Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 6A / c / Solution

For a positive equilibrium, $Y^{\prime }=0$ gives $X^{\ast }=N_c$. The equations $Z^{\prime }=0$ and $X^{\ast }+Y^{\ast }+Z^{\ast }=N$ then give the Endemic equilibrium of the SIR model with demographywhich 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 isIts trace is negative and its determinant is $\beta Y^{\ast }(\mu +\nu )>0$. The two eigenvalues therefore have negative real part, so the endemic equilibrium is locally asymptotically stable.