一类具有饱和发生率的时滞SIS模型动力学分析

兰州理工大学学报 ›› 2025, Vol. 51 ›› Issue (6): 157-166.

一类具有饱和发生率的时滞SIS模型动力学分析

袁海龙¹, 樊雨¹, 王玮明^*2

1.陕西科技大学数学与数据科学学院, 陕西西安 710021;
2.淮阴师范学院计算机科学与技术学院, 江苏淮安 223300

收稿日期:2023-11-13 发布日期:2025-12-31
通讯作者: 王玮明(1968-),男,甘肃庆阳人,博士,教授.Email:wangwm_math@hytc.edu.cn
基金资助:
国家自然科学基金(12171192,12071173,12571531)

Dynamic analysis of a time-delayed SIS model with saturation incidence

YUAN Hai-long¹, FAN Yu¹, WANG Wei-ming²

1. School of Mathematics &Data Science, Shaanxi University of Science &Technology, Xi’an 710021, China;
2. School of Computer Science and Technology, Huaiyin Normal University, Huai’an 223300, China

Received:2023-11-13 Published:2025-12-31

摘要/Abstract

摘要： 以时滞为分支参数,研究一类具有饱和发生率的时滞SIS模型.分析平衡点的稳定性,讨论模型Turing不稳定性和Hopf分支的存在性,通过相关分支理论确定Hopf分支的方向和稳定性,并给出数值模拟.

关键词: 时滞, SIS模型, Turing不稳定, Hopf分支, 稳定性

Abstract: Taking the time delay as the bifurcation parameter, a time-delayed SIS model with saturation incidence is studied. The stability of the equilibrium is analyzed, and the Turing instability of the model and the existence of the Hopf bifurcation are discussed. The direction and stability of the Hopf bifurcation are determined by the correlation bifurcation theory, and numerical simulations are also presented.

Key words: time delay, SIS model, Turing instability, Hopf bifurcation, stability

中图分类号:

O175.12

袁海龙, 樊雨, 王玮明. 一类具有饱和发生率的时滞SIS模型动力学分析[J]. 兰州理工大学学报, 2025, 51(6): 157-166.

YUAN Hai-long, FAN Yu, WANG Wei-ming. Dynamic analysis of a time-delayed SIS model with saturation incidence[J]. Journal of Lanzhou University of Technology, 2025, 51(6): 157-166.

参考文献

[1] 王玮明,蔡永丽.生物数学模型斑图动力学型[M].北京:科学出版社,2020.
[2] KERMACK W O,MCKENDRICK A G.A contribution to the mathematical theory of epidemics[J].Proceedings of the Royal Society of London Series A,1927,115:700-721.
[3] 霍海峰,邹明轩.一类具有接种和隔离治疗的结核病模型的稳定性[J].兰州理工大学学报,2016,42(3):150-154.
[4] CAI Y L,LI J X,KANG Y,et al.The fluctuation impact of human mobility on the influenza transmission[J].Journal of the Franklin Institute,2020,357(13):8899-8924.
[5] ANDERSON R M,MAY R M.Regulation and stability of host-parasite population interactions[J].Journal of Animal Ecology,1978,47(1):219-267.
[6] CAPASSO V,SERIO G.A generalization of the Kermack-McKendrick deterministic epidemic model[J].Mathematical Nioences,1978,42(1/2):43-61.
[7] 黄灿云,史维选,惠富春.一类具有脉冲接种和垂直传染的时滞SEIRS传染病模型[J].兰州理工大学学报,2016,42(3):155-161.
[8] 王雅迪,袁海龙.一类具有时滞的营养—微生物扩散模型的Hopf分支研究[J].西南师范大学学报(自然科学版),2023,48(5):1-13.
[9] WANG Z C,LI W T,RUAN S G.Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay[J].Journal of Differential Equations,2007,238(1):153-200.
[10] 王雅迪,袁海龙.时滞Lengyel-Epstein反应扩散系统的Hopf分支[J].山东大学学报(理学版),2023,58(8):92-103.
[11] HUTCHINSON G E.Circular causal systems in ecology[J].Annals of the New York Academy of Sciences,1948,50(4):221-246.
[12] WANG W D,ZHAO X Q.A nonlocal and time delayed reaction-diffusion model of dengue transmission[J].SIAM Journal on Applied Mathematics,2011,71(1):147-168.
[13] XU R,MA Z.Global stability of a SIR epidemic model with nonlinear incidence rate and time delay[J].Nonlinear Analysis Real World Applications,2009,10(5):3175-3189.
[14] ZHANG T,TENG Z.Global asymptotic stability of a delayed SEIRS epidemic model with saturation incidence[J].Chaos Solitons and Fractals,2008,37(5):1456-1468.
[15] XUE Y K,LI T T.Stability and hopf bifurcation for a delayed SIR epidemic model with logistic growth[J].Abstract and Applied Analysis,2013(2013):1-11.
[16] AVILA-VALES E,PEREZ A G C.Dynamics of a time delayed SIR epidemic model with logistic growth and saturated treatment[J].Chaos Solitons and Fractals,2019,127:55-69.
[17] MA Z E,ZHOU Y C,WU J H.Modeling and dynamics of infectious diseases[M].Singapore:Higher Education Press,2009.
[18] RASS L,RADCLIFFE J.Spatial deterministic epidemics[M].USA:American Mathematical Society,2003.
[19] WU J H.Theory and applications of partial functional differential equations[M].New York:Springer,1996.
[20] LIN X D,SO W H,WU J H.Center manifolds for partial differential equations with delays[J].Proceedings of the Royal Society of Edinburgh,1992,122(3/4):237-254.
[21] HASSARD B D,KAZARINOFF N D,WAN Y H.Theory and applications of Hopf bifurcation[M].Cambridge:Cambridge University Press,1981.