The Discrete SIR Epidemic Reaction–Diffusion Model: Finite-Time Stability and Numerical Simulations

Abstract

This paper investigates the finite-time stability (FTS) of a discrete SIR epidemic reaction–diffusion (R-D) model. The study begins with discretizing a continuous R-D system using finite difference methods (FDMs), ensuring that essential characteristics like positivity and consistency are maintained. The resulting discrete model captures the interplay between spatial heterogeneity, diffusion rates, and reaction dynamics, enabling a robust framework for theoretical analysis. Employing Lyapunov-based techniques and eigenvalue analysis, we derive sufficient conditions for achieving FTS, which is crucial for rapid epidemic containment. The theoretical findings are validated through comprehensive numerical simulations that examine the effects of varying diffusion coefficients, reaction rates, and boundary conditions on system stability. The results highlight the critical role of these factors in achieving FTS of epidemic dynamics. This work contributes to developing efficient computational tools and theoretical insights for modeling and controlling infectious diseases in spatially extended populations, providing a foundation for future research on fractional-order models and complex boundary conditions.