  Despite the progress in medical data collection the actual burden of
SARS-CoV-2 remains unknown due to under-ascertainment of cases. This was
apparent in the acute phase of the pandemic and the use of reported deaths has
been pointed out as a more reliable source of information, likely less prone to
under-reporting. Since daily deaths occur from past infections weighted by
their probability of death, one may infer the total number of infections
accounting for their age distribution, using the data on reported deaths. We
adopt this framework and assume that the dynamics generating the total number
of infections can be described by a continuous time transmission model
expressed through a system of non-linear ordinary differential equations where
the transmission rate is modelled as a diffusion process allowing to reveal
both the effect of control strategies and the changes in individuals behavior.
We develop this flexible Bayesian tool in Stan and study 3 pairs of European
countries, estimating the time-varying reproduction number($R_t$) as well as
the true cumulative number of infected individuals. As we estimate the true
number of infections we offer a more accurate estimate of $R_t$. We also
provide an estimate of the daily reporting ratio and discuss the effects of
changes in mobility and testing on the inferred quantities.
