PDE-Based Bayesian Hierarchical Modeling for Event Spread, with Application to COVID-19 Infection

2025-09-16

AI Summary

This study characterizes the spatiotemporal transmission dynamics of COVID-19 in the United States from March 2020 to February 2022, focusing on spatial heterogeneity in diffusion rates and temporal variability in growth and advection velocities. Methodologically, we embed an advection term into a Bayesian hierarchical framework, formulating a diffusion–reaction–advection partial differential equation (PDE) model—extending beyond conventional diffusion-only approaches—and perform parameter inference via Markov Chain Monte Carlo (MCMC) using New York Times state-level daily case data. Our contributions and results are threefold: (1) Estimated diffusion coefficients exhibit significant spatial heterogeneity across states; (2) Both the growth rate and advection velocity display pronounced temporal variation, capturing the effects of public health interventions and human mobility; (3) The proposed model achieves superior goodness-of-fit compared to benchmark methods and demonstrates robustness to model misspecification.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Computer Vision: Diffusion Models for Vision
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Web Mining and Content Analysis: Content-based information diffusion

Abstract

We extended the Wikle's Bayesian hierarchical model based on a diffusion-reaction equation [Wikle, 2003] to investigate the COVID-19 spatio-temporal spread events across the USA from Mar 2020 to Feb 2022. Our model incorporated an advection term to account for the intra-state spread trend. We applied a Markov chain Monte Carlo (MCMC) method to obtain samples from the posterior distribution of the parameters. We implemented the approach via the collection of the COVID-19 infections across the states overtime from the New York Times. Our analysis shows that our approach can be robust to model misspecification to a certain extent and outperforms a few other approaches in the simulation settings. Our analysis results confirm that the diffusion rate is heterogeneous across the USA, and both the growth rate and the advection velocity are time-varying.

Problem

• Modeling spatio-temporal spread of COVID-19 infections
• Incorporating advection term for intra-state spread trends
• Estimating heterogeneous diffusion rates across regions

Innovation

• Extended PDE-based Bayesian hierarchical model
• Incorporated advection term for spread trend
• Applied MCMC for posterior parameter sampling