 Solution

Let $u$ be the infection time and $s=t-u\ge 0$ its incubation delay, with independent delays having probability density function $f(s)$. Each infection contributes to the symptom-onset intensity according to its delay density. Back-calculation of infection incidence uses the convolutionObserved onset intensity is infection intensity convolved with the incubation distribution. If infections begin at a known $t_0$, take $h(u)=0$ for $u<t_0$, reducing the first integral's lower limit to $t_0$. Otherwise past infections must be included; the observation window's start need not be the infection process's start.