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

Both weights now equal farm size, giving the farm-size stratified SIR modelThe next-generation matrix is $K_{nm}=(\beta /\gamma )n{P}_{n}m$, whose nonzero eigenvalue isThe variance matters because large farms are both more likely to acquire infection and more effective sources afterward. This positive correlation increases the basic reproduction number relative to the homogeneous SIR model. In particular the heterogeneous population can permit invasion when the homogeneous comparison is subcritical.

Put $\Lambda =\beta {\int }_0^{t}J(u)du$. Apart from the negligible initial seed, $S_n=P_n{e}^{-n\Lambda }$. The susceptible mean obeys ${\dot{\mu }}_S=-\beta J{\mathrm{Var}}_S(n)$. For infectives, the incidence mean is again ${\mu }_{\mathrm{inc}}={\sum }_n{n}^2{S}_n/{\sum }_{n}n{S}_n$, andThe growing mode $I_n\propto n{P}_n$ has mean ${\mu }_2/\mu$, not $\mu$. As the larger susceptible farms are depleted, the mean size of incoming infectives falls; the current infected mean follows it with a delay. These equations give the direction of change for any initial condition without imposing false monotonicity of the infected mean.

The weighted infected total satisfies $\dot{J}=(\beta {\sum }_n{n}^2{S}_n-\gamma )J$, so the effective reproduction number declines from $\beta {\mu }_2/\gamma$ as the susceptible size distribution changes. The final size relation for an epidemic becomesThe right-column sketch illustrates the larger initial growth rate and preferential depletion of large farms. The homogeneous comparison has size $\mu$ in both transmission factors; it must not be assigned the heterogeneous second moment. The general conclusions are enhanced invasion and selective removal of large susceptible farms. The final unweighted fraction of farms infected has no universal ordering relative to the homogeneous comparison: higher transmission and stronger heterogeneity-induced depletion compete, and the displayed final-size equations determine the answer for a specified distribution. For example, with $P_1=P_3=1/2$, the infinitesimal-seed attack fractions are approximately $0.367$ versus $0.176$ when the homogeneous reproduction number is $1.1$, but $0.740$ versus $0.797$ when it is $2$; the two comparisons have opposite ordering.