= Solution
For a positive equilibrium, $Y'=0$ gives $X^*=N_c$. The equations $Z'=0$ and $X^*+Y^*+Z^*=N$ then give the <Endemic equilibrium of the SIR model with demography>
$$
X^*=N_c,
\qquad
Y^*=\frac{\mu}{\mu+\nu}(N-N_c),
\qquad
Z^*=\frac{\nu}{\mu+\nu}(N-N_c),
$$
which is strictly positive when $N>N_c$.
Use $Z=N-X-Y$ to reduce to the $(X,Y)$ system. At the endemic equilibrium its Jacobian is
$$
J=\begin{pmatrix}
-\mu-\beta Y^*&-(\mu+\nu)\\
\beta Y^*&0
\end{pmatrix}.
$$
Its trace is negative and its <determinant> is $\beta Y^*(\mu+\nu)>0$. The two <eigenvalues> therefore have negative real part, so the endemic equilibrium is locally asymptotically stable.
Back to article page
