Spatial and temporal regularization to estimate COVID-19 Reproduction Number R(t): Promoting piecewise smoothness via convex optimization

abry, p.; pustelnik, n.; roux, s.; jensen, p.; Flandrin, P.; gribonval, r.; Lucas, C. G.; guichard, e.; Borgnat, P.; garnier, n.; audit, b.

Among the different indicators that quantify the spread of an epidemic, such as the on-going COVID-19, stands first the reproduction number which measures how many people can be contaminated by an infected person. In order to permit the mon- itoring of the evolution of this number, a new estimation procedure is proposed here, assuming a well-accepted model for current incidence data, based on past observa- tions. The novelty of the proposed approach is twofold: 1) the estimation of the repro- duction number is achieved by convex optimization within a proximal-based inverse problem formulation, with constraints aimed at promoting piecewise smoothness; 2) the approach is developed in a multivariate setting, allowing for the simultaneous handling of multiple time series attached to different geographical regions, together with a spatial (graph-based) regularization of their evolutions in time. The effectiveness of the approach is first supported by simulations, and two main applications to real COVID-19 data are then discussed. The first one refers to the comparative evolution of the reproduction number for a number of countries, while the second one focuses on French counties and their joint analysis, leading to dynamic maps revealing the temporal co-evolution of their reproduction numbers.