The SIR epidemic model for an infectious disease divides the population $N$ into three categories of susceptible $S(t)$, infected $I(t)$ and recovered (non-infectious) $R(t)$. It is supposed that the disease is non-lethal, so that the population does not change in time.

Explain the reasons for the terms in the following model equations:

$$\frac{d S}{d t}=p R-r I S, \quad \frac{d I}{d t}=r I S-a I, \quad \frac{d R}{d t}=a I-p R$$

At time $t=0, S \approx N$ while $I, R \ll 1$.

(a) Show that if $r N<a$ no epidemic occurs.

(b) Now suppose that $p>0$ and there is an epidemic. Show that the system has a nontrivial fixed point, and that it is stable to small disturbances. Show also that for both small and large $p$ both the trace and the determinant of the Jacobian matrix are $O(p)$, and deduce that the matrix has complex eigenvalues for sufficiently small $p$, and real eigenvalues for sufficiently large $p$.