Worksheet 1: Prediction using SIR models Math 408R: Goal: To use the SIR model to predict (approximately) the evolution of a disease Recall: we saw, in class this week that the spread of a disease can, under appropriate assumptions, be modeled by the SIR. equations 3' = a.SI, I' = aSIbI, R' = III. Here 3, I, and R denote the number of individuals susceptible, infected, and recovered, respectively, at any given time t. We agree that t is measured in days, and that S, I, R are measured in people. Then S', I ' , and R', which denote the rates of change of S, I , and R respectively, are measured in people per day. Finally, a and b are positive constants, called the transmission coe'icicnt and the recovery coefcient, respectively. Question 1. What are the units for a and for b? Please explain. Now, to make things concrete, we will choose particular values for a and b, let's say a = 0.00002 and b = 0.1. Question 2. Once you get the disease, for how long do you stay infected before you recover? Please explain. OK, so now, and for the rest of this tutorial, the above SIR equations will look like this: .5" = 0.00002SI, I' = 0.0000231r 0.11, R' = 0.1L If we're going to predict how the disease evolves, we'd better know how it starts. So let's assume it starts with 95,000 susceptible, 5,000 infected, and none recovered. Question 3. How large is the overall population we're working with? Please explain. We are now going to try and Construct a table of (approximate) values of S, I, R, S", I', and R', over the course of a few days. Here's the start of that table. Estimates over the rst four days 8' I l R S' I' R' 95,000 5,000 l 0 unmeas- Question 4. In the above table, ll in the values of .5", I', and R' at t = 0. (Use the SIR equations at the bottom of the rst page of this tutorial.) Question 5. Fill in the blanks: Suppose we want to know (or approximate) 3(2), meaning the value of S at t = 2. Since we know 8(0) (meaning the value of S at t = 0), all we have to do, to gure out 3(2), is to gure out the total change in S from t = to t = . Let's call this change AS (pronounced \"delta 3\"). Then we know (ll in the blank with a NUMBER, taken from your table above) that 3(2) = 3(0) + AS = + AS. Question 6. Fill in the blanks again: Let's suppose that the rate of change 5" of 3 doesn't vary too much over the rst two days. Then we can assume that this rate of change over the rst two days is equal to the value of .5\" at the start of these two days. That is, we can assume that, over the rst two days, 3' is roughly equal to (ll in the blank with a NUMBER, taken from your table above) S'(0) = people per day. NOW: note that the total change AS, over the rst two days, can be computed by taking the rate of change of 3 per day, over those two days, and multiplying it by At, where At denotes the elapsed time (in days). In other words, we have, in the present situation, AS = 9(0) >
