(10 pt) (i) Consider the spread of an infectious disease through a population. Let $S(t) > 0$, $I(t) \ge 0$ and $R(t) \ge 0$ denote the number of susceptible (not yet infected), infected, and recovered (and now immune) individuals respectively. Suppose that each infected case comes in contact with another individual an average of $\beta > 0$ times per day. Further, a fixed fraction $\gamma > 0$ of infected individuals recover per day from the disease. Assume that the number of individuals born per day, denoted $\mu$, is equal to the number of individual deaths per day, and that deaths are proportional to the number of people in each state. Figure 1 shows a block diagram of the system.
$\beta$
$\gamma$
$S(t)$
$I(t)$
$R(t)$
Figure 1: Block diagram of simple epidemiological model.
Write down a system of differential equations that models this process. Is this system linear or nonlinear?
