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

Yifei Pan; Heejung Byun; Feng Li; Yifei Pan; Heejung Byun; Feng Li

doi:10.3934/nhm.2026055

Networks and Heterogeneous Media

2026, Volume 21, Issue 4: 1450-1478. doi: 10.3934/nhm.2026055

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

1.
Department of Information Communication Technology, The University of Suwon, Hwaseong 18323, South Korea
2.
School of Mathematics and Statistic, Linyi University, Linyi 276005, China

Received: 21 April 2026 Revised: 15 June 2026 Accepted: 17 June 2026 Published: 06 July 2026

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 $ \mathcal{R}_0 $ acts as the definitive threshold for disease control: the infection is destined for extinction when $ \mathcal{R}_0 < 1 $, whereas the system exhibits spatial persistence and the emergence of at least one non-trivial endemic equilibrium when $ \mathcal{R}_0 > 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.
- SIS epidemic model,
- logistic growth,
- boundary outflow,
- Principal eigenvalue,
- threshold dynamics
Citation: Yifei Pan, Heejung Byun, Feng Li. Threshold dynamics of a Reaction–Diffusion SIS Model with logistic growth and boundary outflow[J]. Networks and Heterogeneous Media, 2026, 21(4): 1450-1478. doi: 10.3934/nhm.2026055

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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 $ \mathcal{R}_0 $ acts as the definitive threshold for disease control: the infection is destined for extinction when $ \mathcal{R}_0 < 1 $, whereas the system exhibits spatial persistence and the emergence of at least one non-trivial endemic equilibrium when $ \mathcal{R}_0 > 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.

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