 Solution

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