An Approximate Joint Distribution for Stochastic Differential Equations, with Application to Modeling the COVID-19 Epidemic in Pennsylvania, Rhode Island, and Massachusetts

• Activity Number: 12 - High-Dimensional Parameter Learning on Spatio-Temporal Hidden Markov Models and Its Applications in Epidemiology
• Type: Invited
• Date/Time: Sunday, August 8, 2021 : 1:30 PM to 3:20 PM
• Sponsor: IMS
• Author(s): Ephraim Hanks*
• Companies: Penn State University
• Keywords: SDE; infectious disease; differential equation; hidden markov model; mechanistic models; stochastic differential equation

Abstract

Stochastic differential equations (SDEs) are important modeling tools in a wide variety of disciplines from epidemiology to ecology and climate science. One challenge to inference in this class of models is that an accurate representation of latent dynamics requires a numerical approximation to the SDE with a fine temporal resolution. This results in the need to infer or integrate over a large latent state space. In this work, we develop a novel approach for statistical inference where the joint distribution of the latent SDE at an arbitrary set of time points can be approximated as a multivariate normal distribution with mean and covariance function a function only of the deterministic ODE limit of the SDE. This approximation removes the need to infer stochastic infill points; it instead only requires deterministic infill, and also allows for the direct evaluation of the joint likelihood of the SDE at all observation times. We formally define error bounds for our approximation and show that inference is at least as accurate as standard existing approaches. We apply these methods to the COVID-19 epidemic in three U.S. states.