A Koopman Operator Framework for Nonlinear Epidemic Dynamics: Application to an SIRSD Model

TL;DR

This paper introduces a Koopman operator-based framework for analyzing complex nonlinear epidemic models, specifically an extended SIRSD model, enabling better prediction of epidemic dynamics using data-driven methods.

Contribution

It develops a Koopman operator approach with EDMD for the SIRSD model, incorporating epidemiological insights and comparing different observables for improved epidemic prediction.

Findings

Koopman approach accurately predicts epidemic peaks and dynamics.

Enriched observables improve the approximation of nonlinear epidemic behavior.

Synthetic data experiments validate the effectiveness of the Koopman-based modeling.

Abstract

We develop and analyze an SIRSD epidemic model, which extends the classical SIR framework by incorporating waning immunity and disease-induced mortality. A rigorous well-posedness analysis ensures the existence, uniqueness, positivity, and boundedness of solutions, guaranteeing the model's epidemiological feasibility. To facilitate theoretical investigations and data-driven modeling, we reformulated the system in normalized variables. To capture and predict complex nonlinear epidemic dynamics, we use the Koopman operator framework with extended dynamic mode decomposition (EDMD) and an epidemiologically informed dictionary of observables. We compare two Koopman approximations: one based on a minimal epidemiological dictionary and another enriched with nonlinear and cross terms. We generate synthetic data using a nonstandard finite difference (NSFD) scheme for four representative epidemics: SARS-CoV-2, seasonal influenza, Ebola, and measles. Numerical experiments demonstrate that the Koopman-based approach effectively identifies dominant epidemic modes and accurately predicts key outbreak characteristics, including peak infection dynamics.