  At the onset of the Covid-19 pandemic, a number of non-pharmaceutical
interventions have been implemented in order to reduce transmission, thus
leading to multiple phases of transmission. The disease reproduction number
$R_t$, a way of quantifying transmissibility, has been a key part in assessing
the impact of such interventions. We discuss the distinct types of transmission
models used and how they are linked. We consider a hierarchical stochastic
epidemic model with piece-wise constant $R_t$, appropriate for modelling the
distinct phases of the epidemic and quantifying the true disease magnitude. The
location and scale of $R_t$ changes are inferred directly from data while the
number of transmissibility phases is allowed to vary. We determine the model
complexity via appropriate Poisson point process and Dirichlet process-type
modelling components. The models are evaluated using synthetic data sets and
the methods are applied to freely available data from California and New York
states as well as the United Kingdom and Greece. We estimate the true infected
cases and the corresponding $R_t$, among other quantities, and independently
validate the proposed approach using a large seroprevalence study.
