Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 6A (Mathematical Biology)

In the SIR model with demography and permanent immunity, $\mu$ is the common per-capita death rate and, through the term $\mu N$, the replacement birth rate. The coefficient $\beta$ is the mass-action transmission rate per susceptible-infective pair, and $\nu$ is the per-capita recovery rate from infection into permanent immunity.

The infective equation isSince $X\le N$, the threshold population isIf $N<N_c$ and $\delta =\mu +\nu -\beta N>0$, thenAlsoAny 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.

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.