= Infection-triggered vaccination model
{title2=$q_t=q(s-1),\quad s_t=-(1+\eta)qs$}
A fatal-infection model with <vaccination> proportional to infective-susceptible encounters adds $v_t=\eta qs$ and $m_t=q$. The total count including deaths is conserved. Integrating $dq/ds$ and $dv/ds$ gives $q=q_0+[s_0-s+\log(s/s_0)]/(1+\eta)$ and $v=\eta(s_0-s)/(1+\eta)$. The selected final root has $0<s_\infty<\min(s_0,1)$, and deaths are $q_0+(s_0-s_\infty)/(1+\eta)$. The model is a specified deterministic reaction scheme, not a general <vaccination> policy rule.
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