Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 57 / 3 / b

With susceptibility independent of size and infectivity proportional to size, define $J={\sum }_{n}n{I}_n$. The farm-size stratified SIR model isAn infectious size-$m$ farm has next-generation column $K_{nm}=(\beta /\gamma )P_{n}m$. Its sole nonzero eigenvalue gives the basic reproduction numberagain equal to the comparison SIR model.

Since every susceptible farm experiences the same force of infection, all ratios $S_n/S_m$ remain fixed. For a representative initial seed, $S_n(0)=(1-i_0)P_n$ and $I_n(0)=i_0{P}_n$, uniqueness of the differential equations gives $S_n=P_{n}S$, $I_n=P_{n}I$, $R_n=P_{n}R$ for all time. Consequently $J=\mu I$ andFor this representative seed, the heterogeneous and homogeneous epidemics coincide exactly, and both susceptible and infected farm means stay at $\mu$. This is the overlapping middle-column sketch above, not merely an approximation based on equal reproduction numbers.

With a nonrepresentative seed the susceptible mean remains its initial value, but the infected mean need not. The influx of new infectives has the constant susceptible mean, soThus it is pulled toward ${\mu }_S$ while incidence continues. Infectivity variation alone does not selectively remove large susceptible farms. The exact homogeneous reduction depends on the representative initial condition; specifying only $P_n$ and ${\mathcal{R}}_0$ does not determine a trajectory for every possible seed.