Infection-triggered vaccination model ($q_t=q(s-1),s_t=-(1+\eta )qs$)

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.