 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 yieldsLet $s_{\infty }$ be the final susceptible level. Using $q_{\infty }=0$ givesThis 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:The remaining susceptible loss is split between infection and vaccination in fixed ratio $1:\eta$; this explains the factor multiplying $s_0-s_{\infty }$.