Computing the daily reproduction number of COVID-19 by inverting the renewal equation using a variational technique

Abstract

The COVID-19 pandemic has undergone frequent and rapid changes in its local and global infection rates, driven by governmental measures, or the emergence of new viral variants. The reproduction number Rt indicates the average number of cases generated by an infected person at time t and is a key indicator of the spread of an epidemic. A timely estimation of Rt is a crucial tool to enable governmental organizations to adapt quickly to these changes and assess the consequences of their policies. The EpiEstim method is the most widely accepted method for estimating Rt. But it estimates Rt with a significant temporal delay. Here, we propose a new method, EpiInvert, that shows good agreement with EpiEstim, but that provides estimates of Rt several days in advance. We show that Rt can be estimated by inverting the renewal equation linking Rt with the observed incidence curve of new cases, it. Our signal processing approach to this problem yields both Rt and a restored it corrected for the “weekend effect” by applying a deconvolution + denoising procedure. The implementations of the EpiInvert and EpiEstim methods are fully open-source and can be run in real-time on every country in the world, and every US state through a web interface at www.ipol.im/epiinvert.Significance StatementBased on a signal processing approach we propose a method to compute the reproduction number Rt, the transmission potential of an epidemic over time. Rt is estimated by minimizing a functional that enforces: (i) the ability to produce an incidence curve it corrected of the weekly periodic bias produced by the “weekend effect”, obtained from Rt through a renewal equation; (ii) the regularity of Rt. A good agreement is found between our Rt estimate and the one provided by the currently accepted method, EpiEstim, except our method predicts Rt several days closer to present. We provide the mathematical arguments for this shift. Both methods, applied every day on each country, can be compared at www.ipol.im/epiinvert.