A class of virus dynamic model with inhibitory effect on thegrowth of uninfected T cells caused by infected T cells and its stability analysis

Wenbo  Cheng; Wanbiao Ma; Songbai  Guo

doi:10.3934/cpaa.2016.15.795

Article Contents

2016, Volume 15, Issue 3: 795-806. Doi: 10.3934/cpaa.2016.15.795

This issue Previous Article A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space Next Article A Liouville-type theorem for higher order elliptic systems of Hé non-Lane-Emden type

A class of virus dynamic model with inhibitory effect on the growth of uninfected T cells caused by infected T cells and its stability analysis

1.
Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing100083

Received: December 31, 2014

Revised: September 30, 2015

Published: April 2016

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Abstract

A class of virus dynamic model with inhibitory effect on the growth of uninfected T cells caused by infected T cells is proposed. It is shown that the infection-free equilibrium of the model is globally asymptotically stable, if the reproduction number $R_0$ is less than one, and that the infected equilibrium of the model is locally asymptotically stable, if the reproduction number $R_0$ is larger than one. Furthermore, it is also shown that the model is uniformly persistent, and some explicit formulae for the lower bounds of the solutions of the model are obtained.

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Mathematics Subject Classification: Primary: 34K20, 34D23; Secondary: 92D30.

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References

[1]	H. T. Banks, D. M. Bortz and S. E. Holte, Incorporation of variability into the modeling of viral delays in HIV infection dynamics, Mathematical Biosciences, 183 (2003), 63-91.doi: 10.1016/S0025-5564(02)00218-3.
[2]	S. Bonhoeffer, R. M. May, G. M. Shaw and M. A. Nowak, Virus dynamics and drug therapy, Proceedings of National Academy of Sciences of the United States of America, 94 (1997), 6971-6976.
[3]	R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4$^+$ T-cells, Mathematical Biosciences, 165 (2000), 27-39.
[4]	R. V. Culshaw, S. Ruan and G. Webb, A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay, Journal of Mathematical Biolgy, 46 (2003), 425-444.doi: 10.1007/s00285-002-0191-5.
[5]	D. C. Douek, M. Roederer and R. A. Koup, Emerging Concepts in the Immunopathogenesis of AIDS, Annual Review of Medicine, 60 (2009), 471-484.
[6]	R. J. De Boer and A. S. Perelson, Target cell limited and immune control models of HIV infection: a comparison, Journal of Theoretical Biology, 190 (1998), 201-214.
[7]	N. M. Dixit and S. Perelson, Complex patterns of viral load decay under antiretroviral therapy: influence of pharmacokinetics and intracellular delay, Journal of Theoretical Biology, 226 (2004), 95-109.doi: 10.1016/j.jtbi.2003.09.002.
[8]	T. Gao, W. Wang and X. Liu, Mathematical analysis of an HIV model with impulsive antiretroviral drug doses, Mathematics and Computers in Simulation, 82 (2011), 653-665.doi: 10.1016/j.matcom.2011.10.007.
[9]	J. E. Mittler, B. Sulzer, A. U. Neumann and A. S. Perelson, Influence of delayed viral production on viral dynamics in HIV-1 infected patients, Mathematical Biosciences, 152 (1998), 143-163.
[10]	T. H. Finkel, G. T.-Williams, N. K. Banda, M. F. Cotton, T. Curiel, C. Monks, T. W. Baba, R. M. Ruprecht and A. Kupfer, Apoptosis occurs predominantly in bystander cells and not in productively infected cells of HIV- and SIV-infected lymph nodes, Nature Medicine, 1 (1995), 129-134.
[11]	J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.doi: 10.1007/978-1-4612-4342-7.
[12]	A. V. M. Herz, S. Bonhoeffer, R. M. Anderson, R. M. May and M. A. Nowak, Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay, Proceedings of the National Academy of Sciences of the United States of America, 93 (1996), 7247-7251.
[13]	G. Huang, W. Ma and Y. Takeuchi, Global properties for virus dynamics model with Beddington-DeAngelis functional response, Applied Mathematics Letters, 22 (2009), 1690-1693.doi: 10.1016/j.aml.2009.06.004.
[14]	S. Iwami, S. Nakaoka and Y. Takeuchi, Viral diversity limits immune diversity in asymptomatic phase of HIV infection, Theoretical Population Biology, 73 (2008), 332-341.
[15]	Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993.
[16]	P. D. Leenheer and H. L. Smith, Virus dynamics: a global analysis, SIAM Journal on Applied Mathematics, 63 (2003), 1313-1327.doi: 10.1137/S0036139902406905.
[17]	D. Li and W. Ma, Asymptotic properties of a HIV-1 infection model with time delay, Journal of Mathematical Analysis and Applications, 335 (2007), 683-691.doi: 10.1016/j.jmaa.2007.02.006.
[18]	M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay, Bulletin of mathematical biology, 72 (2010), 1492-1505.doi: 10.1007/s11538-010-9503-x.
[19]	A. Murase, T. Sasaki and T. Kajiwara, Stability analysis of pathogen-immune interaction dynamics, Journal of Mathematical Biology, 51 (2005), 247-267.doi: 10.1007/s00285-005-0321-y.
[20]	Y. Nakata, Global dynamics of a viral infection model with a latent period and Beddington-DeAngelis response, Nonlinear Analysis: Theory, Methods & Applications, 74 (2011), 2929-2940.doi: 10.1016/j.na.2010.12.030.
[21]	P. W. Nelson, J. D. Murray and A. S. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay, Mathematical Biosciences, 163 (2000), 201-215.doi: 10.1016/S0025-5564(99)00055-3.
[22]	P. W. Nelson and A. S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection, Mathematical Biosciences, 179 (2002), 73-94.doi: 10.1016/S0025-5564(02)00099-8.
[23]	M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79.
[24]	M. A. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, Oxford, 2000.
[25]	H. Pang, W. Wang and K. Wang, Global properties of virus dynamics with CTL immune response, Journal of Southwest China Normal Normal University, 30 (2005), 796-799.
[26]	K. A. Pawelek, S. Liu, F. Pahlevani and L. Rong, A model of HIV-1 infection with two time delays: mathematical analysis and comparison with patient data, Mathematical Biosciences, 235 (2012), 98-109.doi: 10.1016/j.mbs.2011.11.002.
[27]	A. S. Perelson, A. U. Neumann, M. Markowitz, J. M. Leonard and D. D. Ho, HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time, Science, 271 (1996), 1582-1586.
[28]	A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Review, 41 (1999), 3-44.doi: 10.1137/S0036144598335107.
[29]	L. Rong, Z. Feng and A. S. Perelson, Emergence of HIV-1 drug resistance during antiretroviral treatment, Bulletin of Mathematical Biology, 69 (2007), 2027-2060.doi: 10.1007/s11538-007-9203-3.
[30]	B. G. Sampath Aruna Pradeep, Wanbiao Ma and Songbai Guo, Stability properties of a delayed HIV model with nonlinear functional response and absorption effect, Journal of the National Science Foundation of Sri Lanka, 43 (2015), 235-245.
[31]	N. Selliah and T. H. Finkel, Biochemical mechanisms of HIV induced T cell apoptosis, Cell Death and Differentiation, 8 (2001), 127-136.
[32]	H. Shu and L. Wang, Role of CD4$^+$T-cell proliferation in HIV infection under antiretroviral therapy, Journal of Mathematical Analysis and Applications, 394 (2012), 529-544.doi: 10.1016/j.jmaa.2012.05.027.
[33]	H. L. Smith and X. Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Analysis: Theory, Methods & Applications, 47 (2001), 6169-6179.doi: 10.1016/S0362-546X(01)00678-2.
[34]	X. Song and A. U. Neumann, Global stability and periodic solution of the viral dynamics, Journal of Mathematical Analysis Applications, 329 (2007), 281-297.doi: 10.1016/j.jmaa.2006.06.064.
[35]	J. Tam, Delay effect in a model for virus replication, Mathematical Medicine and Biology: A Journal of the IMA, 16 (1999), 29-37.
[36]	W. Wang, Global behavior of an SEIRS epidemic model with time delay, Applied Mathematics Letters, 15 (2002), 423-428.doi: 10.1016/S0893-9659(01)00153-7.
[37]	Y. Wang, Y. Zhou, J. Wu and J. Heffernan, Oscillatory viral dynamics in a delayed HIV pathogenesis model, Mathematical Biosciences, 219 (2009), 104-112.doi: 10.1016/j.mbs.2009.03.003.
[38]	R. Xu, Global stability of an HIV-1 infection model with saturation infection and intracellular delay, Journal of Mathematical Analysis and Applications, 375 (2011), 75-78.doi: 10.1016/j.jmaa.2010.08.055.
[39]	H. Zhu and X. Zou, Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay, Discrete and Continuous Dynamical Systems, 12 (2009), 511-524.doi: 10.3934/dcdsb.2009.12.511.