Among the different indicators that quantify the spread of an epidemic such as the ongoing COVID-19, stands first the reproduction number which measures how many people can be contaminated by an infected person. In order to permit the monitoring 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 observations. The novelty of the proposed approach is twofold: 1) the estimation of the reproduction 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 departments and their joint analysis, leading to dynamic maps revealing the temporal coevolution of their reproduction numbers.

Because of the ever-increasing collections of multivariate data, multivariate selfsimilarity has become a widely used model for scale-free dynamics, with successful applications in numerous different fields. Multivariate selfsimilarity exponent estimation has therefore received considerable attention, with notably an original procedure recently proposed and based on the eigenvalues of the covariance random matrices of the wavelet coefficients at fixed scales. Expanding on preliminary work aiming to test for the equality of the selfsimilarity exponents in bivariate time series, we propose and study here a truly multivariate procedure that permits, from a single observation of multivariate time series, to test for the equality of several, possibly many, selfsimilarity exponents. It is based on an original bootstrap procedure, applied in a multivariate time-scale domain and designed to effectively capture the scale-dependent joint covariance structure of multivariate wavelet coefficients as well as the associated wavelet eigenvalue structures. Extensive simulations conducted on synthetic data, modeled by operator fractional Brownian motions, the reference multivariate selfsimilarity model, permit to show that the proposed multivariate timescale domain bootstrap based test yields the targeted significance level under the null hypothesis (all selfsimilarity exponents are equal) and to assess the power of the test for several alternative hypotheses. This analysis leads us to conclude that the proposed test for the equality of multivariate selfsimilarity exponents is effective and ready for use on (a single time series of) real data.