Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 3 / 6B / b / Solution

For $rN>a$ and $p>0$, a nonzero infected equilibrium requires $S_{\ast }=a/r$. Using $a{I}_{\ast }=p{R}_{\ast }$ and population conservation givesEliminate $R=N-S-I$ before assessing stability, since perturbations must preserve the total population. The resulting Jacobian matrix at this endemic equilibrium isIts trace is negative and its determinant positive, so both eigenvalues have negative real part: the endemic equilibrium is locally asymptotically stable on the fixed-population plane. The full three-variable system has a neutral direction corresponding to changing the conserved population.

Writing $\kappa =rN-a>0$, as $p\to 0$ the trace is $-prN/a+O(p^2)$ and the determinant is $p\kappa$. Hence the discriminant $(\mathrm{tr}J{)}^2-4\det J=-4p\kappa +O(p^2)<0$. As $p\to \infty$, the trace is $-p-\kappa +O(p^{-1})$ and the determinant is again $p\kappa$, giving a positive discriminant $p^2+O(p)$. Both quantities are $O(p)$ in both limits, but the eigenvalues are complex for sufficiently small $p$ and real for sufficiently large $p$.