2016, 13(2): 343-367. doi: 10.3934/mbe.2015006

Bifurcation analysis of HIV-1 infection model with cell-to-cell transmission and immune response delay

1. 

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049

Received  April 2015 Revised  October 2015 Published  November 2015

A within-host viral infection model with both virus-to-cell and cell-to-cell transmissions and time delay in immune response is investigated. Mathematical analysis shows that delay may destabilize the infected steady state and lead to Hopf bifurcation. Moreover, the direction of the Hopf bifurcation and the stability of the periodic solutions are investigated by normal form and center manifold theory. Numerical simulations are done to explore the rich dynamics, including stability switches, Hopf bifurcations, and chaotic oscillations.
Citation: Jinhu Xu, Yicang Zhou. Bifurcation analysis of HIV-1 infection model with cell-to-cell transmission and immune response delay. Mathematical Biosciences & Engineering, 2016, 13 (2) : 343-367. doi: 10.3934/mbe.2015006
References:
[1]

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.

[2]

P. De Leenheer and H. L. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327. doi: 10.1137/S0036139902406905.

[3]

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 mathematical models, Math. Biosci., 200 (2006), 1-27. doi: 10.1016/j.mbs.2005.12.006.

[4]

V. Herz, S. Bonhoeffer and R. Anderson, et al., Viral dynamics in vivo: Limitations on estimations on intracellular delay and virus delay, Proc. Natl. Acad. Sci. USA, 93 (1996), 7247-7251. doi: 10.1073/pnas.93.14.7247.

[5]

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.

[6]

R. V. Culshaw and S. G. 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.

[7]

J. Mittler, B. Sulzer, A. Neumann and A. S. Perelson, Influence of delayed virus production on viral dynamics in HIV-1 infected patients, Math. Biosci., 152 (1998), 143-163. doi: 10.1016/S0025-5564(98)10027-5.

[8]

P. W. Nelson, J. 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.

[9]

M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay, Bull. Math. Biol., 72 (2010), 1492-1505. doi: 10.1007/s11538-010-9503-x.

[10]

S. 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.

[11]

R. V. Culshaw, S. Ruan and G. Webb, A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay, J. Math. Biol., 46 (2003), 425-444. doi: 10.1007/s00285-002-0191-5.

[12]

H. Zhu and X. Zou, Dynamics of a HIV-1 infection model with cell-mediated immume response and intracellular delay, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 513-526. doi: 10.3934/dcdsb.2009.12.511.

[13]

H. Y. Zhu, Y. Luo and M. L. Chen, Stability and Hopf bifurcation of a HIV infection model with CTL-response delay, Comput. Math. Appl., 62 (2011), 3091-3102. doi: 10.1016/j.camwa.2011.08.022.

[14]

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.

[15]

M. Y. Li and H. Shu, Multiple stable periodic oscillations in a mathematical model of CTL response to HTLV-I infection, Bull. Math. Biol., 73 (2011), 1774-1793. doi: 10.1007/s11538-010-9591-7.

[16]

H. Song, W. Jiang and S. Liu, Virus dynamics model with intracellular delays and immune response, Math. Biosci. Eng.,12 (2015), 185-208. doi: 10.3934/mbe.2015.12.185.

[17]

M. Sourisseau, N. Sol-Foulon, F. Porrot, F. Blanchet and O. Schwartz, Inefficient human immunodeficiency virus replication in mobile lymphocytes, J. Virol., 81 (2007), 1000-1012. doi: 10.1128/JVI.01629-06.

[18]

D. S. Dimitrov, R. L. Willey and H. Sato, et al., Quantitation of human immunodeficiency virus type 1 infection kinetics, J. Virol., 67 (1993), 2182-2190.

[19]

H. Sato, J. Orenstein, D. S. Dimitrov and M. A. Martin, Cell-to-cell spread of HIV-1 occurs with minutes and may not involve the participation of virus particles, Virology, 186 (1992), 712-724. doi: 10.1016/0042-6822(92)90038-Q.

[20]

S. Gummuluru, C. M. Kinsey and M. Emerman, An in vitro rapid-turnover assay for human immunodeficiency virus type 1 replication selects for cell-to-cell spread of virus, J. Virol., 74 (2000), 10882-10891. doi: 10.1128/JVI.74.23.10882-10891.2000.

[21]

J. Witteveldt, M. J. Evans and J. Bitzegeio, et al., CD81 is dispensable for hepatitis C virus cell-to-cell transmission in hepatoma cells, J. Gen. Virol., 90 (2009), 48-58. doi: 10.1099/vir.0.006700-0.

[22]

J. Lupberger, J. M. B. Zeisel and F. Xiao et al., EGFR and EphA2 are hepatitis C virus host entry factors and targets for antiviral therapy, Nat. Med., 17 (2011), 589-595. doi: 10.1038/nm.2341.

[23]

I. Fofana, F. Xiao and C. Thumann, et al., A novel monoclonal anti-CD81 antibody produced by genetic immunization efficiently inhibits Hepatitis C virus cell-cell transmission, PloS one., 8 (2013), e64221. doi: 10.1371/journal.pone.0064221.

[24]

S. Imai, J. Nishikawa ang K. Takada, Cell-to-cell contact as an efficient mode of Epstein-Barr virus infection of diverse human epithelial cells, J. Virol., 72 (1998), 4371-4378.

[25]

Q. J. Sattentau, The direct passage of animal viruses between cells, Current opinion in virology, 1 (2011), 396-402. doi: 10.1016/j.coviro.2011.09.004.

[26]

S. Benovic, T. Kok, A. Stephenson and J. McInnes, et al., De novo reverse transcription of HTLV-1 following cell-to-cell transmission of infection, Virology., 244 (1998), 294-301. doi: 10.1006/viro.1998.9111.

[27]

M. L. Dustin and J. A. Cooper, The immunological synapse and the actin cytoskeleton: Molecular hardware for T cell signaling, Nat Immunol., 1 (2000), 23-29. doi: 10.1038/76877.

[28]

N. Martin and Q. Sattentau, Cell-to-cell HIV-1 spread and its implications for immune evasion, Curr Opin HIV AIDS., 4 (2009), 143-149. doi: 10.1097/COH.0b013e328322f94a.

[29]

Q. Sattentau, Avoiding the void: Cell-to-cell spread of human viruses, Nat Rev Microbiol, 6 (2008), 815-826. doi: 10.1038/nrmicro1972.

[30]

G. Carloni, A. Crema and M. B Valli, et al., HCV infection by cell-to-cell transmission: Choice or necessity?, Current molecular medicine., 12 (2012), 83-95. doi: 10.2174/156652412798376152.

[31]

X. L. Lai and X. F. Zou, Modeling HIV-1 virus dynamics with both virus-to-cell infection and cell-to-cell transmission, SIAM J. Appl. Math., 74 (2014), 898-917. doi: 10.1137/130930145.

[32]

X. L. Lai and X. F. Zou, Modeling cell-to-cell spread of HIV-1 with logistic target cell growth, J. Math. Anal. Appl., 426 (2015), 563-584. doi: 10.1016/j.jmaa.2014.10.086.

[33]

M. Nowak and C. Bangham, Population dynamics of immune responses to persistent viruses, Science., 272 (1996), 74-79. doi: 10.1126/science.272.5258.74.

[34]

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.

[35]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993. doi: 10.1007/978-1-4612-4342-7.

[36]

W. M. Hirsch, H. Hanisch and J. P. Gabril, Differential equation models of some parasitic infections: Methods for the study of asymptotic behavior, Commun. Pure. Appl. Math., 38 (1985), 733-753. doi: 10.1002/cpa.3160380607.

[37]

J. Hale and W. Z. Huang, Global geometry of the stable regions for two delay differential equations, J. Math. Anal. Appl., 178 (1993), 344-362. doi: 10.1006/jmaa.1993.1312.

[38]

S. Busenberg and K. L. Cooke, Vertically Transmitted Diseases: Models and Dynamics, vol. 23. Biomathematics. New York: Springer, 1993. doi: 10.1007/978-3-642-75301-5.

[39]

X. Li and J. Wei, On the zeros of a fourth degree exponential polynomial with applications to an eural network model with delays, Chaos Solitions Fractals., 26 (2005), 519-526. doi: 10.1016/j.chaos.2005.01.019.

[40]

J. Hale, Theory of Function Differential Equations, Springer, Heidelberg, 1977.

[41]

B. D. Hassard, N. D. Kazariniff and Y. H. Wan, Theory and Application of Hopf Bifurcation, London math society lecture note series, vol. 41. Cambridge University Press, 1981.

show all references

References:
[1]

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.

[2]

P. De Leenheer and H. L. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327. doi: 10.1137/S0036139902406905.

[3]

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 mathematical models, Math. Biosci., 200 (2006), 1-27. doi: 10.1016/j.mbs.2005.12.006.

[4]

V. Herz, S. Bonhoeffer and R. Anderson, et al., Viral dynamics in vivo: Limitations on estimations on intracellular delay and virus delay, Proc. Natl. Acad. Sci. USA, 93 (1996), 7247-7251. doi: 10.1073/pnas.93.14.7247.

[5]

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.

[6]

R. V. Culshaw and S. G. 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.

[7]

J. Mittler, B. Sulzer, A. Neumann and A. S. Perelson, Influence of delayed virus production on viral dynamics in HIV-1 infected patients, Math. Biosci., 152 (1998), 143-163. doi: 10.1016/S0025-5564(98)10027-5.

[8]

P. W. Nelson, J. 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.

[9]

M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay, Bull. Math. Biol., 72 (2010), 1492-1505. doi: 10.1007/s11538-010-9503-x.

[10]

S. 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.

[11]

R. V. Culshaw, S. Ruan and G. Webb, A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay, J. Math. Biol., 46 (2003), 425-444. doi: 10.1007/s00285-002-0191-5.

[12]

H. Zhu and X. Zou, Dynamics of a HIV-1 infection model with cell-mediated immume response and intracellular delay, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 513-526. doi: 10.3934/dcdsb.2009.12.511.

[13]

H. Y. Zhu, Y. Luo and M. L. Chen, Stability and Hopf bifurcation of a HIV infection model with CTL-response delay, Comput. Math. Appl., 62 (2011), 3091-3102. doi: 10.1016/j.camwa.2011.08.022.

[14]

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.

[15]

M. Y. Li and H. Shu, Multiple stable periodic oscillations in a mathematical model of CTL response to HTLV-I infection, Bull. Math. Biol., 73 (2011), 1774-1793. doi: 10.1007/s11538-010-9591-7.

[16]

H. Song, W. Jiang and S. Liu, Virus dynamics model with intracellular delays and immune response, Math. Biosci. Eng.,12 (2015), 185-208. doi: 10.3934/mbe.2015.12.185.

[17]

M. Sourisseau, N. Sol-Foulon, F. Porrot, F. Blanchet and O. Schwartz, Inefficient human immunodeficiency virus replication in mobile lymphocytes, J. Virol., 81 (2007), 1000-1012. doi: 10.1128/JVI.01629-06.

[18]

D. S. Dimitrov, R. L. Willey and H. Sato, et al., Quantitation of human immunodeficiency virus type 1 infection kinetics, J. Virol., 67 (1993), 2182-2190.

[19]

H. Sato, J. Orenstein, D. S. Dimitrov and M. A. Martin, Cell-to-cell spread of HIV-1 occurs with minutes and may not involve the participation of virus particles, Virology, 186 (1992), 712-724. doi: 10.1016/0042-6822(92)90038-Q.

[20]

S. Gummuluru, C. M. Kinsey and M. Emerman, An in vitro rapid-turnover assay for human immunodeficiency virus type 1 replication selects for cell-to-cell spread of virus, J. Virol., 74 (2000), 10882-10891. doi: 10.1128/JVI.74.23.10882-10891.2000.

[21]

J. Witteveldt, M. J. Evans and J. Bitzegeio, et al., CD81 is dispensable for hepatitis C virus cell-to-cell transmission in hepatoma cells, J. Gen. Virol., 90 (2009), 48-58. doi: 10.1099/vir.0.006700-0.

[22]

J. Lupberger, J. M. B. Zeisel and F. Xiao et al., EGFR and EphA2 are hepatitis C virus host entry factors and targets for antiviral therapy, Nat. Med., 17 (2011), 589-595. doi: 10.1038/nm.2341.

[23]

I. Fofana, F. Xiao and C. Thumann, et al., A novel monoclonal anti-CD81 antibody produced by genetic immunization efficiently inhibits Hepatitis C virus cell-cell transmission, PloS one., 8 (2013), e64221. doi: 10.1371/journal.pone.0064221.

[24]

S. Imai, J. Nishikawa ang K. Takada, Cell-to-cell contact as an efficient mode of Epstein-Barr virus infection of diverse human epithelial cells, J. Virol., 72 (1998), 4371-4378.

[25]

Q. J. Sattentau, The direct passage of animal viruses between cells, Current opinion in virology, 1 (2011), 396-402. doi: 10.1016/j.coviro.2011.09.004.

[26]

S. Benovic, T. Kok, A. Stephenson and J. McInnes, et al., De novo reverse transcription of HTLV-1 following cell-to-cell transmission of infection, Virology., 244 (1998), 294-301. doi: 10.1006/viro.1998.9111.

[27]

M. L. Dustin and J. A. Cooper, The immunological synapse and the actin cytoskeleton: Molecular hardware for T cell signaling, Nat Immunol., 1 (2000), 23-29. doi: 10.1038/76877.

[28]

N. Martin and Q. Sattentau, Cell-to-cell HIV-1 spread and its implications for immune evasion, Curr Opin HIV AIDS., 4 (2009), 143-149. doi: 10.1097/COH.0b013e328322f94a.

[29]

Q. Sattentau, Avoiding the void: Cell-to-cell spread of human viruses, Nat Rev Microbiol, 6 (2008), 815-826. doi: 10.1038/nrmicro1972.

[30]

G. Carloni, A. Crema and M. B Valli, et al., HCV infection by cell-to-cell transmission: Choice or necessity?, Current molecular medicine., 12 (2012), 83-95. doi: 10.2174/156652412798376152.

[31]

X. L. Lai and X. F. Zou, Modeling HIV-1 virus dynamics with both virus-to-cell infection and cell-to-cell transmission, SIAM J. Appl. Math., 74 (2014), 898-917. doi: 10.1137/130930145.

[32]

X. L. Lai and X. F. Zou, Modeling cell-to-cell spread of HIV-1 with logistic target cell growth, J. Math. Anal. Appl., 426 (2015), 563-584. doi: 10.1016/j.jmaa.2014.10.086.

[33]

M. Nowak and C. Bangham, Population dynamics of immune responses to persistent viruses, Science., 272 (1996), 74-79. doi: 10.1126/science.272.5258.74.

[34]

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.

[35]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993. doi: 10.1007/978-1-4612-4342-7.

[36]

W. M. Hirsch, H. Hanisch and J. P. Gabril, Differential equation models of some parasitic infections: Methods for the study of asymptotic behavior, Commun. Pure. Appl. Math., 38 (1985), 733-753. doi: 10.1002/cpa.3160380607.

[37]

J. Hale and W. Z. Huang, Global geometry of the stable regions for two delay differential equations, J. Math. Anal. Appl., 178 (1993), 344-362. doi: 10.1006/jmaa.1993.1312.

[38]

S. Busenberg and K. L. Cooke, Vertically Transmitted Diseases: Models and Dynamics, vol. 23. Biomathematics. New York: Springer, 1993. doi: 10.1007/978-3-642-75301-5.

[39]

X. Li and J. Wei, On the zeros of a fourth degree exponential polynomial with applications to an eural network model with delays, Chaos Solitions Fractals., 26 (2005), 519-526. doi: 10.1016/j.chaos.2005.01.019.

[40]

J. Hale, Theory of Function Differential Equations, Springer, Heidelberg, 1977.

[41]

B. D. Hassard, N. D. Kazariniff and Y. H. Wan, Theory and Application of Hopf Bifurcation, London math society lecture note series, vol. 41. Cambridge University Press, 1981.

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