Stability analysis and numerical simulations of a discrete-time epidemic model

Abstract

This research investigates a discrete epidemic reaction–diffusion model, focusing on the nuances of both local and global stability. By employing second-order difference schemes alongside L1 approximations, we establish a robust numerical framework for simulating disease spread. The analysis begins with a thorough examination of two crucial equilibrium points: the disease-free equilibrium, which signifies complete eradication of the disease, and the endemic equilibrium, where the disease persists within the population. Through this exploration, we seek to identify the specific conditions that can lead to either successful containment or ongoing infection. To assess the global stability of the system, we utilize the Lyapunov method, a powerful analytical technique that enables us to derive sufficient conditions for global asymptotic stability. This rigorous methodology guarantees that, under defined conditions, the system will ultimately reach a stable equilibrium, irrespective of any initial perturbations. Complementing our theoretical framework, we conduct numerical simulations that validate our stability results. These simulations provide deeper insights into the system’s dynamic behavior, illustrating how various parameters and conditions influence its evolution. Moreover, the numerical simulations not only reinforce our theoretical findings but also facilitate the visualization and interpretation of the system’s complex dynamics. This synergy between analytical rigor and numerical validation enhances the reliability of our model, establishing it as a critical tool for understanding epidemic propagation and developing effective control strategies. Our comprehensive investigation thus enriches both the theoretical landscape of reaction–diffusion systems and their practical implications for managing disease outbreaks.