SIR model with demography and permanent immunity

With susceptible, infective, and immune populations $X,Y,Z$, equal per-capita birth and death rate $\mu$, mass-action transmission coefficient $\beta$, and recovery rate $\nu$, the equations areThe conserved population is $N=X+Y+Z$, and the threshold population is $N_c=(\mu +\nu )/\beta$.

When $N>N_c=(\mu +\nu )/\beta$, the positive equilibrium isOn the invariant plane $X+Y+Z=N$, the $(X,Y)$ Jacobian at this equilibrium has negative trace $-(\mu +\beta Y^{\ast })$ and positive determinant $\beta Y^{\ast }(\mu +\nu )$, so both eigenvalues have negative real part and the equilibrium is locally asymptotically stable.