= Solution
With susceptibility independent of size and infectivity proportional to size, define $J=\sum_nnI_n$. The <farm-size stratified SIR model> is
$$
\dot S_n=-\beta S_nJ,\qquad \dot I_n=\beta S_nJ-\gamma I_n,\qquad \dot R_n=\gamma I_n.
$$
An infectious size-$m$ farm has next-generation column $K_{nm}=(\beta/\gamma)P_nm$. Its sole nonzero <eigenvalue> gives the <basic reproduction number>
$$
\boxed{\mathcal R_0=\beta\mu/\gamma,}
$$
again 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 $I_n(0)=i_0P_n$, uniqueness of the <differential equations> gives $S_n=P_nS$, $I_n=P_nI$, $R_n=P_nR$ for all time. Consequently $J=\mu I$ and
$$
\dot S=-\beta\mu SI,\qquad \dot I=\beta\mu SI-\gamma I,\qquad \dot R=\gamma I.
$$
\b[For 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>, so
$$
\dot\mu_I=\frac{\beta SJ}{I}(\mu_S-\mu_I).
$$
Thus 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.
Back to article page
