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May  2016, 15(3): 795-806. doi: 10.3934/cpaa.2016.15.795

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

Wenbo Cheng 1, , Wanbiao Ma 1, and  Songbai Guo 1,

1. 

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

Received  January 2015 Revised  October 2015 Published  February 2016

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.
Keywords: time delay, persistence., Virus dynamic model, stability.
Mathematics Subject Classification: Primary: 34K20, 34D23; Secondary: 92D3.
Citation: Wenbo Cheng, Wanbiao Ma, Songbai Guo. 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. Communications on Pure and Applied Analysis, 2016, 15 (3) : 795-806. doi: 10.3934/cpaa.2016.15.795
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.

show all references

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.

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