Virus dynamics model with intracellular delays and immune response

Haitao Song; Weihua Jiang; Shengqiang Liu

doi:10.3934/mbe.2015.12.185

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2015, 12(1): 185-208. doi: 10.3934/mbe.2015.12.185

Virus dynamics model with intracellular delays and immune response

Haitao Song ^1, , Weihua Jiang ^1, and Shengqiang Liu ^2,

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

Received October 2013 Revised October 2014 Published December 2014

Abstract
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In this paper, we incorporate an extra logistic growth term for uninfected CD4$^+$ T-cells into an HIV-1 infection model with both intracellular delay and immune response delay which was studied by Pawelek et al. in [26]. First, we proved that if the basic reproduction number $R_0<1$, then the infection-free steady state is globally asymptotically stable. Second, when $R_0>1$, then the system is uniformly persistent, suggesting that the clearance or the uniform persistence of the virus is completely determined by $R_0 $. Furthermore, given both the two delays are zero, then the infected steady state is asymptotically stable when the intrinsic growth rate of the extra logistic term is sufficiently small. When the two delays are not zero, we showed that both the immune response delay and the intracellular delay may destabilize the infected steady state by leading to Hopf bifurcation and stable periodic oscillations, on which we analyzed the direction of the Hopf bifurcation as well as the stability of the bifurcating periodic orbits by normal form and center manifold theory introduced by Hassard et al [15]. Third, we engaged numerical simulations to explore the rich dynamics like chaotic oscillations, complicated bifurcation diagram of viral load due to the logistic term of target cells and the two time delays.

Keywords: uniform persistence., logistic growth, HIV-1 model, intracellular delay, Hopf bifurcation.

Mathematics Subject Classification: 34C23, 34D23, 92D3.

Citation: Haitao Song, Weihua Jiang, Shengqiang Liu. Virus dynamics model with intracellular delays and immune response. Mathematical Biosciences & Engineering, 2015, 12 (1) : 185-208. doi: 10.3934/mbe.2015.12.185

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. Google Scholar
[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. Google Scholar
[3]	S. Bonhoeffer, J. M. Coffin and M. A. Nowak, Human immunodeficiency virus drug therapy and virus load, J. Virol., 71 (1997), 3275-3278. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	K. L. Cooke and P. van den Driessche, On zeros of some transcendental equations, Funkcialaj Ekvacioj, 29 (1986), 77-90. Google Scholar
[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. Google Scholar
[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. Google Scholar
[10]	R. J. De Boer and A. S. Perelson, Towards a general function describing T cell proliferation, J. Theor. Biol., 175 (1995), 567-576. Google Scholar
[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. Google Scholar
[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. Google Scholar
[13]	H. I. Freedman and Y. Kuang, Stability switches in linear scalar neutral delay equations, Funkcialaj Ekvacioj, 34 (1991), 187-209. Google Scholar
[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. Google Scholar
[15]	B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Application of Hopf Bifurcation, Cambrige University Press, Cambrige, 1981. Google Scholar
[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. Google Scholar
[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. Google Scholar
[18]	J. P. Lasalle, The stability of dynamical systems, in Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA, 1976. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[24]	M. A. Nowak and C. R. Bangham, Population dynamics of immune responses to persistent virus, Science, 272 (1996), 74-79. Google Scholar
[25]	M. A. Nowak and R. M. May, Virus dynamics: Mathematical principles of immunology and virology, Oxford University, Oxford, 2000. Google Scholar
[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. Google Scholar
[27]	A. S. Perelson, Modelling viral and immune system dynamics, Nat. Rev. Immunol., 2 (2002), 28-36. doi: 10.1038/nri700. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[3]	S. Bonhoeffer, J. M. Coffin and M. A. Nowak, Human immunodeficiency virus drug therapy and virus load, J. Virol., 71 (1997), 3275-3278. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	K. L. Cooke and P. van den Driessche, On zeros of some transcendental equations, Funkcialaj Ekvacioj, 29 (1986), 77-90. Google Scholar
[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. Google Scholar
[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. Google Scholar
[10]	R. J. De Boer and A. S. Perelson, Towards a general function describing T cell proliferation, J. Theor. Biol., 175 (1995), 567-576. Google Scholar
[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. Google Scholar
[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. Google Scholar
[13]	H. I. Freedman and Y. Kuang, Stability switches in linear scalar neutral delay equations, Funkcialaj Ekvacioj, 34 (1991), 187-209. Google Scholar
[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. Google Scholar
[15]	B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Application of Hopf Bifurcation, Cambrige University Press, Cambrige, 1981. Google Scholar
[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. Google Scholar
[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. Google Scholar
[18]	J. P. Lasalle, The stability of dynamical systems, in Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA, 1976. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[24]	M. A. Nowak and C. R. Bangham, Population dynamics of immune responses to persistent virus, Science, 272 (1996), 74-79. Google Scholar
[25]	M. A. Nowak and R. M. May, Virus dynamics: Mathematical principles of immunology and virology, Oxford University, Oxford, 2000. Google Scholar
[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. Google Scholar
[27]	A. S. Perelson, Modelling viral and immune system dynamics, Nat. Rev. Immunol., 2 (2002), 28-36. doi: 10.1038/nri700. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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2018 Impact Factor: 1.313

American Institute of Mathematical Sciences

Virus dynamics model with intracellular delays and immune response

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American Institute of Mathematical Sciences

Virus dynamics model with intracellular delays and immune response

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