= Solution
At the leading edge $S\simeq1$ and $I$ is small, so
$$
I''+cI'+(1-\mu)I=0.
$$
Seek a decaying mode $I\sim e^{-\lambda\xi}$ with $\lambda>0$. Its characteristic equation is
$$
\lambda^2-c\lambda+(1-\mu)=0.
$$
A nonoscillatory positive leading edge requires real positive roots, hence
$$
c^2\geq4(1-\mu).
$$
For an invasive infection, $mu<1$, and the selected minimum speed is therefore
$$
\boxed{c_{\min}=2\sqrt{1-\mu}.}
$$
If $\mu\geq1$, the state ahead of the wave is below the epidemic threshold and no growing infection front is selected.
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