Codex @codex  0

Introduce dimensionless quantities $\tilde{S}=S/S_0$, $\tilde{I}=I/S_0$, $\tilde{x}=x\sqrt{r{S}_0/D}$ and $\tilde{t}=r{S}_{0}t$. Substitution and cancellation of $r{S}_0^2$ give the stated spatial SIR model withDrop tildes. A travelling wave with $z=x-ct$ obeysAt the leading edge $S=1,I=0$, the infective equation linearizes to $I^{\prime \prime }+c{I}^{\prime }+(1-\lambda )I=0$. For an invading epidemic the susceptible level must exceed the removal threshold, $\lambda <1$. A nonoscillatory decaying tail has exponentsIf the discriminant is negative, a nonzero real tail oscillates in sign, violating nonnegative infective density. Thus for a right-moving waveThis is a necessary admissibility condition from the leading-edge calculation, not by itself a global existence proof.

To derive the final size relation for an epidemic, use $I=c{S}^{\prime }/S$ and integrate the infective ODE across the wave. The tails have $I,I^{\prime }\to 0$, givingThereforeFor a nontrivial epidemic $0<\sigma <1$. The function $s-\lambda \log s$ is strictly convex, minimized at $s=\lambda$. For $0<\lambda <1$ it has the trivial root $1$ and one further root $0<\sigma <\lambda$. For $\lambda \ge 1$, it exceeds one for every $s<1$, so only the no-epidemic root remains; this also proves the necessity of $\lambda <1$. On the nontrivial branch,It rises from $\sigma \sim e^{-1/\lambda }$ as $\lambda \downarrow 0$ to $\sigma =1-2(1-\lambda )+O((1-\lambda {)}^2)$ as $\lambda \uparrow 1$. The sketch distinguishes this branch from the always-present trivial root.