时滞非局部扩散 SIR 系统的周期行波解

数学物理学报 ›› 2026, Vol. 46 ›› Issue (1): 157-173.

时滞非局部扩散 SIR 系统的周期行波解

贾梦璇(), 杨赟瑞^*()

兰州交通大学数理学院兰州 730070

收稿日期:2024-12-12 修回日期:2025-03-04 出版日期:2026-02-26 发布日期:2026-01-19
通讯作者: 杨赟瑞 E-mail:jmx5690@163.com;lily1979101@163.com
作者简介:贾梦璇, Email: jmx5690@163.com
基金资助:
国家自然科学基金(12361038);甘肃省基础研究创新群体项目(25JRRA805)

Periodic Traveling Waves for a Delayed SIR System with Nonlocal Dispersal

Mengxuan Jia(), Yunrui Yang^*()

School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070

Received:2024-12-12 Revised:2025-03-04 Online:2026-02-26 Published:2026-01-19
Contact: Yunrui Yang E-mail:jmx5690@163.com;lily1979101@163.com
Supported by:
NSFC(12361038);Foundation for Innovative Fundamental Research Group Project of Gansu Province(25JRRA805)

摘要/Abstract

摘要：

研究一类时滞非局部扩散 SIR 系统的周期行波解. 首先, 通过次代算子法定义基本再生数 $\Re_{0}$. 其次, 基于非紧性 Kuratowski 测度理论与渐近不动点定理建立当基本再生数 $ \Re_{0}>1$ 时该系统周期行波解的存在性. 最后, 借助反证法和分析技术研究当基本再生数 $ \Re_{0}<1$ 时该系统周期行波解的不存在性.

关键词: 周期行波解, 非局部扩散, 时滞, 渐近不动点定理

Abstract:

The periodic traveling waves for a class of delayed SIR system with nonlocal dispersal are considered. Firstly, the basic reproduction number $\Re_{0}$ is defined by the method of subalgebraic operators. Secondly, the existence of periodic traveling waves of the system when the basic reproduction number $\Re_{0}>1$ is established based on the non-compactness Kuratowski measure theory and the asymptotic fixed point theorem. Finally, the non-existence of periodic traveling waves when the basic reproduction number $\Re_{0}<1$ is investigated using the method of contradiction proof and the analysis technique.

Key words: periodic traveling waves, nonlocal dispersal, delay, asymptotic fixed point theorem

中图分类号:

O175.14

贾梦璇, 杨赟瑞. 时滞非局部扩散 SIR 系统的周期行波解[J]. 数学物理学报, 2026, 46(1): 157-173.

Mengxuan Jia, Yunrui Yang. Periodic Traveling Waves for a Delayed SIR System with Nonlocal Dispersal[J]. Acta mathematica scientia,Series A, 2026, 46(1): 157-173.

参考文献 23

[1]	Hosono Y, Ilyas B. Traveling waves for a simple diffusive epidemic model. Mathematical Models and Methods in Applied Sciences, 1995, 5(7): 935-966 doi: 10.1142/S0218202595000504
[2]	杨瑜. 一类非局部扩散的 SIR 模型的行波解. 数学物理学报, 2022, 42(5): 1409-1415
	Yang Y. Traveling wave of a nonlocal dispersal SIR model. Acta Math Sci, 2022, 42(5): 1409-1415
[3]	Li W T, Yang F Y. Traveling waves for a nonlocal dispersal SIR model with standard incidence. Journal of Integral Equations and Applications, 2014, 26: 243-273
[4]	Zhang S P, Yang Y R, Zhou Y H. Traveling waves in a delayed SIR model with nonlocal dispersal and nonlinear incidence. Journal of Mathematical Physics, 2018, 59(1): Art 11513
[5]	张广新, 杨瑞, 宋雪. 具有非局部效应的时滞 SEIR 系统的周期行波解. 数学物理学报, 2024, 44(1): 140-159
	Zhang G X, Yang Y R, Song X. Periodic traveling wave solutions of delayed SEIR systems with nonlocal effects. Acta Math Sci, 2024, 44(1): 140-159
[6]	宋雪, 杨瑞, 杨璐. 带有外部输入项的时间周期 SIR 传染病模型的周期行波解. 应用数学和力学, 2022, 43(10): 1164-176
	Song X, Yang Y R, Yang L. Periodic traveling wave solutions of time-period SIR epidemic model with external supplies. Applied Mathematics and Mechanics, 2022, 43(10): 1164-1176
[7]	Zhang L, Wang Z C, Zhao X Q. Time periodic traveling wave solutions for a Kermack-McKendrick epidemic model with diffusion and seasonality. Journal of Evolution Equations, 2020, 20(3): 1029-1059 doi: 10.1007/s00028-019-00544-2
[8]	Wu W X, Teng Z D. The periodic traveling waves in a diffusive periodic SIR epidemic model with nonlinear incidence. Chaos, Solitons and Fractals, 2021, 144: Art 110683
[9]	Cui J A, Tao X, Zhu H. An SIS infection model incorporating media coverage. The Rocky Mountain Journal of Mathematics, 2008, 38(5): 1323-1334
[10]	王雅琪. 一类 SIR 传染病模型的周期行波解. 长春: 东北师范大学, 2022
	Wang Y Q. Periodic Traveling Wave Solutions for A Class of SIR Epidemic Model. Changchun: Northeast Normal University, 2022
[11]	Yang L, Li Y K. Periodic traveling waves in a time periodic SEIR model with nonlocal dispersal and delay. Discrete and Continuous Dynamical Systems-Series B, 2023, 28(9): 5087-5104 doi: 10.3934/dcdsb.2023056
[12]	Zhao X Q. Dynamical Systems in Population Biology. New York: Springer, 2003
[13]	Nussbaum R D. Some asymptotic fixed point theorems. Transactions of the American Mathematical Society, 1972, 171: 349-375 doi: 10.1090/tran/1972-171-00
[14]	Zhao X Q. Basic reproduction ratios for periodic compartmental models with time delay. Journal of Dynamics and Differential Equations, 2017, 29(1): 67-82 doi: 10.1007/s10884-015-9425-2
[15]	Xu D, Zhao X Q. Dynamics in a periodic competitive model with stage structure. Journal of Mathematical Analysis and Applications, 2005, 311(2): 417-438 doi: 10.1016/j.jmaa.2005.02.062
[16]	Jin Y, Zhao X Q. Spatial dynamics of a nonlocal periodic reaction-diffusion model with stage structure. SIAM Journal on Mathematical Analysis, 2009, 40(6): 2496-2516 doi: 10.1137/070709761
[17]	Zhang L, Wang Z C, Zhao X Q. Propagation dynamics of a time periodic and delayed reaction-diffusion model without quasi-monotonicity. Transactions of the American Mathematical Society, 2019, 372(3): 1751-1782 doi: 10.1090/tran/2019-372-03
[18]	Wu S L, Pang L Y, Ruan S G. Propagation dynamics in periodic predator-prey systems with nonlocal dispersal. Journal de Mathématiques Pures et Appliquées, 2023, 170: 57-95 doi: 10.1016/j.matpur.2022.12.003
[19]	Li W T, Wang J B, Zhao X Q. Propagation dynamics in a time periodic nonlocal dispersal model with stage structure. Journal of Dynamics and Differential Equations, 2020, 32: 1027-1064 doi: 10.1007/s10884-019-09760-3
[20]	Deimling K. Nonlinear Functional Analysis. Berlin: Springer-Verlag, 1985
[21]	Banas J. On measures of noncompactness in Banach spaces. Commentationes Mathematicae Universitatis Carolinae, 1980, 21(1): 131-143
[22]	Bates P W, Chen F. Periodic traveling waves for a nonlocal integro-differential model. Electronic Journal of Differential Equations, 1999, 26: 1-19
[23]	Yang F Y, Li W T, Wang Z C. Traveling waves in a nonlocal dispersal SIR epidemic model. Nonlinear Analysis: Real World Applications, 2015, 23: 129-147 doi: 10.1016/j.nonrwa.2014.12.001