Consider an epidemic model with host demographics (natural births and deaths).

The system is given by

$$\begin{aligned} &\frac{d S}{d t}=-\beta I S-\mu S+\mu N \\ &\frac{d I}{d t}=+\beta I S-\nu I-\mu I \end{aligned}$$

where $S(t)$ are the susceptibles, $I(t)$ are the infecteds, $N$ is the total population size and the parameters $\beta, \mu$ and $\nu$ are positive. The basic reproduction ratio is defined as $R_{0}=\beta N /(\mu+\nu) .$

Show that the system has an endemic equilibrium (where the disease is present) for $R_{0}>1$. Show that the endemic equilibrium is stable.

Interpret the meaning of the case $\nu \gg \mu$ and show that in this case the approximate period of (decaying) oscillation around the endemic equilibrium is given by

$$T=\frac{2 \pi}{\sqrt{\mu \nu\left(R_{0}-1\right)}}$$

Suppose now a vaccine is introduced which is given to some proportion of the population at birth, but not enough to eradicate the disease. What will be the effect on the period of (decaying) oscillations?