Research Highlights: Spatial and temporal regularization to estimate COVID-19 reproduction number R(t): Promoting piecewise smoothness via convex optimization

  • Basic reproductive number is the expected number of cases directly generated by one case in a population where all individuals are susceptible to infection.
  • For example, if an infected person transmits the disease to three other people, then the reproductive number for that infectious disease is three.
  • The higher the reproductive number, the more the disease is contagious.
  • 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 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.
  • There are two new proposed approach here.
  • First approach: 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.
  • Second approach: 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 co-evolution of their reproduction numbers.


Fraser, Christophe; Donnelly, Christl A.; Cauchemez, Simon; Hanage, William P.; Van Kerkhove, Maria D.; Hollingsworth, T. Déirdre; Griffin, Jamie; Baggaley, Rebecca F.; Jenkins, Helen E.; Lyons, Emily J.; Jombart, Thibaut (June 19, 2009). “Pandemic Potential of a Strain of Influenza A (H1N1): Early Findings”Science324 (5934): 1557–1561. Bibcode:2009Sci…324.1557Fdoi:10.1126/science.1176062ISSN 0036-8075PMC 3735127PMID 19433588.

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