An epidemic model is given by

$$\begin{aligned} &\frac{d S}{d t}=-r I S \\ &\frac{d I}{d t}=+r I S-a I \end{aligned}$$

where $S(t)$ are the susceptibles, $I(t)$ are the infecteds, and $a$ and $r$ are positive parameters. The basic reproduction ratio is defined as $R_{0}=r N / a$, where $N$ is the total population size. Find a condition on $R_{0}$ for an epidemic to be possible if, initially, $S \approx N$ and $I$ is small but non-zero.

Now suppose a proportion $p$ of the population was vaccinated (with a completely effective vaccine) so that initially $S \approx(1-p) N$. On a sketch of the $\left(R_{0}, p\right)$ plane, mark the regions where an epidemic is still possible, where the vaccination will prevent an epidemic, and where no vaccination was necessary.

For the case when an epidemic is possible, show that $\sigma$, the proportion of the initially susceptible population that has not been infected by the end of an epidemic, satisfies

$$\sigma-\frac{1}{(1-p) R_{0}} \log \sigma \approx 1$$