= Solution
With the onset means $\mu_k(\theta,\lambda,\kappa)$ defined by the discrete <back-calculation of infection incidence>, the <independent> <Poisson observation model> yields \b[the product likelihood]
$$
\boxed{L(\theta,\lambda,\kappa;y_{1:N})
=\prod_{k=1}^N\frac{\mu_k(\theta,\lambda,\kappa)^{y_k}
e^{-\mu_k(\theta,\lambda,\kappa)}}{y_k!}.}
$$
For 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 $t_0$.
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