Vaccinated epidemic-wave final size ($\log (s^{\ast }/s_{-})=s^{\ast }-s_{-}$)

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