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SARS-CoV-2 virus has spread over the world rapidly creating one of the largest pandemic ever. The absence of immunity, presymptomatic transmission, and the relatively high level of virulence of the COVID-19 infection led to a massive flow of patients in critical care units (CCU). This unprecedented situation called for rapid and accurate mathematical models to best inform public health policies.
Here, we introduce an original discrete-time model -- developped in March 2020 and gradually improved since -- that combines the computational and tractability benefits of deterministic systems and the short-term accuracy of non-Markovian dynamics, taking into account the effect of age of infection on the natural history of the disease. By analysing the hospital time series of COVID-19 in France, we were able to provide early estimates of the main epidemiological parameters with only limited publicly available data. Since then, the model provided an everyday robust framework used to nowcast the epidemic in France, investigate counterfactual scenarios of public health interventions and anticipate CCU overload, while gradually updating hospital care kinetics, variant of concern dynamics and vaccine rollout. The model translated into an online forecasting application and several published projections and support to decision-making at both local and national scale. In the light of almost three years of the pandemic, through failures and successes, we review the real-time improvement history of the model and its related communication and implications with health institutions, authorities and medias and we argue for parsimony as a guideline for the development of projective models in the context of an emerging infectious disease.