Journal of Guangxi Normal University(Natural Science Edition) ›› 2023, Vol. 41 ›› Issue (2): 106-117.doi: 10.16088/j.issn.1001-6600.2022033102

Previous Articles     Next Articles

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

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

CLC Number: 

  • 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] ZHENG Tao, ZHOU Xinran, ZHANG Long. Global Asymptotic Stability of Predator-Competition-Cooperative Hybrid Population Models of Three Species [J]. Journal of Guangxi Normal University(Natural Science Edition), 2020, 38(5): 64-70.
[2] WU Lei, YANG Li, LI Qishang, XIAO Huapeng. Chaos Control of Synchronous Reluctance Motor Based on Small Gain Theorem [J]. Journal of Guangxi Normal University(Natural Science Edition), 2019, 37(2): 44-51.
[3] ZHAO Wei-qi, LIANG Jia-rong, LI Xia. Finite-time Terminal Sliding Mode Control for Singular Systems [J]. Journal of Guangxi Normal University(Natural Science Edition), 2011, 29(4): 73-78.
[4] ZHANG Jing-hua, LIANG Jia-rong, DU Fu. Quadratic Stability of Discretes T-S Descriptor Fuzzy Systems [J]. Journal of Guangxi Normal University(Natural Science Edition), 2010, 28(4): 49-53.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] ZHOU Zhengchun. Research Progress of Complementary Sequences[J]. Journal of Guangxi Normal University(Natural Science Edition), 2023, 41(1): 1 -16 .
[2] YANG Shuozhen, ZHANG Long, WANG Jianhua, ZHANG Hengyuan. Review of Sound Event Detection[J]. Journal of Guangxi Normal University(Natural Science Edition), 2023, 41(2): 1 -18 .
[3] YANG Shenglong, MU Qingchuang, ZHANG Zhihua, LIU Kui. Technical Progress in Recovery and Utilization of Spent Lithium-ion Batteries[J]. Journal of Guangxi Normal University(Natural Science Edition), 2023, 41(2): 19 -26 .
[4] LI Kangliang, QIU Caixiong, HE Shuang, HUANG Chunhua, WU Guanyi. Research Progress of IL-31 in Itch[J]. Journal of Guangxi Normal University(Natural Science Edition), 2023, 41(2): 27 -35 .
[5] LU Xumeng, NAN Xinyuan, XIA Sibo. Trajectory Tracking Control Based on Model-Free Coordinate Compensation Integral Sliding Mode Constraints[J]. Journal of Guangxi Normal University(Natural Science Edition), 2023, 41(2): 36 -48 .
[6] ZHANG Weijian, BING Qichun, SHEN Fuxin, HU Yanran, GAO Peng. Travel Time Estimation Method of Urban Expressway Section[J]. Journal of Guangxi Normal University(Natural Science Edition), 2023, 41(2): 49 -57 .
[7] YANG Xiu, WEI Duqu. Chaos Tracking Control of Permanent Magnet Synchronous Motor Based on Single State Variable[J]. Journal of Guangxi Normal University(Natural Science Edition), 2023, 41(2): 58 -66 .
[8] ZHAO Yuan, SONG Shuxiang, LIU Zhenyu, CEN Mingcan, CAI Chaobo, JIANG Pinqun. Design of a Novel Current-Mirror Operational Transconductance Amplifier[J]. Journal of Guangxi Normal University(Natural Science Edition), 2023, 41(2): 67 -75 .
[9] WANG Luna, DU Hongbo, ZHU Lijun. Stacked Capsule Autoencoders Optimization Algorithm Based on Manifold Regularization[J]. Journal of Guangxi Normal University(Natural Science Edition), 2023, 41(2): 76 -85 .
[10] ZHAO Ming, LUO Qiulian, CHEN Weimeng, CHEN Jiani. Influence of Control Timing and Strength on the Spreading of Epidemic[J]. Journal of Guangxi Normal University(Natural Science Edition), 2023, 41(2): 86 -97 .
Copyright © Journal of Guangxi Normal University(Natural Science Edition), All Rights Reserved.
Tel: 0773-5857325　E-mail: gxsdzkb@mailbox.gxnu.edu.cn
Powered by Beijing Magtech Co. Ltd