PDE-based Bayesian hierarchical modeling for event spread, with application to COVID-19 infection

Motivated partly by the ongoing COVID-19 pandemic, this project aims to develop a tool for investigating spatio-temporal spread of events. We use the records of American coronavirus disease cases over time from The New York Times to motivate and illustrate the methodological development. Wikle (2003) considers a Bayesian hierarchical model based on a diffusion-reaction equation with a space-varying diffusion rate to describe the latent spatiotemporal process underlying a collection of bird migration data. We extend the model by adding an advection term to account for the additional trend of the transmission, and considering time-varying reaction and advection terms. A Markov chain Monte Carlo (MCMC) method is applied to obtain samples from the posterior distribution of the parameters. The proposed approach is implemented via the COVID-19 data from The New York Times. The analysis results indicate that the diffusion rate is heterogeneous across USA, and the growth rate and the advection velocity are time-varying. We verify the findings from the analysis by simulation. The proposed approach appears robust to model misspecification and outperforms other approaches in the simulation settings.