Delay differential equations for the spatially-resolved simulation of epidemics with specific application to COVID-19

TL;DR

This paper introduces delay differential equations (DDEs) for spatially-resolved epidemic modeling, specifically applied to COVID-19, offering a natural way to incorporate delays like incubation periods with computational and modeling advantages.

Contribution

The paper develops a novel DDE-based epidemic model in both ODE and PDE frameworks, demonstrating stability analysis and validating against real COVID-19 data.

Findings

DDE models effectively capture delays such as incubation periods.

Mathematical stability of the models is established.

Numerical experiments confirm the model's predictive capacity.

Abstract

In the wake of the 2020 COVID-19 epidemic, much work has been performed on the development of mathematical models for the simulation of the epidemic, and of disease models generally. Most works follow the susceptible-infected-removed (SIR) compartmental framework, modeling the epidemic with a system of ordinary differential equations. Alternative formulations using a partial differential equation (PDE) to incorporate both spatial and temporal resolution have also been introduced, with their numerical results showing potentially powerful descriptive and predictive capacity. In the present work, we introduce a new variation to such models by using delay differential equations (DDEs). The dynamics of many infectious diseases, including COVID-19, exhibit delays due to incubation periods and related phenomena. Accordingly, DDE models allow for a natural representation of the problem dynamics, in addition to offering advantages in terms of computational time and modeling, as they eliminate the need for additional, difficult-to-estimate, compartments (such as exposed individuals) to incorporate time delays. Here, we introduce a DDE epidemic model in both an ordinary- and partial differential equation framework. We present a series of mathematical results assessing the stability of the formulation. We then perform several numerical experiments, validating both the mathematical results and establishing model's ability to reproduce measured data on realistic problems.