 Solution

From the preceding conservation identity,The nontrivial root $s_{-}$ depends only on $s^{\ast }$, through $\log (s^{\ast }/s_{-})=s^{\ast }-s_{-}$. Therefore the coefficient $s^{\ast }-s_{-}$ is independent of $\eta$, proving the exact requested proportionality to $(1+\eta {)}^{-1}$, conditional on existence of the positive wave. This is stronger than an asymptotic estimate and differs from the spatially homogeneous problem with a fixed positive initially infected count.