A bootstrap particle filter for viral Rt inference and forecasting using wastewater data

Wastewater is increasingly being recognized as an important data stream that can contribute to infectious disease surveillance and forecasting. With this recognition, a growing number of statistical inference approaches are being developed to use wastewater data to provide quantitative insights into epidemiological dynamics. However, few existing approaches have allowed for systematic integration of data streams for inference, for example by combining case incidence data and/or serological data with wastewater data. Furthermore, only a subset of existing approaches have been able to handle missing data without imputation and to handle datasets with different sampling times or intervals. Here, we develop a statistically rigorous, yet lightweight, approach to infer and forecast time-varying effective reproduction numbers (Rt values) using longitudinal wastewater virus concentrations either alone or jointly with additional data streams including case incidence data and serological data. Our approach relies on a state-space modeling approach for inference and forecasting, within the context of a simple bootstrap particle filter. We first describe the structure of our underlying disease transmission process model as well as our observation models. Using a mock dataset, we then show that Rt can be accurately estimated by interfacing this model with case incidence data, wastewater data, or a combination of these two data streams using the bootstrap particle filter. Of note, we show that these data streams alone do not allow for reconstruction of underlying infection dynamics due to structural parameter unidentifiability. We then apply our particle filter to a previously analyzed SARS-CoV-2 dataset from Zurich that includes case data and wastewater data. Our analyses of these real-world datasets indicate that incorporation of process noise (in the form of environmental stochasticity) into the state space model greatly improves our ability to reconstruct the latent variables of the model. We further show that underlying infection dynamics can be made identifiable through the incorporation of serological data and that the bootstrap particle filter can be used to make forecasts of Rt, case incidence, and wastewater virus concentrations. We hope that the inference approach presented here will lead to greater reliance on wastewater data for disease surveillance and forecasting that will aid public health practitioners in responding to infectious disease threats.