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Virus dynamics model with intracellular delays and immune response
1. | Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China |
2. | Academy of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology, 3041#, 2 Yi-Kuang Street, Harbin, 150080 |
References:
[1] |
R. Arnaout, M. A. Nowak and D. Wodarz, HIV-1 dynamics revisited: Biphasic decay by cytotoxic lymphocyte killing?, Proc. R. Soc. Lond. B, 267 (2000), 1347-1354.
doi: 10.1098/rspb.2000.1149. |
[2] |
H. T. Banks and D. M. Bortz, A parameter sensitivity methodology in the context of HIV delay equation models, J. Math. Biol., 50 (2005), 607-625.
doi: 10.1007/s00285-004-0299-x. |
[3] |
S. Bonhoeffer, J. M. Coffin and M. A. Nowak, Human immunodeficiency virus drug therapy and virus load, J. Virol., 71 (1997), 3275-3278. |
[4] |
L. M. Cai and X. Z. Li, Stability and Hopf bifurcation in a delayed model for HIV infection of CD4$^+$ T-cells, Chaos, Solitons and Fractals, 42 (2009), 1-11.
doi: 10.1016/j.chaos.2008.04.048. |
[5] |
D. S. Callaway and A. S. Perelson, HIV-1 infection and low steady state viral loads, Bull. Math. Biol., 64 (2002), 29-64.
doi: 10.1006/bulm.2001.0266. |
[6] |
M. S. Ciupe, B. L. Bivort, D. M. Bortz and P. W. Nelson, Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathmatical models, Math. Biosci., 200 (2006), 1-27.
doi: 10.1016/j.mbs.2005.12.006. |
[7] |
K. L. Cooke and P. van den Driessche, On zeros of some transcendental equations, Funkcialaj Ekvacioj, 29 (1986), 77-90. |
[8] |
R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4+ T-cells, Math. Biosci., 165 (2000), 27-39.
doi: 10.1016/S0025-5564(00)00006-7. |
[9] |
E. S. Daar, T. Moudgil, R. D. Meyer and D. D. Ho, Transient high levels of viremia in patients with primary human immunodefiniency virus type 1 infection, N. Engl. J. Med., 324 (1991), 961-964.
doi: 10.1056/NEJM199104043241405. |
[10] |
R. J. De Boer and A. S. Perelson, Towards a general function describing T cell proliferation, J. Theor. Biol., 175 (1995), 567-576. |
[11] |
R. J. De Boer and A. S. Perelson, Target cell limited and immune control models of HIV infection: A comparison, J. Theor. Biol., 190 (1998), 201-214. |
[12] |
N. M. Dixit and A. S. Perelson, Complex patterns of viral load decay under antiretroviral therapy: influence of pharmacokinetics and intracellur delay, J. Theor. Biol., 226 (2004), 95-109.
doi: 10.1016/j.jtbi.2003.09.002. |
[13] |
H. I. Freedman and Y. Kuang, Stability switches in linear scalar neutral delay equations, Funkcialaj Ekvacioj, 34 (1991), 187-209. |
[14] |
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4342-7. |
[15] |
B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Application of Hopf Bifurcation, Cambrige University Press, Cambrige, 1981. |
[16] |
J. M. Heffernan and L. M. Wahl, Natural variation in HIV infection: Monte carlo estimates that include CD8 effector cells, J. Theor. Biol., 243 (2006), 191-204.
doi: 10.1016/j.jtbi.2006.05.032. |
[17] |
A. V. 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, Proc. Natl. Acad. Sci. USA, 93 (1996), 7247-7251.
doi: 10.1073/pnas.93.14.7247. |
[18] |
J. P. Lasalle, The stability of dynamical systems, in Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA, 1976. |
[19] |
M. Y. Li and H. Y. Shu, Global dynamics of a mathematical model for HTLV-I infection of CD4+ T cells with delayed CTL response, Nonlinear Analysis: Real World Applications, 13 (2012), 1080-1092.
doi: 10.1016/j.nonrwa.2011.02.026. |
[20] |
M. Y. Li and H. Shu, Impact of intracelluar delays and target-cell dynamics on in vivo viral infections, SIAM J. Appl. Math., 70 (2010), 2434-2448.
doi: 10.1137/090779322. |
[21] |
S. Q. Liu and L. Wang, Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy, Math. Biosci. Eng., 7 (2010), 675-685.
doi: 10.3934/mbe.2010.7.675. |
[22] |
P .W. Nelson, J. D. Murray and A. S. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay, Math. Biosci., 163 (2000), 201-215.
doi: 10.1016/S0025-5564(99)00055-3. |
[23] |
P. W. Nelson and A. S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection, Math. Biosci., 179 (2002), 73-94.
doi: 10.1016/S0025-5564(02)00099-8. |
[24] |
M. A. Nowak and C. R. Bangham, Population dynamics of immune responses to persistent virus, Science, 272 (1996), 74-79. |
[25] |
M. A. Nowak and R. M. May, Virus dynamics: Mathematical principles of immunology and virology, Oxford University, Oxford, 2000. |
[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, Math. Biosci., 235 (2012), 98-109.
doi: 10.1016/j.mbs.2011.11.002. |
[27] |
A. S. Perelson, Modelling viral and immune system dynamics, Nat. Rev. Immunol., 2 (2002), 28-36.
doi: 10.1038/nri700. |
[28] |
A. S. Perelson, P. Essunger, Y. Cao, M. Vesanen, A. Hurley, K. Saksela, M. Markowitz and D. D. Ho, Decay characteristics of HIV-1 infected compartments during combination therepy, Nature, 387 (1997), 188-191.
doi: 10.1038/387188a0. |
[29] |
A. S. Perelson, D. E. Kirschner and R. De Boer, Dynamics of HIV infection of CD4+ T cells, Math. Biosci., 114 (1993), 81-125.
doi: 10.1016/0025-5564(93)90043-A. |
[30] |
A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44.
doi: 10.1137/S0036144598335107. |
[31] |
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.
doi: 10.1126/science.271.5255.1582. |
[32] |
A. N. Phillips, Reduction of HIV concentration during acute infection: Independence from a specific immune response, Science, 271 (1996), 497-499.
doi: 10.1126/science.271.5248.497. |
[33] |
B. Ramratnam, S. Bonhoeffer, J. Binley, A. Hurley, L. Zhang, J. E. Mittler, M. Minarkowitz, J. P. Moore, A. S. Perelson and D. D. Ho, Rapid production and clearance of HIV-1 and hepatitis C virus assessed by large volume plasma apheresis, Lancet, 354 (1999), 1782-1785.
doi: 10.1016/S0140-6736(99)02035-8. |
[34] |
L. Rong, Z. Feng and A. S. Perelson, Emergence of HIV-1 drug resistence during antiretroviral treatment, Bull. Math. Biol., 69 (2007), 2027-2060.
doi: 10.1007/s11538-007-9203-3. |
[35] |
H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, in Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995. |
[36] |
H. Smith and X. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.
doi: 10.1016/S0362-546X(01)00678-2. |
[37] |
M. A. Stafford, L. Corey, Y. Z. Cao, E. S. Daar, D. D. Ho and A. S. Perelson, Modeling Plasma Virus Concentration during Primary HIV Infection, J. Theor. Biol., 203 (2000), 285-301.
doi: 10.1006/jtbi.2000.1076. |
[38] |
J. Wang, G. Huang and Y. Takeuchi, Global asymptotic stability for HIV-1 dynamics with two distributed delays, Mathematical Medicine and Biology, 29 (2012), 283-300.
doi: 10.1093/imammb/dqr009. |
[39] |
L. Wang and M. Y. Li, Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells, Math. Biosci., 200 (2006), 44-57.
doi: 10.1016/j.mbs.2005.12.026. |
[40] |
K. Wang, W. Wang and X. Liu, Global stability in a viral infection model with lytic and nonlytic immune response, Comput. Math. Appl., 51 (2006), 1593-1610.
doi: 10.1016/j.camwa.2005.07.020. |
[41] |
Y. Wang, Y. Zhou, J. Wu and J. Heffernan, Oscillatory viral dynamics in a delayed HIV pathogenesis model, Math, Biosci., 219 (2009), 104-112.
doi: 10.1016/j.mbs.2009.03.003. |
[42] |
D. Wodarz, J. P. Christensen and A. R. Thomsen, The importance of lytic and nonlytic immune responses viral infections, Trends. Immunol., 23 (2002), 194-200.
doi: 10.1016/S1471-4906(02)02189-0. |
[43] |
D. Wodarz, K. Page, R. Arnaout, A. Thomsen, J. Lifson and M. A. Nowak, A new theiry of cytotoxic T-lymphocyte memory: Implications for HIV treatment, Philosophical Transactions of the Royal Society B: Biological Sciences, 355 (2000), 329-343.
doi: 10.1098/rstb.2000.0570. |
[44] |
Z. Wu, Z. Y. Liu and R. Detels, HIV-1 infection in commercial plasma donors in China, Lancet, 346 (1995), 61-62.
doi: 10.1016/S0140-6736(95)92698-4. |
[45] |
H. Zhu and X. Zou, Dynamics of an HIV-1 infection model with cell-mediated immune response and intracellular delay, Discrete Contin. Dyn. Syst. B, 12 (2009), 511-524.
doi: 10.3934/dcdsb.2009.12.511. |
[46] |
H. Zhu and X. Zou, Impact of delays in cell infection and virus production on HIV-1 dynamics, Math. Med. Biol., 25 (2008), 99-112.
doi: 10.1093/imammb/dqm010. |
show all references
References:
[1] |
R. Arnaout, M. A. Nowak and D. Wodarz, HIV-1 dynamics revisited: Biphasic decay by cytotoxic lymphocyte killing?, Proc. R. Soc. Lond. B, 267 (2000), 1347-1354.
doi: 10.1098/rspb.2000.1149. |
[2] |
H. T. Banks and D. M. Bortz, A parameter sensitivity methodology in the context of HIV delay equation models, J. Math. Biol., 50 (2005), 607-625.
doi: 10.1007/s00285-004-0299-x. |
[3] |
S. Bonhoeffer, J. M. Coffin and M. A. Nowak, Human immunodeficiency virus drug therapy and virus load, J. Virol., 71 (1997), 3275-3278. |
[4] |
L. M. Cai and X. Z. Li, Stability and Hopf bifurcation in a delayed model for HIV infection of CD4$^+$ T-cells, Chaos, Solitons and Fractals, 42 (2009), 1-11.
doi: 10.1016/j.chaos.2008.04.048. |
[5] |
D. S. Callaway and A. S. Perelson, HIV-1 infection and low steady state viral loads, Bull. Math. Biol., 64 (2002), 29-64.
doi: 10.1006/bulm.2001.0266. |
[6] |
M. S. Ciupe, B. L. Bivort, D. M. Bortz and P. W. Nelson, Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathmatical models, Math. Biosci., 200 (2006), 1-27.
doi: 10.1016/j.mbs.2005.12.006. |
[7] |
K. L. Cooke and P. van den Driessche, On zeros of some transcendental equations, Funkcialaj Ekvacioj, 29 (1986), 77-90. |
[8] |
R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4+ T-cells, Math. Biosci., 165 (2000), 27-39.
doi: 10.1016/S0025-5564(00)00006-7. |
[9] |
E. S. Daar, T. Moudgil, R. D. Meyer and D. D. Ho, Transient high levels of viremia in patients with primary human immunodefiniency virus type 1 infection, N. Engl. J. Med., 324 (1991), 961-964.
doi: 10.1056/NEJM199104043241405. |
[10] |
R. J. De Boer and A. S. Perelson, Towards a general function describing T cell proliferation, J. Theor. Biol., 175 (1995), 567-576. |
[11] |
R. J. De Boer and A. S. Perelson, Target cell limited and immune control models of HIV infection: A comparison, J. Theor. Biol., 190 (1998), 201-214. |
[12] |
N. M. Dixit and A. S. Perelson, Complex patterns of viral load decay under antiretroviral therapy: influence of pharmacokinetics and intracellur delay, J. Theor. Biol., 226 (2004), 95-109.
doi: 10.1016/j.jtbi.2003.09.002. |
[13] |
H. I. Freedman and Y. Kuang, Stability switches in linear scalar neutral delay equations, Funkcialaj Ekvacioj, 34 (1991), 187-209. |
[14] |
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4342-7. |
[15] |
B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Application of Hopf Bifurcation, Cambrige University Press, Cambrige, 1981. |
[16] |
J. M. Heffernan and L. M. Wahl, Natural variation in HIV infection: Monte carlo estimates that include CD8 effector cells, J. Theor. Biol., 243 (2006), 191-204.
doi: 10.1016/j.jtbi.2006.05.032. |
[17] |
A. V. 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, Proc. Natl. Acad. Sci. USA, 93 (1996), 7247-7251.
doi: 10.1073/pnas.93.14.7247. |
[18] |
J. P. Lasalle, The stability of dynamical systems, in Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA, 1976. |
[19] |
M. Y. Li and H. Y. Shu, Global dynamics of a mathematical model for HTLV-I infection of CD4+ T cells with delayed CTL response, Nonlinear Analysis: Real World Applications, 13 (2012), 1080-1092.
doi: 10.1016/j.nonrwa.2011.02.026. |
[20] |
M. Y. Li and H. Shu, Impact of intracelluar delays and target-cell dynamics on in vivo viral infections, SIAM J. Appl. Math., 70 (2010), 2434-2448.
doi: 10.1137/090779322. |
[21] |
S. Q. Liu and L. Wang, Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy, Math. Biosci. Eng., 7 (2010), 675-685.
doi: 10.3934/mbe.2010.7.675. |
[22] |
P .W. Nelson, J. D. Murray and A. S. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay, Math. Biosci., 163 (2000), 201-215.
doi: 10.1016/S0025-5564(99)00055-3. |
[23] |
P. W. Nelson and A. S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection, Math. Biosci., 179 (2002), 73-94.
doi: 10.1016/S0025-5564(02)00099-8. |
[24] |
M. A. Nowak and C. R. Bangham, Population dynamics of immune responses to persistent virus, Science, 272 (1996), 74-79. |
[25] |
M. A. Nowak and R. M. May, Virus dynamics: Mathematical principles of immunology and virology, Oxford University, Oxford, 2000. |
[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, Math. Biosci., 235 (2012), 98-109.
doi: 10.1016/j.mbs.2011.11.002. |
[27] |
A. S. Perelson, Modelling viral and immune system dynamics, Nat. Rev. Immunol., 2 (2002), 28-36.
doi: 10.1038/nri700. |
[28] |
A. S. Perelson, P. Essunger, Y. Cao, M. Vesanen, A. Hurley, K. Saksela, M. Markowitz and D. D. Ho, Decay characteristics of HIV-1 infected compartments during combination therepy, Nature, 387 (1997), 188-191.
doi: 10.1038/387188a0. |
[29] |
A. S. Perelson, D. E. Kirschner and R. De Boer, Dynamics of HIV infection of CD4+ T cells, Math. Biosci., 114 (1993), 81-125.
doi: 10.1016/0025-5564(93)90043-A. |
[30] |
A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44.
doi: 10.1137/S0036144598335107. |
[31] |
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.
doi: 10.1126/science.271.5255.1582. |
[32] |
A. N. Phillips, Reduction of HIV concentration during acute infection: Independence from a specific immune response, Science, 271 (1996), 497-499.
doi: 10.1126/science.271.5248.497. |
[33] |
B. Ramratnam, S. Bonhoeffer, J. Binley, A. Hurley, L. Zhang, J. E. Mittler, M. Minarkowitz, J. P. Moore, A. S. Perelson and D. D. Ho, Rapid production and clearance of HIV-1 and hepatitis C virus assessed by large volume plasma apheresis, Lancet, 354 (1999), 1782-1785.
doi: 10.1016/S0140-6736(99)02035-8. |
[34] |
L. Rong, Z. Feng and A. S. Perelson, Emergence of HIV-1 drug resistence during antiretroviral treatment, Bull. Math. Biol., 69 (2007), 2027-2060.
doi: 10.1007/s11538-007-9203-3. |
[35] |
H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, in Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995. |
[36] |
H. Smith and X. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.
doi: 10.1016/S0362-546X(01)00678-2. |
[37] |
M. A. Stafford, L. Corey, Y. Z. Cao, E. S. Daar, D. D. Ho and A. S. Perelson, Modeling Plasma Virus Concentration during Primary HIV Infection, J. Theor. Biol., 203 (2000), 285-301.
doi: 10.1006/jtbi.2000.1076. |
[38] |
J. Wang, G. Huang and Y. Takeuchi, Global asymptotic stability for HIV-1 dynamics with two distributed delays, Mathematical Medicine and Biology, 29 (2012), 283-300.
doi: 10.1093/imammb/dqr009. |
[39] |
L. Wang and M. Y. Li, Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells, Math. Biosci., 200 (2006), 44-57.
doi: 10.1016/j.mbs.2005.12.026. |
[40] |
K. Wang, W. Wang and X. Liu, Global stability in a viral infection model with lytic and nonlytic immune response, Comput. Math. Appl., 51 (2006), 1593-1610.
doi: 10.1016/j.camwa.2005.07.020. |
[41] |
Y. Wang, Y. Zhou, J. Wu and J. Heffernan, Oscillatory viral dynamics in a delayed HIV pathogenesis model, Math, Biosci., 219 (2009), 104-112.
doi: 10.1016/j.mbs.2009.03.003. |
[42] |
D. Wodarz, J. P. Christensen and A. R. Thomsen, The importance of lytic and nonlytic immune responses viral infections, Trends. Immunol., 23 (2002), 194-200.
doi: 10.1016/S1471-4906(02)02189-0. |
[43] |
D. Wodarz, K. Page, R. Arnaout, A. Thomsen, J. Lifson and M. A. Nowak, A new theiry of cytotoxic T-lymphocyte memory: Implications for HIV treatment, Philosophical Transactions of the Royal Society B: Biological Sciences, 355 (2000), 329-343.
doi: 10.1098/rstb.2000.0570. |
[44] |
Z. Wu, Z. Y. Liu and R. Detels, HIV-1 infection in commercial plasma donors in China, Lancet, 346 (1995), 61-62.
doi: 10.1016/S0140-6736(95)92698-4. |
[45] |
H. Zhu and X. Zou, Dynamics of an HIV-1 infection model with cell-mediated immune response and intracellular delay, Discrete Contin. Dyn. Syst. B, 12 (2009), 511-524.
doi: 10.3934/dcdsb.2009.12.511. |
[46] |
H. Zhu and X. Zou, Impact of delays in cell infection and virus production on HIV-1 dynamics, Math. Med. Biol., 25 (2008), 99-112.
doi: 10.1093/imammb/dqm010. |
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