8 Sierra Leone: the number of infections from our model compared to the data . 27 vii 2.9 Sierra Leone: the numerical solution for the (S)EIR model . 28 2.10 Comparing R0 between countries and over time . 30 2.11 testing the validity of the new SEI1I2R model . 33 2.12 Guinea: modeling the recovered number of cases with vaccination . 34 2.13 Guinea: modeling the infected number of cases with vaccination . 34 2.14 Liberia: modeling the recovered number of cases with vaccination . 35 2.15 Liberia: modeling the infected number of cases with vaccination . 35 2.16 Sierra Leone: modeling the recovered number of cases with vaccination 36 2.17 Sierra Leone: modeling the infected number of cases with vaccination . 36 2.18 Guinea: R(t) with vaccination from the beginning . 38 viii Introduction The idea for this thesis originated while a friend, Brian Tran, and I competed in the 2015 Mathematical Contest in Modeling (MCM). 1 The following is the original problem, 2015 MCM Problem A: Eradicating Ebola (this thesis changes focuses from the original problem, as outlined in the abstract): The world medical association has announced that their new medication could stop Ebola and cure patients whose disease is not advanced. Build a realistic, sensible, and useful model that considers not only the spread of the disease, the quantity of the medicine needed, possible feasible delivery systems (sending the medicine to where it is needed), (geographical) loca­ tions of delivery, speed of manufacturing of the vaccine or drug, but also any other critical factors your team considers necessary as part of the model to optimize the eradication of Ebola, or at least its current strain. In addition to your modeling approach for the contest, prepare a 1-2 page non-technical letter for the world medical association to use in their announcement. [3] 1For more on the MCM, please visit http://www.comap.com/undergraduate/contests/. 1 Chapter 1 Modeling an Epidemic “All models are wrong, but some are useful.” - George E.P. Box In this chapter, we will go over some of the mathematical background essential for un­ derstanding how we build a model for the Ebola outbreak. What is a Mathematical Model? A model is a mathematical representation of something. This may sound like it needs to be complicated, but not necessarily. Ideally, we want to use the easiest representation of a system we can, while still providing us with sufficient insight. There are many types of mathematical models with even more applications. For the purpose of this paper, we will focus on deterministic biological (epidemiological) systems of differential equations. If these big words make you nervous, don’t fret! We will do some learning along the way! Instead of belabouring the definition of a model, let’s propagate the idea with an example. 2 1.1 An SI Model Consider the most boring of scenarios: a zombie apocalypse. We define: • S(t) to be the population of humans at time t (or individuals Susceptible to becom­ ing zombies). • I(t) to be the population of zombies at time t (or individuals who have become Infected with the zombie disease). • N(t) to be the total population [of humans and zombies] at time t (that is, N(t) = S(t) + I(t)). Figure 1.1: flow chart for the SI model We will describe the apocalypse using the following system of first order ordinary differential equations: S˙ = dS = −b S(t)I(t) dt N(t) : (1.1) ˙ dI S(t)I(t) I =dt = b N(t) 3 Some things to note: • There are two variables: S and I (we will show why N does not vary below). The population is closed. An individual can only move from being a human to being a zombie. Zombies cannot go back to being human again. • Birth, death, emigration, and immigration are all ignored. The population is closed. So, the only thing our system allows for is humans to become zombies. This will approximately represent a real system if the spread of zombies happens relatively fast (when compared to the demographics we are ignoring). • This type of model (due to the derivatives) allows there to be non-integer values of a population. Obviously, having 1.573 humans doesn’t make sense in the real world. These type of models can be good approximations for large populations, but for smaller populations the solutions may be deceptive. • This type of model assumes uniform mixing (all individuals have an equal chance of coming in contact with each other).