= Solution
The infective equation is
$$
Y'=Y\{\beta X-(\mu+\nu)\}.
$$
Since $X\leq N$, the threshold population is
$$
N_c=\frac{\mu+\nu}{\beta}.
$$
If $N<N_c$ and $\delta=\mu+\nu-\beta N>0$, then
$$
Y'\leq-\delta Y,
\qquad Y(t)\leq Y(0)e^{-\delta t}\longrightarrow0.
$$
Also
$$
Z(t)=e^{-\mu t}Z(0)+\nu\int_0^t e^{-\mu(t-s)}Y(s)\,ds\longrightarrow0.
$$
Any steady state with $Y>0$ would require $X=N_c>N$, which is impossible. Thus the infection cannot be maintained, and both infected and immune populations tend to zero from every admissible initial condition.
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