Solution
= Solution
For $f(s)=\sigma e^{-\sigma s}$,
$$
\mu(t)=p\sigma\int_0^te^{-\sigma(t-s)}\lambda(s)ds.
$$
Differentiating the convolution, equivalently integrating by parts, gives
$$
\mu'(t)=p\sigma\lambda(t)-p\sigma^2
\int_0^te^{-\sigma(t-s)}\lambda(s)ds
=\sigma\{p\lambda(t)-\mu(t)\}.
$$
Thus admissions follow a first-order lag: they move toward $p$ times current infection incidence at adjustment rate $\sigma$.