Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 6C (Mathematical Biology)

The cross-infection terms transfer susceptible individuals into the infected class: $\beta S\hat{I}$ is male incidence caused by infected females and $\hat{\beta }\hat{S}I$ is female incidence caused by infected males. The terms $\nu I$ and $\hat{\nu }\hat{I}$ are recovery flows back to susceptibility. Adding each susceptible-infected pair shows that $N$ and $\hat{N}$ are conserved.

Substitute $S=N-I$ and $\hat{S}=\hat{N}-\hat{I}$:Besides the origin, solve both nullcline equations. Eliminating one infected population givesThe assumed epidemic threshold inequality makes both positive.

At the origin the Jacobian isThus it is a saddle and unstable. At the positive equilibrium the trace is negative. The product of the off-diagonal entries is $\nu \hat{\nu }$ by the nullcline identities, while the product of the magnitudes of the diagonal entries is strictly larger, so the determinant is positive. Both eigenvalues therefore have negative real part and the endemic equilibrium is stable.