High order discretization methods for spatial-dependent epidemic models

TL;DR

This paper develops high-order numerical methods for a spatially dependent epidemic model, analyzing stability, positivity, and accuracy, with computational experiments confirming their effectiveness.

Contribution

It introduces high-order discretization schemes for spatial epidemic models with integral terms, ensuring stability and accuracy while preserving key properties.

Findings

High-order schemes maintain stability under specific conditions.

Numerical methods accurately approximate the continuous model.

Computational experiments confirm convergence and effectiveness.

Abstract

In this paper, an epidemic model with spatial dependence is studied and results regarding its stability and numerical approximation are presented. We consider a generalization of the original Kermack and McKendrick model in which the size of the populations differs in space. The use of local spatial dependence yields a system of partial-differential equations with integral terms. The uniqueness and qualitative properties of the continuous model are analyzed. Furthermore, different spatial and temporal discretizations are employed, and step-size restrictions for the discrete model's positivity, monotonicity preservation, and population conservation are investigated. We provide sufficient conditions under which high-order numerical schemes preserve the stability of the computational process and provide sufficiently accurate numerical approximations. Computational experiments verify the convergence and accuracy of the numerical methods.