(a) Consider a population of size $N(t)$ whose per capita rates of birth and death are $b e^{-a N}$ and $d$, respectively, where $b>d$ and all parameters are positive constants.

(i) Write down the equation for the rate of change of the population.

(ii) Show that a population of size $N^{*}=\frac{1}{a} \log \frac{b}{d}$ is stationary and that it is asymptotically stable.

(b) Consider now a disease introduced into this population, where the number of susceptibles and infectives, $S$ and $I$, respectively, satisfy the equations

$$\begin{aligned} &\frac{d S}{d t}=b e^{-a S} S-\beta S I-d S \\ &\frac{d I}{d t}=\beta S I-(d+\delta) I \end{aligned}$$

(i) Interpret the biological meaning of each term in the above equations and comment on the reproductive capacity of the susceptible and infected individuals.

(ii) Show that the disease-free equilibrium, $S=N^{*}$ and $I=0$, is linearly unstable if

$$N^{*}>\frac{d+\delta}{\beta}$$

(iii) Show that when the disease-free equilibrium is unstable there exists an endemic equilibrium satisfying

$$\beta I+d=b e^{-a S}$$

and that this equilibrium is linearly stable.