A fungal disease is introduced into an isolated population of frogs. Without disease, the normalised population size $x$ would obey the logistic equation $\dot{x}=x(1-x)$, where the dot denotes differentiation with respect to time. The disease causes death at rate $d$ and there is no recovery. The disease transmission rate is $\beta$ and, in addition, offspring of infected frogs are infected from birth.

(a) Briefly explain why the population sizes $x$ and $y$ of uninfected and infected frogs respectively now satisfy

$$\begin{aligned} \dot{x} &=x[1-x-(1+\beta) y] \\ \dot{y} &=y[(1-d)-(1-\beta) x-y] \end{aligned}$$

(b) The population starts at the disease-free population size $(x=1)$ and a small number of infected frogs are introduced. Show that the disease will successfully invade if and only if $\beta>d$.

(c) By finding all the equilibria in $x \geqslant 0, y \geqslant 0$ and considering their stability, find the long-term outcome for the frog population. State the relationships between $d$ and $\beta$ that distinguish different final populations.

(d) Plot the long-term steady total population size as a function of $d$ for fixed $\beta$, and note that an intermediate mortality rate is actually the most harmful. Explain why this is the case.