The dynamics of a directly transmitted microparasite can be modelled by the system

$$\begin{aligned} \frac{d X}{d t} &=b N-\beta X Y-b X \\ \frac{d Y}{d t} &=\beta X Y-(b+r) Y \\ \frac{d Z}{d t} &=r Y-b Z \end{aligned}$$

where $b, \beta$ and $r$ are positive constants and $X, Y$ and $Z$ are respectively the numbers of susceptible, infected and immune (i.e. infected by the parasite, but showing no further symptoms of infection) individuals in a population of size $N$, independent of $t$, where $N=X+Y+Z$.

Consider the possible steady states of these equations. Show that there is a threshold population size $N_{c}$ such that if $N<N_{c}$ there is no steady state with the parasite maintained in the population. Show that in this case the number of infected and immune individuals decreases to zero for all possible initial conditions.

Show that for $N>N_{c}$ there is a possible steady state with $X=X_{s}<N$ and $Y=Y_{s}>0$, and find expressions for $X_{s}$ and $Y_{s}$.

By linearising the equations for $d X / d t$ and $d Y / d t$ about the steady state $X=X_{s}$ and $Y=Y_{s}$, derive a quadratic equation for the possible growth or decay rate in terms of $X_{s}$ and $Y_{s}$ and hence show that the steady state is stable.