Consider an epidemic model in which $S(x, t)$ is the local population density of susceptibles and $I(x, t)$ is the density of infectives

$$\begin{aligned} &\frac{\partial S}{\partial t}=-r I S \\ &\frac{\partial I}{\partial t}=D \frac{\partial^{2} I}{\partial x^{2}}+r I S-a I \end{aligned}$$

where $r, a$, and $D$ are positive. If $S_{0}$ is a characteristic population value, show that the rescalings $I / S_{0} \rightarrow I, S / S_{0} \rightarrow S,\left(r S_{0} / D\right)^{1 / 2} x \rightarrow x, r S_{0} t \rightarrow t$ reduce this system to

$$\begin{aligned} &\frac{\partial S}{\partial t}=-I S \\ &\frac{\partial I}{\partial t}=\frac{\partial^{2} I}{\partial x^{2}}+I S-\lambda I \end{aligned}$$

where $\lambda$ should be found.

Travelling wavefront solutions are of the form $S(x, t)=S(z), I(x, t)=I(z)$, where $z=x-c t$ and $c$ is the wave speed, and we seek solutions with boundary conditions $S(\infty)=1, S^{\prime}(\infty)=0, I(\infty)=I(-\infty)=0$. Under the travelling-wave assumption reduce the rescaled PDEs to ODEs, and show by linearisation around the leading edge of the advancing front that the requirement that $I$ be non-negative leads to the condition $\lambda<1$ and hence the wave speed relation

$$c \geqslant 2(1-\lambda)^{1 / 2}, \quad \lambda<1$$

Using the two ODEs you have obtained, show that the surviving susceptible population fraction $\sigma=S(-\infty)$ after the passage of the front satisfies

$$\sigma-\lambda \ln \sigma=1,$$

and sketch $\sigma$ as a function of $\lambda$.