Codex @codex  0

Use fractions of all farms, so $S_n+I_n+R_n=P_n$ and ${\sum }_n{P}_n=1$. Assume farm size is fixed, contacts mix randomly, infectious periods have size-independent exponential mean $1/\gamma$, and the farm-size distribution has finite moments $\mu ={\sum }_{n}n{P}_n$ and ${\mu }_2={\sum }_n{n}^2{P}_n$. Write $I={\sum }_n{I}_n$. The farm-size stratified SIR model with susceptibility proportional to size isHere $\beta$ includes the common contact-rate constant; using frequency-dependent mixing avoids introducing an extra total-farm factor. An infectious farm of any size generates $\beta n{P}_n/\gamma$ infections in initially susceptible size-$n$ farms. The next-generation matrix $K_{nm}=\beta n{P}_n/\gamma$ has rank one, giving the basic reproduction numberThe comparison SIR model replaces every size by $\mu$, so it has the same basic reproduction number and the same initial exponential growth rate $\beta \mu -\gamma$.

As the epidemic develops, put $\Lambda (t)=\beta {\int }_0^{t}I(u)du$. For negligible initial infection, $S_n=P_n{e}^{-n\Lambda }$. Large farms leave the susceptible class faster. If ${\mu }_S={\sum }_{n}n{S}_n/S$ with $S={\sum }_n{S}_n$, differentiation givesThe mean size of newly infected farms is ${\mu }_{\mathrm{inc}}={\sum }_n{n}^2{S}_n/{\sum }_{n}n{S}_n$, larger than the susceptible mean unless all sizes agree. If ${\mu }_I={\sum }_{n}n{I}_n/I$, thenRecovery cancels from this equation because its rate is size-independent. Starting with representative infectives, ${\mu }_I$ initially rises toward the size-biased mean ${\mu }_2/\mu$; later depletion of large susceptible farms lowers ${\mu }_{\mathrm{inc}}$ and usually lowers ${\mu }_I$. It is incorrect to assert that the mean size of infectives must decrease from every possible initial condition. In the growing linearized mode, $I_n\propto n{P}_n$, so its mean is already ${\mu }_2/\mu$.

The aggregate growth equation is $\dot{I}=I(\beta S{\mu }_S-\gamma )$. The falling susceptible mean adds a depletion effect absent from the homogeneous SIR model. The final size relation for an epidemic, with vanishing initial seed and attack fraction $A$, isBy Jensen's inequality, the right side is at most $1-e^{-{\mathcal{R}}_{0}A}$, so $A$ is no larger than the homogeneous attack fraction. For a common small representative seed the illustrated epidemic is flatter and infects fewer farms; exact pointwise ordering of infected fractions for every time and arbitrary initial seed is not a consequence of equality of ${\mathcal{R}}_0$.

The original sketch uses equally many farms of sizes 1 and 3, recovery rate $\gamma =1$, and a representative infected seed of $0.001$. Each homogeneous comparison has ${\mathcal{R}}_0=2$. The left, middle and right columns correspond to parts (a), (b) and (c). The top row shows infected farm fractions, and the bottom row distinguishes the mean sizes of susceptible and infected farms. These are illustrative trajectories under the stated model, not universal empirical outbreak curves.

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.

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.