= Solution
Rearrange the final-size equation as $s_\infty=s_0\exp[s_\infty-s_0-(1+\eta)q_0]$. With fixed $q_0>0,s_0>0$ and $\eta\to\infty$, the selected root is exponentially small:
$$
s_\infty\sim s_0e^{-s_0-(1+\eta)q_0}.
$$
Substitution into the death count yields
$$
\boxed{m_\infty=q_0+\frac{s_0}{\eta}+O(\eta^{-2})+O(e^{-\eta q_0}/\eta).}
$$
Thus strong <vaccination> suppresses most secondary fatal infections but cannot save the $q_0$ animals already infected, as required by the stated reaction scheme. The limit holds for the fixed positive initial <infection> assumed here; it need not be uniform if $q_0$ is allowed to shrink with increasing $\eta$.
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