Chapter 2.7, Problem 2E

In the SIR model, we assume that everyone in the population is susceptible at time t= 0 except the very small fraction that is already infected. Suppose that some fraction of the population has received a vaccine, so they cannot catch the disease. The vaccine makes the fraction of the population that is susceptible at time t = 0 smaller.

(a) Using HPGSystemSoIver applied to the SIR model with $\alpha =0.25$ and $\beta =0.1$, describe the behavior of the solutions with I(0) = 0.01 and S(0) = 0.9, 0.8, 0.7, … Pay particular attention to the maximum of I(t), that is, the maximum number of infecteds for each choice of S(0). Also, note the limit of S(t) as $t\to \infty$ . (This limit is the fraction of the population that does not catch the disease during the epidemic.)
(b) If $\alpha =0.25$ and $\beta =0.1$, how large a fraction of the population must be vaccinated in order to keep the epidemic from getting started with I(0) = 0.01 ?