  Daily pandemic surveillance, often achieved through the estimation of the
reproduction number, constitutes a critical challenge for national health
authorities to design countermeasures. In an earlier work, we proposed to
formulate the estimation of the reproduction number as an optimization problem,
combining data-model fidelity and space-time regularity constraints, solved by
nonsmooth convex proximal minimizations. Though promising, that first
formulation significantly lacks robustness against the Covid-19 data low
quality (irrelevant or missing counts, pseudo-seasonalities,.. .) stemming from
the emergency and crisis context, which significantly impairs accurate pandemic
evolution assessments. The present work aims to overcome these limitations by
carefully crafting a functional permitting to estimate jointly, in a single
step, the reproduction number and outliers defined to model low quality data.
This functional also enforces epidemiology-driven regularity properties for the
reproduction number estimates, while preserving convexity, thus permitting the
design of efficient minimization algorithms, based on proximity operators that
are derived analytically. The explicit convergence of the proposed algorithm is
proven theoretically. Its relevance is quantified on real Covid-19 data,
consisting of daily new infection counts for 200+ countries and for the 96
metropolitan France counties, publicly available at Johns Hopkins University
and Sant{\'e}-Publique-France. The procedure permits automated daily updates of
these estimates, reported via animated and interactive maps. Open-source
estimation procedures will be made publicly available.
