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.

翻译：尽管医学数据收集取得了进展，但由于病例漏报，SARS-CoV-2的实际负担仍未知。这在疫情急性期尤为明显，而报告死亡数已被指出是更可靠的信息来源，其漏报可能性较低。由于每日死亡人数源于既往感染及其死亡概率的加权，因此可利用报告死亡数据，根据年龄分布推断总感染人数。我们采用这一框架，假设总感染人数的动态变化可由连续时间传播模型描述，该模型通过非线性常微分方程组表示，其中传播速率被建模为扩散过程，从而揭示控制策略效果和个体行为变化。我们在Stan中开发了这一灵活贝叶斯工具，并研究了3对欧洲国家，估算了时变再生数($R_t$)及真实累积感染人数。通过估算真实感染人数，我们提供了更准确的$R_t$估计值。我们还估算了每日报告率，并讨论了流动性和检测变化对推断量的影响。