Consider a model for the common cold in which the population is partitioned into susceptible $(S)$, infective $(I)$, and recovered $(R)$ categories, which satisfy

$$\begin{aligned} \frac{d S}{d t} &=\alpha R-\beta S I \\ \frac{d I}{d t} &=\beta S I-\gamma I \\ \frac{d R}{d t} &=\gamma I-\alpha R \end{aligned}$$

where $\alpha, \beta$ and $\gamma$ are positive constants.

(i) Show that the sum $N \equiv S+I+R$ does not change in time.

(ii) Determine the condition, in terms of $\beta, \gamma$ and $N$, for an endemic steady state to exist, that is, a time-independent state with a non-zero number of infectives.

(iii) By considering a reduced set of equations for $S$ and $I$ only, show that the endemic steady state identified in (ii) above, if it exists, is stable.