Appropriately smoothing prevalence data to inform estimates of growth rate and reproduction number

The time-varying reproduction number (R_t) can change rapidly over the course of a pandemic due to changing restrictions, behaviours, and levels of population immunity. Many methods exist that allow the estimation of R_t from case data. However, these are not easily adapted to point prevalence data nor can they infer R_t across periods of missing data. We developed a Bayesian P-spline model suitable for fitting to a wide range of epidemic time-series, including point-prevalence data. We demonstrate the utility of the model by fitting to periodic daily SARS-CoV-2 swab-positivity data in England from the first 7 rounds (May 2020-December 2020) of the REal-time Assessment of Community Transmission-1 (REACT-1) study. Estimates of R_t over the period of two subsequent rounds (6-8 weeks) and single rounds (2-3 weeks) inferred using the Bayesian P-spline model were broadly consistent with estimates from a simple exponential model, with overlapping credible intervals. However, there were sometimes substantial differences in point estimates. The Bayesian P-spline model was further able to infer changes in R_t over shorter periods tracking a temporary increase above one during late-May 2020, a gradual increase in R_t over the summer of 2020 as restrictions were eased, and a reduction in R_t during England's second national lockdown followed by an increase as the Alpha variant surged. The model is robust against both under-fitting and over-fitting and is able to interpolate between periods of available data; it is a particularly versatile model when growth rate can change over small timescales, as in the current SARS-CoV-2 pandemic. This work highlights the importance of pairing robust methods with representative samples to track pandemics.