An epidemical model with nonlocal spatial infections

TL;DR

This paper extends the SIR epidemiological model into a nonlocal PDE framework, providing tools for analyzing spatial infection dynamics, stability, and data-driven identification of epidemic quantities in one- and two-dimensional settings.

Contribution

It introduces a nonlocal, nonlinear PDE extension of the SIR model and develops analytical, computational, and data-driven tools for understanding spatial epidemic spread.

Findings

Analysis of stationary states and their stability.

Visualization of spatio-temporal infection progression.

Development of moment-based diagnostics and SINDy-based modeling.

Abstract

The SIR model is one of the most prototypical compartmental models in epidemiology. Generalizing this ordinary differential equation (ODE) framework into a spatially distributed partial differential equation (PDE) model is a considerable challenge. In the present work, we extend a recently proposed model based on nearest-neighbor spatial interactions by one of the authors in~\cite{vaziry2022modelling} towards a nonlocal, nonlinear PDE variant of the SIR prototype. We then seek to develop a set of tools that provide insights for this PDE framework. Stationary states and their stability analysis offer a perspective on the early spatial growth of the infection. Evolutionary computational dynamics enable visualization of the spatio-temporal progression of infection and recovery, allowing for an appreciation of the effect of varying parameters of the nonlocal kernel, such as, e.g., its width parameter. These features are explored in both one- and two-dimensional settings. At a model-reduction level, we develop a sequence of interpretable moment-based diagnostics to observe how these reflect the total number of infections, the epidemic's epicenter, and its spread. Finally, we propose a data-driven methodology based on the sparse identification of nonlinear dynamics (SINDy) to identify approximate closed-form dynamical equations for such quantities. These approaches may pave the way for further spatio-temporal studies, enabling the quantification of epidemics.