Dynamic Analysis of a Class of HIV-1 Infection Models With Pulsed Immunotherapy

WANG Xiaoe; LIN Xiaolin; LI Jianquan

doi:10.21656/1000-0887.390334

Volume 40 Issue 7

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Article Navigation > Applied Mathematics and Mechanics > 2019 > 40(7): 728-740

WANG Xiaoe, LIN Xiaolin, LI Jianquan. Dynamic Analysis of a Class of HIV-1 Infection Models With Pulsed Immunotherapy[J]. Applied Mathematics and Mechanics, 2019, 40(7): 728-740. doi: 10.21656/1000-0887.390334

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Dynamic Analysis of a Class of HIV-1 Infection Models With Pulsed Immunotherapy

doi: 10.21656/1000-0887.390334

School of Arts and Sciences, Shaanxi University of Science and Technology,Xi’an 710021, P.R.China

Funds: The National Natural Science Foundation of China(11371031;11371369)

Received Date: 2018-11-29
Rev Recd Date: 2019-05-06
Publish Date: 2019-07-01

Abstract

Abstract

Based on a class of HIV-1 infection immunotherapy models, an HIV-1 infection model with pulsed immunotherapy was addressed. The non-negativity and uniform boundedness of the solutions to the pulsed immunotherapy model were studied with the pulsed differential equation theory. According to the Floquet multiplier theory and the comparison theorem for differential equations, the threshold conditions for the local and global asymptotic stability of the infection-free periodic solution as well as uniform persistence of HIV-1 were obtained. Through numerical simulation, the therapeutic effects of 3 different treatment regimens were compared, and the effectiveness of the pulsed immunotherapy was verified. The numerical results show that, the virus can be effectively controlled or eliminated theoretically when the drug input is large enough or the dosing interval is short properly.
- HIV-1 infection model,
- infection-free periodic solution,
- uniform persistence,
- immunotherapy

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References

[1]	AIDS, Information on HIV[EB/OL]. [2018-11-29]. https://aidsinfo.nih.gov/clinical-trials.
[2]	KIRSCHNER D E, WEBB G F. Immunotherapy of HIV-1 infection[J]. Journal of Biological Systems,1998,6(1): 71-83.
[3]	PERELSON A S, NELSON P W. Mathematical analysis of HIV-1 dynamics in vivo[J]. Society for Industrial and Applied Mathematics,1999,41(1): 3-44.
[4]	CALLAWAY D S, PERELSON A S. HIV-1 infection and low steady state viral loads[J]. Bulletin of Mathematical Biology,2002,64(1): 29-64.
[5]	LEENHEER P D, SMITH H L. Virus dynamics: a global analysis[J]. SIAM Journal on Applied Mathematics,2003,63(4): 1313-1327.
[6]	HUANG Y X, ROSENKRANZ S L, WU H L. Modeling HIV dynamics and antiviral response with consideration of time-varying drug exposures, adherence and phenotypic sensitivity[J]. Mathematical Biosciences,2003,184(2): 165-186.
[7]	SMITH R J, WAHL L M. Distinct effects of protease and reverse transcriptase inhibition in an immunological model of HIV-1 infection with impulsive drug effects[J]. Bulletin of Mathematical Biology,2004,66(5): 1259-1283.
[8]	SMITH R J, WAHL L M. Drug resistance in an immunological model of HIV-1 infection with impulsive drug effects[J]. Bulletin of Mathematical Biology,2005,67(4): 783-813.
[9]	GAO T, WANG W D, LIU X N. Mathematical analysis of an HIV model with impulsive antiretroviral drug doses[J].Mathematics and Computers in Simulation,2012, 82(4): 653-665.
[10]	MIRON R E, SMITH R J. Resistance to protease inhibitors in a model of HIV-1 infection with impulsive drug effects[J]. Bulletin of Mathematical Biology,2014,76(1): 59-97.
[11]	宋保军, 娄洁, 文清芝. 使用T-20治疗HIV-1患者的不同策略的数学建模与研究[J]. 应用数学和力学, 2011,32(4): 400-416.(SONG Baojun, LOU Jie, WEN Qingzhi. Modelling two different therapy strategies for drug T-20 on HIV-1 patients[J]. Applied Mathematics and Mechanics,2011,32(4): 400-416.(in Chinese))
[12]	韩溢. 具有脉冲免疫因子的HIV模型的稳定性研究[J]. 重庆工商大学学报(自然科学版), 2013, 30(3): 77-82.(HAN Yi. Research on stability for an HIV model with impulsive releasing immune factor[J]. Journal of Chongqing Technology and Business University(Natural Science ), 2013,30(3): 77-82.(in Chinese))
[13]	ROY P K, CHATTERJEE A N, LI X Z. The effect of vaccination to dendritic cell and immune cell interaction in HIV disease progression[J]. International Journal of Biomathematics,2016,9(1): 1-20.
[14]	CHATTERJEE A N, ROY P K. Anti-viral drug treatment along with immune activator IL-2: a control-based mathematical approach for HIV infection[J]. International Journal of Control,2012, 85(2): 220-237.
[15]	JOLY M, ODLOAK D. Modeling interleukin-2-based immunotherapy in AIDS pathogenesis[J]. Journal of Theoretical Biology,2013,335(4): 57-78.
[16]	ABRAMS D, LEVY Y, LOSSO M H. Interleukin-2 therapy in patients with HIV infection[J]. New England Journal of Medicine,2009,361(16): 1548-1559.
[17]	BELL C J M, SUN Y L, NOWAK U M, et al. Sustained in vivo signaling by long-lived IL-2 induces prolonged increases of regulatory T cells[J]. Journal of Autoimmunity,2015,56: 66-80.
[18]	READ S W, LEMPICKI R A, MASCIO M D, et al. CD4 T cell survival after intermittent interleukin-2 therapy is predictive of an increase in the CD4 T cell count of HIV-infected patients[J]. The Journal of Infectious Diseases,2008,198(6): 843-850.
[19]	胡晓虎, 唐三一. 血管外给药的非线性房室模型解的逼近[J]. 应用数学和力学, 2014,35(9): 1033-1045.(HU Xiaohu, TANG Sanyi. Approximate solutions to the nonlinear compartmental model for extravascular administration[J]. Applied Mathematics and Mechanics,35(9): 1033-1045.(in Chinese))
[20]	KIRSCHNER D E, WEBB G F. A mathematical model of combined drug therapy of HIV infection[J]. Journal of Theoretical Medicine,2014,1(1):25-34.
[21]	宋新宇, 郭红建, 师向云. 脉冲微分方程理论及其应用[M]. 北京: 科学出版社, 2011.(SONG Xinyu, GUO Hongjian, SHI Xiangyun. Impulsive Differential Equation Theory and Its Application [M]. Beijing: Science Press, 2011.(in Chinese))
[22]	陆启韶. 常微分方程的定性方法和分叉[M]. 北京: 北京航空航天大学出版社, 1989.(LU Qishao. Qualitative Methods and Bifurcations of Ordinary Differential Equations [M]. Beijing: Beihang University Press, 1989.(in Chinese))
[23]	FONDA A. Uniformly persistent semidynamical systems[J]. Proceedings of the American Mathematical Society,1988,104(1): 111-116.
[24]	白振国. 周期传染病模型的基本再生数[J]. 工程数学学报, 2013,30(2): 175-183.(BAI Zhenguo. Basic reproduction number of periodic epidemic models[J]. Chinese Journal of Engineering Mathematics,2013,30(2): 175-183.(in Chinese))

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