= Solution
Introduce dimensionless quantities $\widetilde S=S/S_0$, $\widetilde I=I/S_0$, $\widetilde x=x\sqrt{rS_0/D}$ and $\widetilde t=rS_0t$. Substitution and cancellation of $rS_0^2$ give the stated <spatial SIR model> with
$$
\boxed{\lambda=\frac a{rS_0}.}
$$
Drop tildes. A <travelling wave> with $z=x-ct$ obeys
$$
\boxed{cS'=IS,\qquad I''+cI'+(S-\lambda)I=0.}
$$
At the leading edge $S=1,I=0$, the infective equation linearizes to $I''+cI'+(1-\lambda)I=0$. For an invading epidemic the susceptible level must exceed the removal threshold, $\lambda<1$. A nonoscillatory decaying tail has exponents
$$
r_\pm=\frac{-c\pm\sqrt{c^2-4(1-\lambda)}}2.
$$
If the discriminant is negative, a nonzero real tail oscillates in sign, violating nonnegative infective density. Thus for a right-moving wave
$$
\boxed{c\ge2\sqrt{1-\lambda},\qquad\lambda<1.}
$$
This 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 $I=cS'/S$ and integrate the infective ODE across the wave. The tails have $I,I'\to0$, giving
$$
c(1-\sigma)=\lambda\int_{-\infty}^\infty I(z)\,dz,
\qquad \int I\,dz=c\log(1/\sigma).
$$
Therefore
$$
\boxed{\sigma-\lambda\log\sigma=1.}
$$
For 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 $\lambda\ge1$, 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,
$$
\frac{d\sigma}{d\lambda}=\frac{\log\sigma}{1-\lambda/\sigma}>0.
$$
It rises from $\sigma\sim e^{-1/\lambda}$ as $\lambda\downarrow0$ to $\sigma=1-2(1-\lambda)+O((1-\lambda)^2)$ as $\lambda\uparrow1$. The sketch distinguishes this branch from the always-present trivial root.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2010/ii/paper-3-epidemic-final-size.png]
{title=Surviving susceptible fraction on the nontrivial epidemic branch and the no-epidemic root}
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