Asymptotic dynamics of SIRS epidemic model with dispersal budgets and nonlinear rates about heterogenous environments

¹ Department of Mathematics and Informatics, University Ain Temouchent, Belhadj Bouchaib, BP 284 RP, 46000, Algeria Engineering and Sustainable Development Laboratory, Faculty of Science and Technology, University of Ain Temouchent, Ain Temouchent 46000, Algeria
² Department of Mathematics, Faculty of Exact Sciences and Informatics, Hassiba Benbouali University, Chlef 02000, Algeria

^* Corresponding author: soufiane.bentout@univ-temouchent.edu.dz

Received: 28 January 2025
Accepted: 19 May 2025

Abstract

This paper examines an SIRS epidemic model incorporating nonlinear incidence functions and nonlocal diffusion with scaled dispersal to enhance understanding of infectious disease spread in human populations. We establish the well-posedness of the model by proving both the existence and uniqueness of its solution. Additionally, we demonstrate the existence of a global compact attractor that describes the asymptotic behavior of all positive solutions. The basic reproduction number, ℝ₀, is derived as the spectral radius of the linear and compact next-generation operator R(·). When ℝ₀ < 1, the infection-free equilibrium (IFE) is globally asymptotically stable, leading to disease extinction, which has significant implications for public health policies. Conversely, when R0 > 1, persistence theory shows the system is strongly persistent, ensuring at least one positive endemic equilibrium state (PEES). The study investigates the system’s asymptotic behavior under varying costs and scaling parameters of the dispersal kernel, revealing that when the dispersal kernel’s support (σ) is sufficiently small and the cost parameter m < 2, the epidemic persists, posing public health risks. These results highlight the critical influence of scaling and cost parameters on disease dynamics.