Consider an epidemic model where susceptibles are vaccinated at per capita rate $v$, but immunity (from infection or vaccination) is lost at per capita rate $b$. The system is given by

$$\begin{aligned} &\frac{d S}{d t}=-r I S+b(N-I-S)-v S \\ &\frac{d I}{d t}=r I S-a I \end{aligned}$$

where $S(t)$ are the susceptibles, $I(t)$ are the infecteds, $N$ is the total population size and all parameters are positive. The basic reproduction ratio $R_{0}=r N / a$ satisfies $R_{0}>1$.

Find the critical vaccination rate $v_{c}$, in terms of $b$ and $R_{0}$, such that the system has an equilibrium with the disease present if $v<v_{c}$. Show that this equilibrium is stable when it exists.

Find the long-term outcome for $S$ and $I$ if $v>v_{c}$.