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

Abstract

In infectious disease epidemiology, the instantaneous reproduction number R(t) is a timevarying metric defined as the average number of secondary infections generated by individuals who are infectious at time t. It is therefore a crucial epidemiological parameter that assists public health decision makers in the management of an epidemic. We present a new Bayesian tool 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 envelopes of R(t) 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 R(t) in only a few seconds; and (2) an approach based on a 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 excess variability 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 R(t) 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 current SARS-CoV-2 pandemic (2020-2021) for Belgium, Portugal, Denmark and France.Author summaryThe instantaneous reproduction number R(t) is a key metric that provides important insights into an epidemic outbreak. We present a flexible Bayesian approach called EpiLPS (Epidemiological modeling with Laplacian-P-splines) for smooth estimation of the epidemic curve and R(t). Computational speed and absence of arbitrary assumptions on smoothing makes EpiLPS an interesting tool for near real-time estimation of the reproduction number. An R software package is available (https://github.com/oswaldogressani).

An efficient approach to nowcasting the time-varying reproduction number

“…The former data stream ignores the reporting delays, while the latter takes these delays into account but ignores the uncertainty associated with the delay dimension. In both scenarios, estimation of R t is carried out using the estimR() routine of the EpiLPS package (Gressani, 2021). To summarize, the simulation study allows to compare three models, namely M 1 providing R t estimates using reported cases only, M 2 providing R t estimates using nowcasted incidence data and M 3 , our proposed model, based on joint modeling of delay and the time-varying reproduction number.…”

Bayesian nowcasting with Laplacian-P-splines

“…Specifically, we fit the following model log(µ t,d ) = 1), where 6 l=1 β l z l (t, d) represents the day of the week with Monday taken as the reference category. Algorithms to fit the model are available within the EpiLPS package (Gressani, 2021) through the nowcasting() routine.…”

“…While the proposed nowcasting model is rather complex, we have coded the nowcasting() routine in the EpiLPS package (Gressani (2021)) to provide a user-friendly experience.…”

“…Therefore, given the flexibility of previously mentioned P-splines smoothers and the computational benefit of Laplace approximations, it can be a helpful tool in the daily monitoring of new cases during an epidemic period. Laplacian-P-splines already proved to be useful in survival models (Gressani and Lambert, 2018; Gressani et al, 2022), generalized additive models (Gressani and Lambert, 2021) and also in epidemic models for estimating the effective reproduction number (Gressani et al, 2021). We build on the work of Gressani and Lambert (2021) to extend the Laplacian-P-splines methodology to now-casting, thereby providing a fast and flexible (fully) Bayesian alternative to Van de Kassteele et al (2019).…”

Flexible Bayesian estimation of incubation times

“…Finally, Section 5 concludes with a discussion for future research and limitations of our work. A routine reflecting the proposed methodology to estimate the incubation density has been added to the EpiLPS package (Gressani, 2021) and a dedicated repository (https://github.com/oswaldogressani/Incubation) permits to reproduce the results of the manuscript.…”