Learning delay dynamics for multivariate stochastic processes, with application to the prediction of the growth rate of COVID-19 cases in the United States

TL;DR

This paper develops a method to learn delay differential equation models from observed stochastic trajectories, applying it to predict COVID-19 growth rates in U.S. states using functional data analysis.

Contribution

It introduces a framework for inverse modeling of delay differential equations from data, including existence and uniqueness results, and applies it to epidemiological and economic data.

Findings

Successfully models COVID-19 growth rate dynamics across U.S. states.

Demonstrates the effectiveness of delay differential equation models in capturing complex stochastic processes.

Provides a novel approach for inverse modeling in systems with delay and functional predictors.

Abstract

Delay differential equations form the underpinning of many complex dynamical systems. The forward problem of solving random differential equations with delay has received increasing attention in recent years. Motivated by the challenge to predict the COVID-19 caseload trajectories for individual states in the U.S., we target here the inverse problem. Given a sample of observed random trajectories obeying an unknown random differential equation model with delay, we use a functional data analysis framework to learn the model parameters that govern the underlying dynamics from the data. We show existence and uniqueness of the analytical solutions of the population delay random differential equation model when one has discrete time delays in the functional concurrent regression model and also for a second scenario where one has a delay continuum or distributed delay. The latter involves a functional linear regression model with history index. The derivative of the process of interest is modeled using the process itself as predictor and also other functional predictors with predictor-specific delayed impacts. This dynamics learning approach is shown to be well suited to model the growth rate of COVID-19 for the states that are part of the U.S., by pooling information from the individual states, using the case process and concurrently observed economic and mobility data as predictors.