COVID-19 pandemic has brought to the fore epidemiological models which, though describing a wealth of behaviors, have previously received little attention in signal processing literature. In this work, a generalized time-varying autoregressive model is considered, encompassing, but not reducing to, a state-of-the-art model of viral epidemics propagation. The time-varying parameter of this model is estimated via the minimization of a penalized likelihood estimator. A major challenge is that the estimation accuracy strongly depends on hyperparameters fine-tuning. Without available ground truth, hyperparameters are selected by minimizing specifically designed data-driven oracles, used as proxy for the estimation error. Focusing on the time-varying autoregressive Poisson model, Stein's Unbiased Risk Estimate formalism is generalized to construct asymptotically unbiased risk estimators based on the derivation of an original autoregressive counterpart of Stein's lemma. The accuracy of these oracles and of the resulting estimates are assessed through intensive Monte Carlo simulations on synthetic data. Then, elaborating on recent epidemiological models, a novel weekly scaled Poisson model is proposed, better accounting for intrinsic variability of the contaminations while being robust to reporting errors. Finally, the data-driven procedure is particularized to the estimation of COVID-19 reproduction number from weekly infection counts demonstrating its ability to tackle real-world applications.