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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 8' = aSI, I' = aSI M, R, = bI. Here S, 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'icient and the recovery coeicient, respectively. Question 12. Now that you know (approximately) 8(2), 1(2), R(2), and S'(2), I'(2), and R'(2), you can compute (approximately) 8 (4), 1(4), and R(4) at t = 4, by repeating the above procedure. For example, you can observe that, roughly, if AS now denotes the change in S from t = 2 to t = 4, then S(4)=S(2)+AS =S(2)+( XAt) =+(x)= (The last four blanks should be lled in by numbers; the blank before that, by a symbolic quantity.) Put your nal answer here into the correct place in the above table. Repeat this procedure for I and R. Now suppose we had taken things one day, instead of two days, at a time. That is: suppose we had taken At = 1. We would then compute 3(1), I(1), R(1), S'(1), I'(1), and R'(1), based on the values of these quantities at t = 0, and then use the values at t = 1 to compute the values at t = 2, and so 011. Question 13. How would the values at t = 2, computed using At = 1, compare to those you computed previously, using At = 2? Which method is likely to give a better approximation to the true nature of the disease after two days? Please explain. (You don't have to actually compute anything.)