The following ODEs have been proposed as a model of an epidemic:

Where S = the susceptible individuals, I = the infected, R = the recovered, a = the infection rate, and r = the recovery rate. A city has 10,000 people, all of whom are susceptible.
(a) If a single infectious individual enters the city at t = 0, compute the progression of the epidemic until the number of infected individuals falls below 10. Use the following parameters:
a = 0.002/(person?week) and r = 0.15yd. Develop time-series plots of all the state variables. Also generate a phaseplane plot of S versus I versus R.
(b) Suppose that after recovery, there is a loss of immunity that causes recovered individuals to become susceptible. This reinfection mechanism can be computed as rR, where r = the reinfection rate. Modify the model to include this mechanism and repeat the computations in (a) using r = 0.015yd.

= _al dt dI dt