Back-calculation of infection incidence ($\mu (t)={\int }_0^{\infty }h(t-s)f(s)ds$)

Observed symptom-onset intensity is the convolution of past infection intensity with the incubation delay density: $\mu (t)={\int }_0^{\infty }h(t-s)f(s)ds$. Back-calculation infers the infection history from that relation. Include pre-observation infections unless their absence is known.

For equal-width time bins and infections assigned to bin-end time $t_i$, let $q_{\ell }=\mathrm{Pr}((\ell -1)\Delta \le T<\ell \Delta )$. The next onset-bin mean is ${\mu }_k={\sum }_{i<k}{h}_i{q}_{k-i}$. Assigning infections instead to bin starts shifts the index by one. Bin probabilities integrate a delay density; they are not the density itself.