  Monitoring the Covid19 pandemic constitutes a critical societal stake that
received considerable research efforts. The intensity of the pandemic on a
given territory is efficiently measured by the reproduction number, quantifying
the rate of growth of daily new infections. Recently, estimates for the time
evolution of the reproduction number were produced using an inverse problem
formulation with a nonsmooth functional minimization. While it was designed to
be robust to the limited quality of the Covid19 data (outliers, missing
counts), the procedure lacks the ability to output credibility interval based
estimates. This remains a severe limitation for practical use in actual
pandemic monitoring by epidemiologists that the present work aims to overcome
by use of Monte Carlo sampling. After interpretation of the nonsmooth
functional into a Bayesian framework, several sampling schemes are tailored to
adjust the nonsmooth nature of the resulting posterior distribution. The
originality of the devised algorithms stems from combining a Langevin Monte
Carlo sampling scheme with Proximal operators. Performance of the new
algorithms in producing relevant credibility intervals for the reproduction
number estimates and denoised counts are compared. Assessment is conducted on
real daily new infection counts made available by the Johns Hopkins University.
The interest of the devised monitoring tools are illustrated on Covid19 data
from several different countries.
