= Solution
Use fractions of all farms, so $S_n+I_n+R_n=P_n$ and $\sum_nP_n=1$. 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 $\mu=\sum_nnP_n$ and $\mu_2=\sum_nn^2P_n$. Write $I=\sum_nI_n$. The <farm-size stratified SIR model> with susceptibility proportional to size is
$$
\dot S_n=-\beta nS_nI,\qquad \dot I_n=\beta nS_nI-\gamma I_n,\qquad \dot R_n=\gamma I_n.
$$
Here $\beta$ includes the common contact-rate constant; using frequency-dependent mixing avoids introducing an extra total-farm factor. An infectious farm of any size generates $\beta nP_n/\gamma$ infections in initially susceptible size-$n$ farms. The next-generation <matrix> $K_{nm}=\beta nP_n/\gamma$ has rank one, giving the <basic reproduction number>
$$
\boxed{\mathcal R_0=\beta\mu/\gamma.}
$$
The 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 $\Lambda(t)=\beta\int_0^tI(u)\,du$. For negligible initial infection, $S_n=P_ne^{-n\Lambda}$. Large farms leave the susceptible class faster. If $\mu_S=\sum_nnS_n/S$ with $S=\sum_nS_n$, differentiation gives
$$
\dot\mu_S=-\beta I\,\operatorname{Var}_S(n)\leq0.
$$
The <mean> size of newly infected farms is $\mu_{\rm inc}=\sum_nn^2S_n/\sum_nnS_n$, larger than the susceptible <mean> unless all sizes agree. If $\mu_I=\sum_nnI_n/I$, then
$$
\dot\mu_I=\frac{\beta I\sum_nnS_n}{I}(\mu_{\rm inc}-\mu_I).
$$
Recovery 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 $\mu_{\rm inc}$ 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, $I_n\propto nP_n$, 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 $A$, is
$$
A=1-\sum_nP_n e^{-(\beta/\gamma)nA}.
$$
By <Jensen's inequality>, the right side is at most $1-e^{-\mathcal R_0A}$, 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 $\mathcal R_0$.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2002/iii/paper-57-epidemics.png]
{title=Farm-size heterogeneity: epidemic prevalence and mean livestock counts for the three transmission assumptions}
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.
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