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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, China, China |
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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