= Vaccinated epidemic-wave final size
{title2=$\log(s^*/s_-)=s^*-s_-$}
For a pulse in which only infectives diffuse, the travelling-wave equations imply $v_- =\eta(s^*-s_-)/(1+\eta)$ and $m_- =\log(s^*/s_-)/(1+\eta)$. Integrating total <biological population> balance with vanishing infective <derivative> at both infinities gives the displayed final-size relation. Its nontrivial root requires $0<s_-<1<s^*$. Thus deaths are exactly $(s^*-s_-)/(1+\eta)$, while live density behind the wave is $(s_-+\eta s^*)/(1+\eta)$. The root is independent of $\eta$ in this particular pulse model.
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