Chapter 28, Problem 34P

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

$\begin{array}{l}\frac{dS}{dt}=-aSI\\ \frac{dI}{dt}=aSI-rl\\ \frac{dR}{dt}=rl\end{array}$

where $S=\text{the }$ susceptible individuals, $I=\text{the infected, }R=\text{the }$ recovered, $a=\text{the}$ infection rate, and $r=\text{the }$ 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: $a=0.002\left(\text{person}\cdot \text{week}\right)\text{and }r=0.15/\text{d}$. 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 5 the reinfection rate. Modify the model to include this mechanism and repeat the computations in (a) using $\rho =0.015/\text{d}$.