Threshold dynamics of a Reaction–Diffusion SIS Model with logistic growth and boundary outflow

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Abstract

This study presents a reaction-diffusion framework that integrates boundary outflow mechanisms with logistic population dynamics to investigate epidemic evolution in spatially non-uniform environments. By incorporating non-linear, density-dependent regulation and boundary leakage, the model builds on the constraints of population conservation or closed habitats found in classical studies, thus offering a more objective representation of open population dynamics under resource limitations. We provide a rigorous analysis of the system’s foundational mathematical properties, establishing the invariant non-negativity of state variables and uniform upper bounds for all global solutions, thereby ensuring the reliability of the subsequent dynamical analysis. Based on the spectral radius characterizations, the fundamental parameter R₀ acts as the definitive threshold for disease control: the infection is destined for extinction when R₀ < 1, whereas the system exhibits spatial persistence and the emergence of at least one non-trivial endemic equilibrium when R₀ > 1. The findings reveal that the boundary outflow mechanism remains an effective mitigation strategy in resource-constrained environments, thereby significantly mitigating the infection pressure and suppressing the spatial dissemination of the disease.

Keywords

SIS epidemic model / logistic growth / boundary outflow / Principal eigenvalue / threshold dynamics

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