Time-varying reproduction number estimation: Fusing compartmental models with generalised additive models

The reproduction number, the mean number of secondary cases infected by each primary case, is a central metric in infectious disease epidemiology, and played a key role in the COVID-19 pandemic response. This is because it gives an indication of the effort required to control the disease. Beyond the well-known basic reproduction number, there are two natural versions, namely the control and effective reproduction numbers. As behaviour, population immunity and viral characteristics can change with time, these reproduction numbers can vary over time and in different regions. Real world data can be complex, for example with daily variation in numbers due to weekend surveillance biases as well as natural stochastic noise. As such, in this work we consider a Generalised Additive Model to smooth real data through the explicit incorporation of day-of-the-week effects, to provide a simple measure of the time-varying growth rate associated with the data. Converting the resulting spline into an estimator for both the control and effective reproduction numbers requires assumptions on a model structure, which we here assume to be a compartmental model. The reproduction numbers calculated are based on both simulated and real world data, and are compared with estimates from an already existing tool. The derived method for estimating the time-varying reproduction number is effective, efficient and comparable to other methods. It provides a useful alternative approach, which can be included as part of a toolbox of models, that is particularly apt at smoothing out day-of-the-week effects in surveillance. ### Competing Interest Statement The authors have declared no competing interest. ### Funding Statement LP gratefully acknowledges the Wellcome Trust and Royal Society (grant 202562/Z/16/Z). LP, TH and IH are also supported by the JUNIPER modelling consortium (grant MR/V038613/1), by the Alan Turing Institute for Data Science and Artificial Intelligence under the EPSRC grants EP/N510129/1 and EP/V027468/1 and by the UKRI Impact Acceleration Account (IAA 386). YH, LP, TH and IH also acknowledge the UK Health Security Agency (UKHSA) for honorary contracts and funding. The views expressed are those of the author(s) and not necessarily those of the Department of Health or UKHSA. ### Author Declarations I confirm all relevant ethical guidelines have been followed, and any necessary IRB and/or ethics committee approvals have been obtained. Yes I confirm that all necessary patient/participant consent has been obtained and the appropriate institutional forms have been archived, and that any patient/participant/sample identifiers included were not known to anyone (e.g., hospital staff, patients or participants themselves) outside the research group so cannot be used to identify individuals. Yes I understand that all clinical trials and any other prospective interventional studies must be registered with an ICMJE-approved registry, such as ClinicalTrials.gov. I confirm that any such study reported in the manuscript has been registered and the trial registration ID is provided (note: if posting a prospective study registered retrospectively, please provide a statement in the trial ID field explaining why the study was not registered in advance). Yes I have followed all appropriate research reporting guidelines, such as any relevant EQUATOR Network research reporting checklist(s) and other pertinent material, if applicable. Yes All data produced in the present study are available upon reasonable request to the authors