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 R t 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 R t 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 R t . But it estimates R t with a significant temporal delay. Here, we propose a new method, EpiInvert , that shows good agreement with EpiEstim, but that provides estimates of R t several days in advance. We show that R t can be estimated by inverting the renewal equation linking R t with the observed incidence curve of new cases, i t . Our signal processing approach to this problem yields both R t and a restored i t 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 <ext-link xmlns:xlink="http://www.w3.org/1999/xlink" ext-link-type="uri" xlink:href="http://www.ipol.im/epiinvert">www.ipol.im/epiinvert</ext-link> .
Significance Statement
Based on a signal processing approach we propose a method to compute the reproduction number R t , the transmission potential of an epidemic over time. R t is estimated by minimizing a functional that enforces: (i) the ability to produce an incidence curve i t corrected of the weekly periodic bias produced by the “weekend effect”, obtained from R t through a renewal equation; (ii) the regularity of R t . A good agreement is found between our R t estimate and the one provided by the currently accepted method, EpiEstim, except our method predicts R t 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 <ext-link xmlns:xlink="http://www.w3.org/1999/xlink" ext-link-type="uri" xlink:href="http://www.ipol.im/epiinvert">www.ipol.im/epiinvert</ext-link> .
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