= Solution
Both weights now equal farm size, giving the <farm-size stratified SIR model>
$$
\dot S_n=-\beta nS_nJ,\qquad \dot I_n=\beta nS_nJ-\gamma I_n,\qquad J=\sum_nnI_n.
$$
The next-generation <matrix> is $K_{nm}=(\beta/\gamma)nP_nm$, whose nonzero <eigenvalue> is
$$
\boxed{\mathcal R_0=\frac{\beta\mu_2}{\gamma},\qquad
\mathcal R_{0,\rm hom}=\frac{\beta\mu^2}{\gamma},\qquad
\frac{\mathcal R_0}{\mathcal R_{0,\rm hom}}=1+\frac{\operatorname{Var}(n)}{\mu^2}.}
$$
The <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 $\Lambda=\beta\int_0^tJ(u)\,du$. Apart from the negligible initial seed, $S_n=P_ne^{-n\Lambda}$. The susceptible <mean> obeys $\dot\mu_S=-\beta J\operatorname{Var}_S(n)$. For infectives, the incidence <mean> is again $\mu_{\rm inc}=\sum_nn^2S_n/\sum_nnS_n$, and
$$
\dot\mu_I=\frac{\beta J\sum_nnS_n}{I}(\mu_{\rm inc}-\mu_I).
$$
The growing mode $I_n\propto nP_n$ 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 $\dot J=(\beta\sum_nn^2S_n-\gamma)J$, so the effective reproduction number declines from $\beta\mu_2/\gamma$ as the susceptible size distribution changes. The <final size relation for an epidemic> becomes
$$
\Lambda_\infty=\frac\beta\gamma\sum_nnP_n(1-e^{-n\Lambda_\infty}),\qquad
A=1-\sum_nP_ne^{-n\Lambda_\infty}.
$$
The 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. \b[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.
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