Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 13E (Mathematical Biology)

The term ${\lambda }_i{S}_i$ moves susceptible people in group $i$ into its infective class, while $\gamma I_i$ moves infectives into recovery. In$C_{ij}$ is the contact rate of a person in group $i$ with group $j$, and $I_j/N_j$ is the infectious fraction there. The reciprocity condition $N_i{C}_{ij}=N_j{C}_{ji}$ says that the total number of $i$--$j$ contacts agrees when counted from either group. Common $\beta$ and $\gamma$ mean age-independent transmission per infectious contact and recovery rate. There is no transmission between distinct age groups exactly when

With $S_i\simeq N_i$, the infective equations becomewhere reciprocity was used. For a column vector $\mathbf{I}$,Therefore

Since $C$ and $C^T$ have the same eigenvalues, an exponentially growing mode exists precisely when $L$ has an eigenvalue with real part greater than one. For the nonnegative contact matrix this iswhere $\rho (C)$ is the spectral radius.