5. Consider the following set of ODE's, proposed to model an epidemic as we did before- dS -aSI dt dt = aSI-rI dR = rI dt where S = the susceptible individuals, I = the infected, R = the recovered. a = the infection rate and r = the recovery rate. As before, a city has a population of 5,000 people, all of whom are susceptible. If a single infected individual enters the city at t=0, find the progression of the epidemic until the number of infected people falls below 10. Take a =0.000285/(people-day) and r =0.15/day. Plot S, I, R versus time. Use the Heun method. Plot the answer out to 50 days. Use a step of 0.01 day. Consider the number of infected people as a function of time. What is the maximum number? When does it occur? You have done this as part of a previous problem set. Now repeat the problem with a different scenario! You just developed a vaccine that reduces the infection rate such that the infection rate, a, decreases with time according to a(t) = ao exp(-0.12 t) where ao =0.000285/(people-day) is the same as above without the vaccine. How many people were not infected owing to your vaccine and its prompt administration