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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.

For the stated scale-parameterisation of the Weibull distribution, its incubation survivor function is $S_T(s)=\exp [-(s/\lambda {)}^{\kappa }]$ for $s\ge 0$, with $\lambda ,\kappa >0$. The continuous back-calculation of infection incidence equation becomesSet $h(t-s)=0$ before the infection process's start if one is specified. For the equal-width endpoint approximation, integrate the incubation density over each delay bin:Thus the discrete Weibull equation isFor unequal intervals replace $q_{k-i}$ by $S_T(\max \{0,t_{k-1}-t_i\})-S_T(\max \{0,t_k-t_i\})$. These are probabilities, not point evaluations of a density; no additional factor of $\Delta$ is needed after bin integration.

With the onset means ${\mu }_k(\theta ,\lambda ,\kappa )$ defined by the discrete back-calculation of infection incidence, the independent Poisson observation model yields the product likelihoodFor positive means its log-likelihood is ${\sum }_k\{y_k\log {\mu }_k-{\mu }_k-\log (y_k!)\}$. A zero mean assigns probability one to a zero count and zero to a positive count. Any unknown pre-observation infection history must be parameterised or supplied as well; it cannot silently be excluded merely because the observation series starts at $t_0$.