Past exam of the topics in applied mathematics course of the University of Oxford / 2009 / A / 2 / b / ii / 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:Substitution into the death count yieldsThus 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 $\eta$.