Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 6A (Mathematical Biology)

The term $-IS$ removes healthy foxes through mass-action contact with infectives. The Laplacian ${\nabla }^{2}I$ models random migration of infected foxes. The matching term $+IS$ creates newly infected animals, while $-\mu I$ removes infectives at per-capita rate $\mu$. The equation $R_t=\mu I$ records those removed from infectious circulation, for example foxes that have died from rabies; they neither migrate nor return to susceptibility in this model.

In a homogeneous population,An infected fox remains infectious for mean time $1/\mu$ and creates new infections at initial rate $S_0$, so the basic reproductive ratio isInitially $I$ grows exactly when $S_0-\mu >0$, equivalently ${\mathcal{R}}_0>1$. Below that threshold each infective produces fewer than one replacement on average and the infection decays.

With $\xi =x-ct$, time derivatives become $-c$ times $\xi$ derivatives and $I_{xx}=I^{\prime \prime }$. Henceor equivalently

At the leading edge $S\simeq 1$ and $I$ is small, soSeek a decaying mode $I\sim e^{-\lambda \xi }$ with $\lambda >0$. Its characteristic equation isA nonoscillatory positive leading edge requires real positive roots, henceFor an invasive infection, $mu<1$, and the selected minimum speed is thereforeIf $\mu \ge 1$, the state ahead of the wave is below the epidemic threshold and no growing infection front is selected.