EpiLPS: A fast and flexible Bayesian tool for estimation of the time-varying reproduction number

Abstract

In infectious disease epidemiology, the instantaneous reproduction number [Formula: see text] is a time-varying parameter defined as the average number of secondary infections generated by an infected individual at time t. It is therefore a crucial epidemiological statistic that assists public health decision makers in the management of an epidemic. We present a new Bayesian tool (EpiLPS) for robust estimation of the time-varying reproduction number. The proposed methodology smooths the epidemic curve and allows to obtain (approximate) point estimates and credible intervals of [Formula: see text] by employing the renewal equation, using Bayesian P-splines coupled with Laplace approximations of the conditional posterior of the spline vector. Two alternative approaches for inference are presented: (1) an approach based on a maximum a posteriori argument for the model hyperparameters, delivering estimates of [Formula: see text] in only a few seconds; and (2) an approach based on a Markov chain Monte Carlo (MCMC) scheme with underlying Langevin dynamics for efficient sampling of the posterior target distribution. Case counts per unit of time are assumed to follow a negative binomial distribution to account for potential overdispersion in the data that would not be captured by a classic Poisson model. Furthermore, after smoothing the epidemic curve, a "plug-in'' estimate of the reproduction number can be obtained from the renewal equation yielding a closed form expression of [Formula: see text] as a function of the spline parameters. The approach is extremely fast and free of arbitrary smoothing assumptions. EpiLPS is applied on data of SARS-CoV-1 in Hong-Kong (2003), influenza A H1N1 (2009) in the USA and on the SARS-CoV-2 pandemic (2020-2021) for Belgium, Portugal, Denmark and France. Author summary The instantaneous reproduction number R-t is a key statistic that provides important insights into an epidemic outbreak as it informs about the average number of secondary infections engendered by an infectious agent. We present a flexible Bayesian approach called EpiLPS (Epidemiological modeling with Laplacian-P-Splines) for efficient estimation of the epidemic curve and Rt based on daily case count data and the serial interval distribution. Computational speed and absence of arbitrary assumptions on smoothing makes EpiLPS an interesting tool for estimation of the reproduction number. Our methodology is validated through different simulation scenarios by using the associated R software package (https://cran.r-project.org/package=EpiLPS). We also demonstrate the use of EpiLPS on real data from two historical outbreaks and on the SARS-CoV-2 pandemic.