Consider a model of an epidemic consisting of populations of susceptible, $S(t)$, infected, $I(t)$, and recovered, $R(t)$, individuals that obey the following differential equations

$$\begin{aligned} \frac{d S}{d t} &=a R-b S I \\ \frac{d I}{d t} &=b S I-c I \\ \frac{d R}{d t} &=c I-a R \end{aligned}$$

where $a, b$ and $c$ are constant. Show that the sum of susceptible, infected and recovered individuals is a constant $N$. Find the fixed points of the dynamics and deduce the condition for an endemic state with a positive number of infected individuals. Expressing $R$ in terms of $S, I$ and $N$, reduce the system of equations to two coupled differential equations and, hence, deduce the conditions for the fixed point to be a node or a focus. How do small perturbations of the populations relax to the steady state in each case?