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 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 and 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.

Modeling Covid-19 incidence by the renewal equation after removal of administrative bias and noise

“…Figs. 5,A1,A2 show the minimization results for the energy (7). They display for each country (i) the original incidence curve i 0 t , (ii) the incidence curve after bias correction ît , (iii) the restored incidence curve i r t using the renewal equation ( 1), (iv) the weekly bias correction factors q t , (v) the reproduction number estimation R t and (vi) the normalized error defined by…”

“…In [7] we have proven extensively by simulations and experiments on live worldwide Covid-19 incidence data that using the simplified causal renewal equation (11) incurs in a five days delay in the estimation of R t , compared to the Nishiura renewal equation (1). This is why we used here this second model.…”

“…This is the case of [23], [24] [25]. This last paper also involves the Wallinga-Teunis formulation [2], also based on the renewal equation but only allowing a backward estimate of R t (see the discussion in [7]). Some papers like [26] propose a simplified version of (11).…”

“…The EpiInvert variational method proposed in [7] is a deconvolution + denoising procedure estimating both R t and a restored i t corrected for the weekend bias. To remove the weekend effect, it computes a 7-day periodic multiplicative factor q = ( q 0 , q 1 , q 2 , q 3 , q 4 , q 5 , q 6 ).…”

“…These recording delays and subsequent rash corrections result in impulse noise, and in a strong weekly periodic bias observable on the observed incidence curve . In [3] this bias is corrected by a seven days sliding average and in [7] it is corrected by multiplying by a 7-day periodic factor q t . These bias correcting coefficients q t are learned by a variational method that we describe below.…”