Infectious disease modeling Matthew Macauley Department of - PowerPoint PPT Presentation

Infectious disease modeling Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4500, Spring 2015 M. Macauley (Clemson) Infectious disease modeling Math 4500, Spring 2015 1 / 12

Some history John Snow (1813–1858) is widely considered to be the “father of epidemiology.” In 1854, an outbreak of cholera struck the Soho neighborhood of London. Snow identified the source of the outbreak to be the Broad Street water pump. M. Macauley (Clemson) Infectious disease modeling Math 4500, Spring 2015 2 / 12

The SIR model Consider a disease spreading through a population, with the following assumptions: N people. 3 states: S usceptible, I nfected, R ecovered. γ α Transition: S Ý Ñ I Ý Ñ R . S t � I t � R t ✏ N . No births or deaths. Population is homogeneously mixed. This is just one example. These assumptions can be changed in other similar models. SIR model The following is the “standard” SIR model, using ODEs (left) and difference equations (right): S ✶ ✏ ✁ α SI ✩ ✩ ∆ S ✏ ✁ α SI ✬ ✬ ✫ I ✶ ✏ α SI ✁ γ I ✫ ∆ I ✏ α SI ✁ γ I R ✶ ✏ γ I ✬ ✬ ∆ R ✏ γ I ✪ ✪ M. Macauley (Clemson) Infectious disease modeling Math 4500, Spring 2015 3 / 12

SIR model: S ✶ ✏ ✁ α SI ; I ✶ ✏ α SI ✁ γ I ; R ✶ ✏ γ I The following is what we should expect S ♣ t q , I ♣ t q , and R ♣ t q to look like, qualitatively. Key aspect: I ✶ ✏ ♣ α S ✁ γ q I ✏ γ � α ✟ γ S ✁ 1 I . If S → γ ④ α , then I ✶ → 0 (epidemic is growing ). If S ➔ γ ④ α , then I ✶ ➔ 0 (epidemic is shrinking ). The value ρ : ✏ γ ④ α is a threshold , called the “ relative removal rate ”. M. Macauley (Clemson) Infectious disease modeling Math 4500, Spring 2015 4 / 12

SIR model: S ✶ ✏ ✁ α SI ; I ✶ ✏ α SI ✁ γ I ; R ✶ ✏ γ I Definition Initially, I ✶ 0 ✏ ♣ α S 0 ✁ γ q I . Define R 0 : ✏ α γ S 0 , the “ basic reproductive number ”. If R 0 → 1, then I ✶ ♣ 0 q → 0 ù ñ epidemic occurs. If R 0 ➔ 1, then I ✶ ♣ 0 q ➔ 0 ù ñ no epidemic occurs. Key point R 0 represents the expected number of people an initially infected person will infect. We’ll see why this is true shortly. First, here are estimates of R 0 for some well-known diseases: 12–18 for measles 5–7 for smallpox and polio 2–5 for HIV and SARS 2–3 for the 1918 Spanish flu 1.5–2.5 for Ebola M. Macauley (Clemson) Infectious disease modeling Math 4500, Spring 2015 5 / 12

SIR model: ∆ S ✏ ✁ α SI ; ∆ I ✏ α SI ✁ γ I ; ∆ R ✏ γ I Let’s analyze R 0 : ✏ α γ S 0 in the (discrete) setting of difference equations. α S 0 I 0 ✏ # people infected in the first time-step. α S 0 ✏ # people infected per-capita in the first time-step. 1 γ ✓ average duration of infection. Putting this together, we get ✟✁ 1 R 0 ✏ α ✠ � γ S 0 ✏ α S 0 γ ✏ ♣ # new infections per person per day q♣ ave. duration q ✏ # secondary infections from one sick person Policy goal Inact steps to reduce R 0 to be below 1. M. Macauley (Clemson) Infectious disease modeling Math 4500, Spring 2015 6 / 12

An example: ∆ S ✏ ✁ α SI ; ∆ I ✏ α SI ✁ γ I ; ∆ R ✏ γ I Consider an epidemic with the following properties, with a timestep of 1-day: Initially there are S 0 ✏ 500 susceptibles. There is a 0 . 1% chance of transmission ( α ✏ . 001). 10-day illness ( γ ✏ . 1). The basic reproductive number is easily computed: R 0 ✏ α γ S 0 ✏ . 001 . 1 ♣ 500 q ✏ 5 . On day 1, the expected number of infections will be I 1 ✏ α S 0 I 0 � ♣ 1 ✁ γ q I 0 ✏ . 001 ♣ 500 q♣ 1 q � . 9 ♣ 1 q ✏ 1 . 4 . If measures (e.g., vaccinations) can be taken to reduce S 0 ✏ 90, then R 0 ✏ . 9 and the epidemic will be averted. The epidemic subsides when S t ➔ ρ ✏ γ α ✏ 100 (i.e., when four-fifths of the population has contracted the disease). Computationally, lim t Ñ✽ S t ✏ 2 . 15, which means that not everyone gets sick. M. Macauley (Clemson) Infectious disease modeling Math 4500, Spring 2015 7 / 12

Other epidemic models SI model (e.g., herpes, HIV). S ✶ ✏ ✁ α SI ★ α S I I ✶ ✏ α SI SIS model. Disease w/o immunity (e.g., chlamydia). α S ✶ ✏ ✁ α SI � γ I ★ S I I ✶ ✏ α SI ✁ γ I γ SIRS model. Finite-time immunity (e.g., common cold). S ✶ ✏ ✁ α SI � δ R ✩ γ ✬ α ✫ I ✶ ✏ α SI ✁ γ I S I R R ✶ ✏ γ I � δ R ✬ ✪ δ M. Macauley (Clemson) Infectious disease modeling Math 4500, Spring 2015 8 / 12

Other epidemic models SEIR model. E ✏ exposed (incubation period, no symptoms). S ✶ ✏ ✁ α SI ✩ ✬ E ✶ ✏ α SI ✁ ǫ E ✬ γ ✬ α ǫ ✫ S E I R I ✶ ✏ ǫ E ✁ γ I ✬ ✬ R ✶ ✏ γ I ✬ ✪ SIR model with birth and death rate. S ✶ ✏ ✁ β SI � µ ♣ N ✁ S q ✩ β ✬ α ǫ ✫ I ✶ ✏ β SI ✁ γ I ✁ µ I S I R R ✶ ✏ γ I ✁ µ R ✬ µ µ µ ✪ M. Macauley (Clemson) Infectious disease modeling Math 4500, Spring 2015 9 / 12

Other approaches Differential equations are continuous time, continuous space. Difference equations are discrete time, continuous space. Consider an SIR model that is discrete time and discrete space. Let X be a social network: vertices: people edges: contacts edge weights: contact hours Every person (node) has a state: S , I , or R (e.g., x j ✏ 0 , 1 , 2). M. Macauley (Clemson) Infectious disease modeling Math 4500, Spring 2015 10 / 12

How to model agent-based disease transmission? 1. Bernoulli trials (weighted “coin flips”) Suppose i is infectious, and j is susceptible ( x i ✏ I , x j ✏ S ) Let p be the probability that i infects j after 1 hour together: Pr ♣ i infects j q ✏ p ù ñ Pr ♣ i doesn’t infect j q ✏ 1 ✁ p . If i and j spend t → 1 hours together, then Pr ♣ i infects j q ✏ 1 ✁ Pr ♣ i doesn’t infect j q ✏ 1 ✁ ♣ 1 ✁ p q t (assuming each hour is an independent event). Now, suppose j comes in contact with individuals i 1 , . . . i k , for a duration of t 1 , . . . , t k , respectively. k ➵ ♣ 1 ✁ p q t i . Pr ♣ j gets infected q ✏ 1 ✁ Pr ♣ nobody infects j q ✏ 1 ✁ i ✏ 1 M. Macauley (Clemson) Infectious disease modeling Math 4500, Spring 2015 11 / 12

How to model agent-based disease transmission? 2. Exponential distribution. Suppose i is in contact with j for t ij P r 0 , 24 s hours during a given day, and say Pr ♣ i infects j q ✏ 1 ✁ e ✁ rt ij . Assume that the probability of getting infected by two different people are independent events, thus 1 ✁ ♣ 1 ✁ e ✁ rt ij q ➵ � ✟ Pr ♣ i not infected q ✏ edges t i , j ✉ 1 ✁ ♣ 1 ✁ e ✁ rt ij q ➵ � ✟ Pr ♣ i infected q ✏ 1 ✁ edges t i , j ✉ Regardless of which approch is taken, the computations are usually done with the aid of a computer, and rely more on simulation and statistical analysis. Question What are the pros and cons of using an agent-based model versus an ODE-based model ? M. Macauley (Clemson) Infectious disease modeling Math 4500, Spring 2015 12 / 12