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Threshold dynamics of a time-periodic SIS model with anomalous diffusion in heterogeneous environments

Part of: Parabolic equations and systems Partial differential equations

Published online by Cambridge University Press:  24 February 2026

Peng Shi ,
Wan-Tong Li  and
Fei-Ying Yang
Peng Shi
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China
Wan-Tong Li*
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China (wtli@lzu.edu.cn)
Fei-Ying Yang
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China
*
*Corresponding author.
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Abstract

Recent research has demonstrated the importance of spatial diffusion and environmental heterogeneity in influencing the transmission dynamics of infectious diseases. At the same time, human mobility patterns have been shown to exhibit scale-free, nonlocal dynamics characterized by an anomalous Lévy process diffusion, which is mathematically represented by nonlocal equations involving fractional Laplacian operators. To investigate the effects of environmental heterogeneity and long-range geographical disease transmission, we propose a time-periodic susceptible-infectious-susceptible (SIS) epidemic model that incorporates anomalous diffusion and spatial heterogeneity. The key issues of this paper include the existence and stability of both disease-free and endemic periodic equilibria, as well as the impact of diffusion rates and fractional powers on the spatial distribution of these periodic states. Our analytical findings indicate that spatio-temporal heterogeneity promotes disease persistence and that the fractional power can modulate the transmission threshold.

Keywords

epidemic modelfractional diffusionperiodic equilibriaspatio-temporal heterogeneitytransmission threshold

MSC classification

Primary: 35K57: Reaction-diffusion equations
Secondary: 35B40: Asymptotic behavior of solutions 92D25: Population dynamics (general)

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 28
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Copyright
© The Author(s), 2026. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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