 Solution

In discrete back-calculation with endpoint cohorts, for equal-width intervals $I_k=[t_{k-1},t_k)$, write $\Delta =t_k-t_{k-1}$, $h_i$ for the expected infection count assigned to time $t_i$ under the end-of-interval approximation, and ${\mu }_k$ for the expected onset count in $I_k$. DefineAn infection assigned to $t_i$ can first contribute to $I_{i+1}$, so the discrete convolution with this timing convention isThe sum includes any infection cohorts before the observation window. With no infections before $t_0$ and cohorts beginning at $i=1$, it is ${\sum }_{i=1}^{k-1}$ and the first mean is zero in this endpoint approximation.

For unequal intervals use $q_{ik}=\mathrm{Pr}(t_{k-1}-t_i\le T<t_k-t_i)$, truncating negative endpoints at zero, and ${\mu }_k\approx {\sum }_i{h}_i{q}_{ik}$. An alternative approximation assigning cohort $i$ to $t_{i-1}$ gives ${\sum }_{i\le k}{h}_i{q}_{k-i+1}$. Both are legitimate discretisations; the subsequent solutions use the end-of-interval convention explicitly specified in part (d), so the index shift is essential.