广西师范大学学报（自然科学版） ›› 2023, Vol. 41 ›› Issue (2): 106-117.doi: 10.16088/j.issn.1001-6600.2022033102

• 研究论文 • 上一篇    下一篇

具有饱和发生率和分布时滞的HIV免疫模型的稳定性

宋冰, 张育茹, 桑媛, 张龙*   

  1. 新疆大学 数学与系统科学学院,新疆 乌鲁木齐 830046
  • 收稿日期:2022-03-31 修回日期:2022-05-20 出版日期:2023-03-25 发布日期:2023-04-25
  • 通讯作者: 张龙(1977—),男,新疆伊宁人,新疆大学教授,博导。E-mail:longzhang_xj@sohu.com
  • 基金资助:
    国家自然科学基金(11861065,12261087);新疆维吾尔自治区自然科学基金(2019D01C076);新疆维吾尔自治区应用数学重点实验室开放课题(2021D04014);新疆维吾尔自治区高校科研重点项目(XJEDU2021I002);新疆维吾尔自治区优秀青年科技创新人才项目(2019Q017)

Stability of an HIV Immune Model with Saturation Incidence and Distributed Delays

SONG Bing, ZHANG Yuru, SANG Yuan, ZHANG Long*   

  1. College of Mathematics and System Sciences, Xinjiang University, Urumqi Xinjiang 830046, China
  • Received:2022-03-31 Revised:2022-05-20 Online:2023-03-25 Published:2023-04-25

PDF (PC)

116

摘要： 本文提出一类具有饱和发生率和抗体免疫反应的细胞内感染的时滞HIV感染模型,包括未感染细胞、潜伏感染细胞、感染细胞、游离HIV病毒和CTL免疫反应细胞。考虑4种时滞:潜伏感染时滞、细胞内时滞、感染细胞转化病毒时滞和CTL免疫反应时滞。定义2个阈值:感染基本再生数R0和CTL免疫再生数 R1,得到了模型的3类平衡点:无感染平衡点、免疫灭活感染平衡点和免疫激活感染平衡点。通过分析特征方程、构造 Lyapunov 函数和 LaSalle 不变原理,建立了各平衡点局部以及全局渐近稳定性的判定准则。

关键词: CTL免疫反应, 分布时滞, 全局渐近稳定, Lyapunov 函数, 饱和发生率

Abstract: A delayed HIV model of intracellular infection with saturated infection rate and antibody immune response is presented here. Including uninfected cells, latently infected cells, infected cells, free HIV virus, and CTLs. The following four delays are considered: latently infected delay, intracellular delay, the time lag between an infected cell and a virus, and CTL response delay. Two threshold conditions are defined: infection reproduction number R0 and CTL immunity-activated reproduction number R1. It obtains three equilibria of this model: disease-free equilibrium, immune-free infection equilibrium and immune-activated equilibrium. By analyzing the characteristic equation, Lyapunov function and LaSalle's invariance principle, the locally and globally asymptotic stable criteria of each equilibrium are established.

Key words: CTL immune response, distributed delay, globally asymptotically stable, Lyapunov function, saturated infection rate

中图分类号: 

  • O175
[1] BONHOEFFER S, MAY R M, SHAW G M, et al. Virus dynamics and drug therapy[J]. Proceedings of the National Academy of Sciences of the United States of America, 1997, 94(13): 6971-6976. DOI: 10.1073/pnas.94.13.6971.
[2] ANDERSON R M, MAY R M. Infectious diseases of humans: dynamics and control[M]. Oxford: Oxford Universtiy Press, 1991.
[3] ELAIW A M, AZOZ S A. Global properties of a class of HIV infection models with Beddington-DeAngelis functional response[J]. Mathematical Methods in the Applied Sciences, 2013, 36(4): 383-394. DOI: 10.1002/mma.2596.
[4] ROSEN H R. Chronic hepatitis C infection[J]. The New England Journal of Medicine, 2011, 364(25): 2429-2438. DOI: 10.1056/NEJMcp1006613.
[5] DAVIES S E, PORTMANN B C, O'GRADY J G, et al. Hepatic histological findings after transplantation for chronic hepatitis B virus infection, including a unique pattern of fibrosing cholestatic hepatitis[J]. Hepatology, 1991, 13(1): 150-157. DOI: 10.1002/hep.1840130122.
[6] RIHAN F A, ALSAKAJI H J, RAJIVGANTHI C. Stochastic SIRC epidemic model with time delay for COVID-19[J]. Advances in Difference Equations, 2020, 2020(1): 502. DOI: 10.1186/s13662-020-02964-8.
[7] DEANS J A, COHEN S. Immunology of Malaria[J]. Annual Review of Microbiology, 1983, 37(1): 25-50. DOI: 10.1146/annurev.mi.37.100183.000325.
[8] ZHOU X R, ZHANG L, ZHENG T, et al. Global stability for a delayed HIV reactivation model with latent infection and Beddington-DeAngelis incidence[J]. Applied Mathematics Letters, 2021, 117: 107047. DOI: 10.1016/j.aml.2021.107047.
[9] ZHOU X R, ZHANG L, ZHENG T, et al. Global stability for a class of HIV virus-to-cell dynamical model with Beddington-DeAngelis functional response and distributed time delay[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 4527-4543. DOI: 10.3934/mbe.2020250.
[10] CHEN W, TENG Z D, ZHANG L. Global dynamics for a drug-sensitive and drug-resistant mixed strains of HIV infection model with saturated incidence and distributed delays[J]. Applied Mathematics and Computation, 2021, 406: 126284. DOI: 10.1016/j.amc.2021.126284.
[11] ZHENG T, LOU Y T, ZHANG L, et al. Spatial dynamic analysis for COVID-19 epidemic model with diffusion and Beddington-DeAngelis type incidence[J/OL]. Communication on Pure and Applied Analysis. [2022-03-31]. https://www.aimsciences.org/cpaa. DOI: 10.3934/cpaa.2021154.
[12] 眭鑫, 刘贤宁, 周林. 具有潜伏细胞和CTL免疫反应的HIV模型的稳定性分析[J]. 西南大学学报(自然科学版), 2012, 34(5): 23-27. DOI: 10.13718/j.cnki.xdzk.2012.05.004.
[13] 周欣然, 郑涛, 张龙. 具有Beddington-DeAngelis功能反应和分布时滞的病毒动力学模型的稳定性[J]. 华南师范大学学报(自然科学版), 2020, 52(5): 118-123. DOI: 10.6054/j.jscnun.2020084.
[14] 王秀男. HIV感染动力学模型分析[D]. 重庆:西南大学, 2013.
[15] 闫冬雪. 几类结构种群和HIV模型的渐近行为分析[D]. 上海:华东师范大学, 2019.
[16] NOWAK M A, BANGHAM C R. Population dynamics of immune responses to persistent viruses[J]. Science, 1996, 272(5258): 74-79. DOI: 10.1126/science.272.5258.74.
[17] YUAN Z H, MA Z J, TANG X H. Global stability of a delayed HIV infection model with nonlinear incidence rate[J]. Nonlinear Dynamics, 2012, 68(1/2): 207-214. DOI: 10.1007/s11071-011-0219-8.
[18] WANG Z P, XU R. Stability and Hopf bifurcation in a viral infection model with nonlinear incidence rate and delayed immune response[J]. Communications in Nonlinear Science and Numerical Simulation, 2012, 17(2): 964-978. DOI: 10.1016/j.cnsns.2011.06.024.
[19] LIU W M, HETHCOTE H W, LEVIN S A. Dynamical behavior of epidemiological models with nonlinear incidence rates[J]. Journal of Mathematical Biology, 1987, 25(4): 359-380. DOI: 10.1007/BF00277162.
[20] REGOES R R, EBERT D, BONHOEFFER S. Dose-dependent infection rates of parasites produce the Allee effect in epidemiology[J]. Proceedings of the Royal Society B: Biological Sciences, 2002, 269(1488): 271-279. DOI: 10.1098/rspb.2001.1816.
[21] SONG X Y, NEUMANN A U. Global stability and periodic solution of the viral dynamics[J]. Journal of Mathematical Analysis and Applications, 2007, 329(1): 281-297. DOI: 10.1016/j.jmaa.2006.06.064.
[22] WANG K F, WANG W D, PANG H Y. Complex dynamic behavior in a viral model with delayed immune response[J].Physica D:Nonlinear Phenomena, 2007, 226(2): 197-208. DOI: 10.1016/j.physd.2006.12.001.
[23] CANABARRO A A, GLÉRIAl I M, LYRA M L. Periodic solutions and chaos in a non-linear model for the delayed cellular immune response[J]. Physica A:Statistical Mechanics and its Applications, 2004, 342(1-2): 234-241. DOI: 10.1016/j.physa.2004.04.083.
[24] ZHU H Y, LUO Y, CHEN M L. Stability and Hopf bifurcation of an HIV infection model with CTL-response delay[J]. Computers and Mathematics with Applications, 2011, 62(8): 3091-3102. DOI: 10.1016/j.camwa.2011.08.022.
[25] BERETTA E, KUANG Y. Geometric stability switch criteria in delay differential systems with delay dependent parameters[J]. SIAM Journal on Mathematical Analysis, 2002, 33(5): 1144-1165. DOI: 10.1137/S0036141000376086.
[26] WANG T L, HU Z X, LIAO F C, et al. Global stability analysis for delayed virus infection model with general incidence rate and humoral immunity[J]. Mathematics and Computers in Simulation, 2013, 89: 13-22. DOI: 10.1016/j.matcom.2013.03.004.
[27] ZHOU X Y, SONG X Y, SHI X Y. Analysis of stability and Hopf bifurcation for an HIV infection model with time delay[J]. Applied Mathematics and Computation, 2008, 199(1): 23-38. DOI: 10.1016/j.amc.2007.09.030.
[28] HETHCOTE H W, van den DRIESSCHE P. Two SIS epidemiologic models with delays[J]. Journal of Mathematical Biology, 2000, 40(1): 3-26. DOI: 10.1007/s002850050003.
[29] WANG J L, PANG J M, KUNIYA T, et al. Global threshold dynamics in a five-dimensional virus model with cell-mediated, humoral immune responses and distributed delays[J]. Applied Mathematics and Computation, 2014, 241(15): 298-316. DOI: 10.1016/j.amc.2014.05.015.
[30] BAGASRA O, POMERANTZ R J. Human immunodeficiency virus type I provirus is demonstrated in peripheral blood monocytes in vivo: a study utilizing an in situ polymerase chain reaction[J]. Aids Research and Human Retroviruses, 1993, 9(1): 69-76. DOI: 10.1089/aid.1993.9.69.
[31] PACE M J, AGOSTO L, GRAF E H, et al.HIV reservoirs and latency models[J]. Virology, 2011, 411(2): 344-354. DOI: 10.1016/j.virol.2010.12.041.
[32] WANG H B, XU R. Stability and Hopf bifurcation in an HIV-1 infection model with latently infected cells and delayed immune response[J]. Discrete Dynamics in Nature and Society, 2013,2013: 169427. DOI: 10.1155/2013/169427.
[33] XU R. Global dynamics of an HIV-1 infection model with distributed intracellular delays[J]. Computers and Mathematics with Applications, 2011, 61(9): 2799-2805. DOI: 10.1016/j.camwa.2011.03.050.
[34] ZOU D Y, WANG S F, LI X Z. Global dynamics for a live-virus-blood model with distributed time delay[J]. International Journal of Biomathematics, 2017, 10(5): 1750067. DOI: 10.1142/S179352451750067X.
[35] HALE J K, VERDUYN LUNEL S M. Introduction to functional differential equations[M]. New York: Springer, 1993.
[36] SMITH H L. Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems[M]. Providence: American Mathematical Society, 1995.
[37] van den DRIESSCHE P, WATMOUGH J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission[J]. Mathematical Biosciences, 2002, 180(1/2): 29-48. DOI: 10.1016/S0 025-5564(02)00108-6.
[38] SHU H Y, WANG L, WATMOUGH J. Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL immune responses[J]. SIAM Journal on Applied Mathematics, 2013, 73(3): 1280-1302. DOI: 10.1137/120896463.
[39] ZHOU X Y, SHI X Y, ZHANG Z H, et al. Dynamical behavior of a virus dynamics model with CTL immune response[J]. Applied Mathematics and Computation, 2009, 213(2): 329-347. DOI: 10.1016/j.amc.2009.03.026.
[40] 马知恩, 周义仓. 常微分方程稳定性与稳定性方法[M]. 北京: 科学出版社, 2001.
[41] 马知恩, 周义仓, 王稳地,等. 传染病动力学的数学建模与研究[M]. 北京: 科学出版社,2004.
[1] 郑涛, 周欣然, 张龙. 三种群捕食-竞争-合作混杂模型的全局渐近稳定性[J]. 广西师范大学学报（自然科学版）, 2020, 38(5): 64-70.
[2] 苗新艳, 张龙, 罗颜涛, 潘丽君. 一类交替变化的竞争—合作混杂种群模型研究[J]. 广西师范大学学报（自然科学版）, 2018, 36(3): 25-31.
[3] 邢伟, 高晋芳, 颜七笙, 周其华. 具有非线性传染率及脉冲免疫接种的SIQR传染病模型[J]. 广西师范大学学报（自然科学版）, 2017, 35(2): 58-65.
[4] 韩彩虹, 李略, 黄丽丽. 一类差分方程的全局渐近稳定性[J]. 广西师范大学学报（自然科学版）, 2017, 35(1): 53-57.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] 周正春. 互补序列研究进展[J]. 广西师范大学学报（自然科学版）, 2023, 41(1): 1 -16 .
[2] 杨烁祯, 张珑, 王建华, 张恒远. 声音事件检测综述[J]. 广西师范大学学报（自然科学版）, 2023, 41(2): 1 -18 .
[3] 杨生龙, 母庆闯, 张志华, 刘葵. 废旧锂离子电池回收利用技术进展[J]. 广西师范大学学报（自然科学版）, 2023, 41(2): 19 -26 .
[4] 李康良, 邱彩雄, 何爽, 黄春华, 伍冠一. 白介素-31参与瘙痒的研究进展[J]. 广西师范大学学报（自然科学版）, 2023, 41(2): 27 -35 .
[5] 卢许孟, 南新元, 夏斯博. 无模型坐标补偿积分滑模约束的自动驾驶汽车轨迹跟踪控制[J]. 广西师范大学学报（自然科学版）, 2023, 41(2): 36 -48 .
[6] 张伟健, 邴其春, 沈富鑫, 胡嫣然, 高鹏. 城市快速路路段行程时间估计方法[J]. 广西师范大学学报（自然科学版）, 2023, 41(2): 49 -57 .
[7] 杨秀, 韦笃取. 基于单状态变量的永磁同步电机混沌跟踪控制[J]. 广西师范大学学报（自然科学版）, 2023, 41(2): 58 -66 .
[8] 赵媛, 宋树祥, 刘振宇, 岑明灿, 蔡超波, 蒋品群. 一种新型电流镜运算跨导放大器的设计[J]. 广西师范大学学报（自然科学版）, 2023, 41(2): 67 -75 .
[9] 王鲁娜, 杜洪波, 朱立军. 基于流形正则的堆叠胶囊自编码器优化算法[J]. 广西师范大学学报（自然科学版）, 2023, 41(2): 76 -85 .
[10] 赵明, 罗秋莲, 陈蔚萌, 陈嘉妮. 控制时机和力度对传染病传播的影响[J]. 广西师范大学学报（自然科学版）, 2023, 41(2): 86 -97 .
版权所有 © 广西师范大学学报（自然科学版）编辑部
地址：广西桂林市三里店育才路15号　邮编：541004
电话：0773-5857325　E-mail: gxsdzkb@mailbox.gxnu.edu.cn
本系统由北京玛格泰克科技发展有限公司设计开发