= Solution
While $q>0$ and $s>0$, division of the equations gives $dq/ds=(1-s)/[(1+\eta)s]$ and $dv/ds=-\eta/(1+\eta)$. Integrating from the initial data yields
$$
q(s)=q_0+\frac{s_0-s+\log(s/s_0)}{1+\eta},\qquad v(s)=\frac\eta{1+\eta}(s_0-s).
$$
Let $s_\infty$ be the final susceptible level. Using $q_\infty=0$ gives
$$
\boxed{s_\infty-\log s_\infty=s_0-\log s_0+(1+\eta)q_0,\qquad0<s_\infty<\min(s_0,1).}
$$
This is the smaller positive root selected by decreasing susceptibility; the function $s-\log s$ is decreasing on $(0,1)$, so that selected root is unique. Its upper bound below $s_0$ also follows directly because the added positive term moves the decreasing-branch root left.
Conservation of the total count, or integration of $m_t=q$, then gives the number killed, including those initially infected:
$$
\boxed{m_\infty=q_0+\frac{s_0-s_\infty}{1+\eta}.}
$$
The remaining susceptible loss is split between <infection> and <vaccination> in fixed ratio $1:\eta$; this explains the factor multiplying $s_0-s_\infty$.
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