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《山东大学学报(理学版)》 ›› 2022, Vol. 57 ›› Issue (11): 70-77.doi: 10.6040/j.issn.1671-9352.0.2021.452

• • 上一篇    

空间异质的非局部扩散SI传染病模型的动力学

焦战1,靳祯2*   

  1. 1.山西大学复杂系统研究所, 山西 太原 030006;2.山西大学疾病防控的数学技术与大数据分析山西省重点实验室, 山西 太原 030006
  • 发布日期:2022-11-10
  • 作者简介:焦战(1997— ),男,硕士研究生,研究方向为生物数学、复杂网络. E-mail:jiaozhan0215@163.com*通信作者简介:靳祯(1965— ),男,博士,教授,研究方向为生物数学、复杂网络. E-mail:jinzhn@263.net
  • 基金资助:
    国家自然科学基金资助项目(61873154)

Dynamics of a spatially heterogeneous SI epidemic model with nonlocal diffusion

JIAO Zhan1, JIN Zhen2*   

  1. 1. Complex System Research Center, Shanxi University, Taiyuan 030006, Shanxi, China;
    2. Shanxi Key Laboratory of Mathematical Technology and Big Data Analysis on Disease Control and Prevention, Shanxi University, Taiyuan 030006, Shanxi, China
  • Published:2022-11-10

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摘要： 研究了具有标准发生率的空间异质性非局部扩散SI传染病模型。利用下一代算子的谱半径方法计算了系统的基本再生数R0,借助Lyapunov函数证明了R0<1时无病稳态解的全局渐近稳定性;当易感者的扩散率DS=0且R0>1时,利用上、下解等方法证明了系统地方病稳态解的存在性、唯一性与全局渐近稳定性。

关键词: SI传染病模型, 非局部扩散, 标准发生率, 全局渐近稳定性

Abstract: We study a spatially heterogeneous non-local dispersal SI epidemic model with the standard incidence. The basic reproduction number R0 of the system is defined as the spectral radius of the next generation operator, and by means of suitable Lyapunov functional, the global asymptotic stability of the disease-free equilibrium is proved when R0<1; the upper and lower solutions are used to prove the existence, uniqueness and global asymptotic stability of the endemic equilibrium of the system when the dispersal rate DS=0 of susceptible individuals and R0>1.

Key words: SI epidemic model, nonlocal diffusion, standard incidence, global asymptotic stability

中图分类号: 

  • O175.5
[1] BOGOYAM. A nonlocal nonlinear diffusion equation in higher space dimensions[J]. Journal of Mathematical Analysis and Applications, 2008, 344(2):601-615.
[2] MORITA Y, NINOMIYA H. Entire solutions with merging fronts to reaction-diffusion equations[J]. Journal of Differential Equations, 2006, 18:841-861.
[3] KAO C Y, LOU Y, SHEN W X. Random dispersal vs non-local dispersal[J]. Discrete Contin Dyn Syst, 2010, 26(2):551-596.
[4] DU Q, GUNZBURGER M, LEHOUCQ R B. Analysis and approximation of nonlocal diffusion problems with volume constraints[J]. SIAM Review, 2012, 54(4):667-696.
[5] SCHUMACHER K. Travelling front solutions for integro-differential equations[M] //Advances in Applied Probability. Cambridge: Cambridge University Press, 1980: 54-70. DOI:https://doi.org/10.1017/S0001867800035217.
[6] PAN S X. Traveling wave fronts of delayed non-local diffusion systems without quasimonotonicity[J]. Journal of Mathematical Analysis and Applications, 2008, 346(2):415-424.
[7] SHEN W X, ZHANG A J. Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats[J]. Journal of Differential Equations, 2010, 249(4):747-795.
[8] GUO Y J L. Entire solutions for a discrete diffusive equation[J]. Journal of Mathematical Analysis and Applications, 2008, 347(2):450-458.
[9] CHASSEIGNE E, CHAVES M, ROSSI J D. Asymptotic behavior for nonlocal diffusion equation[J]. Journal de Mathématiques Pures et Appliquées, 2006, 86(3):271-291.
[10] CORTAZAR C, ELGUETA M, ROSSI J D, et al. Boundary fluxes for nonlocal diffusion[J]. Journal of Differential Equations, 2006, 234(2):360-390.
[11] KUNIYA T, WANG J L. Global dynamics of an SIR epidemic model with nonlocal diffusion[J]. Nonlinear Analysis: Real World Applications, 2018, 43:262-282.
[12] LI B T, WEINBERGER H F, LEWIS M A, Spreading speeds as slowest wave speeds for cooperative systems[J]. Mathematical Biosciences, 2005, 196(1):82-98.
[13] BATES P W, ZHAO G Y. Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal[J]. Journal of Mathematical Analysis and Applications, 2006, 332(1):428-440.
[14] HAN B S, YANG Y H. On a predator-prey reaction diffusion model with nonlocal effects[J]. Commun Nonlinear Sci Numer Simulat, 2017, 46:49-61
[15] BATES P W, CHEN F X. Spectral analysis and multidimensional stability of traveling waves for nonlocal Allen-Cahn equation[J]. Journal of Mathematical Analysis and Applications, 2002, 237(1):45-57.
[16] TIAN H, JU L L, DU Q. A conservative nonlocal convection diffusion model and asymptotically compatible finite difference discretization[J]. Comput Meth Appl Mech Engin, 2017, 320:46-67.
[17] YANG F Y, LI W T, RUAN S G. Dynamics of a nonlocal dispersal SIS epidemic model with Neumann boundary conditions[J]. Journal of Differential Equations, 2019, 267(3):2011-2051.
[18] FIFE P. Some nonclassical trends in parabolic and parabolic-like evolutions[M]. Berlin: Springer, 2003.
[19] PAZY A. Semigroups of linear operators and applications to partial differential Equations[M]. New York: Springer, 1983.
[20] WEBB G F. Theory of nonlinear age-dependent population dynamics[M]. New York: CRC Press, 1985.
[21] YANG J Y, XU F. The computational approach for the basic reproduction number of epidemic models on complex networks[J]. Browse Journals and Magazines, 2019, 7:26474-26479.
[22] THIEME H R. Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity[J]. SIAM Journal on Applied Mathematics, 2009, 70(1):188-211.
[23] WANG X Y, YANG J Y. Dynamics of a nonlocal dispersal foot-and-mouth disease model in a spatially heterogeneous environment[J]. Acta Mathematica Scientia, 2021, 41(2):552-572.
[24] HADDOCK J R, TERJEKI J. Liapunov-Razumikhin functions and an invariance principle for functional-differential equations[J]. Journal of Differential Equations, 1983, 48(1):95-122.
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