Proximal schemes for the estimation of the reproduction number of Covid19 pandemic: from convex optimization to Monte Carlo sampling

Monitoring the Covid19 pandemic constitutes a critical societal stake that received considerable research efforts. Raw infection counts are not informative enough about the pandemic spread dynamics, and one has to recourse to more advanced epidemiological indicators, the most popular being the reproduction number, defined in Cori's model as the average number of secondary cases caused by an infected individual. Though, the quality of Covid19 data, consisting in daily new infection counts reported by public health authorities, is low (due, e.g., to pseudo-seasonalities, irrelevant or missing counts), making robust estimation of daily reproduction numbers very challenging.
A first approach consists in designing a nonsmooth convex functional, whose minimization performs jointly a correction of erroneous counts and a temporal regularization, yielding epidemiologically relevant piecewise linear estimates of the reproduction number. A second approach aims at enriching the aforementioned pointwise estimates with a level of confidence, which is crucial for sanitary policies design and assessment. The variational approach have thus been reinterpreted in a Bayesian framework permitting credibility interval estimation. Leveraging proximal schemes used for nonsmooth functional minimization, proximal Langevin-based Monte Carlo sampling algorithms have been derived. These Markov chain Monte Carlo methods yield credibility interval estimates of both the reproduction number and of corrected new infection counts for more than 200 countries worldwide. 
The purpose of this talk is to draw a path from variational formulations, designed in an inverse problem spirit, to novel Markov chain Monte Carlo methods, intertwining Langevin proposal mechanisms and nonsmooth convex optimization tools to yield Bayesian estimates. The developed tools will be illustrated at work on real Covid19 data.