Currently, the forward prediction uses simply the last estimated week's Rt to forward simulate the deterministic latent epidemic and case counts. This needs changing on two fronts.
The forward prediction of Rt should be sampled from the posterior for Rt in the next weeks based on the observation of cases. This can be achieved in Stan reasonably simply by sampling extra Rts in the future but never conditioning them on observations.
The forward simulation should be simulated stochastically. The observation model used here is in question. We could either sample observed cases from the negative binomial approximation, or sampled from the delay distribution for each case.
Currently, the forward prediction uses simply the last estimated week's Rt to forward simulate the deterministic latent epidemic and case counts. This needs changing on two fronts.