In epidemiological modelling, the instantaneous reproduction number, $R_t$, is important to understand the transmission dynamics of infectious diseases. Current $R_t$ estimates often suffer from problems such as lagging, averaging and uncertainties demoting the usefulness of $R_t$. To address these problems, we propose a new method in the framework of sequential Bayesian inference where a Data Assimilation approach is taken for $R_t$ estimation, resulting in the state-of-the-art DAR$_t$ system for $R_t$ estimation. With DAR$_t$, the problem of time misalignment caused by lagging observations is tackled by incorporating observation delays into the joint inference of infections and $R_t$; the drawback of averaging is improved by instantaneous updating upon new observations and a model selection mechanism capturing abrupt changes caused by interventions; the uncertainty is quantified and reduced by employing Bayesian smoothing. We validate the performance of DAR$_t$ through simulations and demonstrate its power in revealing the transmission dynamics of COVID-19.
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