August  2014, 19(6): 1749-1768. doi: 10.3934/dcdsb.2014.19.1749

Global stability and backward bifurcation of a general viral infection model with virus-driven proliferation of target cells

1. 

Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, E3B 5A3, Canada

Received  April 2013 Revised  October 2013 Published  June 2014

In this paper, a general viral model with virus-driven proliferation of target cells is studied. Global stability results are established by employing the Lyapunov method and a geometric approach developed by Li and Muldowney. It is shown that under certain conditions, the model exhibits a global threshold dynamics, while if these conditions are not met, then backward bifurcation and bistability are possible. An example is presented to provide some insights on how the virus-driven proliferation of target cells influences the virus dynamics and the drug therapy strategies.
Citation: Hongying Shu, Lin Wang. Global stability and backward bifurcation of a general viral infection model with virus-driven proliferation of target cells. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1749-1768. doi: 10.3934/dcdsb.2014.19.1749
References:
[1]

N. P. Bhatia and G. P. Szegö, Dynamical Systems: Stability Theory and Applications,, Lecture Notes in Math. 35, (1967).   Google Scholar

[2]

G. Butler and P. Waltman, Persistence in dynamical Systems,, J. Differential Equations, 63 (1986), 255.  doi: 10.1016/0022-0396(86)90049-5.  Google Scholar

[3]

W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations,, Health, (1995).   Google Scholar

[4]

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.   Google Scholar

[5]

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

[6]

N. M. Dixit and A. S. Perelson, Complex patterns of viral load decay under antiretroviral therapy: influence of pharmacokineticsand intracellular delay,, J. Theor. Biol., 226 (2004), 95.  doi: 10.1016/j.jtbi.2003.09.002.  Google Scholar

[7]

H. I. Freedman, S. G. Ruan and M. X. Tang, Uniform persistence and flows near a closed positively invariant set,, J. Dynam. Differen. Equat., 6 (1994), 583.  doi: 10.1007/BF02218848.  Google Scholar

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

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J. M. Heffernan and L. M. Wahl, Monte Carlo estimates of natural variation in HIV infection,, J. Theor. Biol., 236 (2005), 137.  doi: 10.1016/j.jtbi.2005.03.002.  Google Scholar

[10]

T. B. Kepler and A. S. Perelson, Drug concentration heterogeneity facilitates the evolution of drug resistance,, Proc. Natl. Acad. Sci. USA, 95 (1998), 11514.  doi: 10.1073/pnas.95.20.11514.  Google Scholar

[11]

D. Kirschner, Using mathematics to understand HIV immune dynamics,, Notices of the AMS, 43 (1996), 191.   Google Scholar

[12]

A. Korobeinikov, Global properties of basic virus dynamics models,, Bull. Math. Biol., 66 (2004), 879.  doi: 10.1016/j.bulm.2004.02.001.  Google Scholar

[13]

J. P. LaSalle, The Stability of Dynamical Systems,, Regional Conference Series in Applied Mathematics, (1976).   Google Scholar

[14]

M. Y. Li, J. R. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size,, Math. Biosci., 160 (1999), 191.  doi: 10.1016/S0025-5564(99)00030-9.  Google Scholar

[15]

M. Y. Li and J. S. Muldowney, A geometric approach to the global-stability problems,, SIAM J. Math. Anal., 27 (1996), 1070.  doi: 10.1137/S0036141094266449.  Google Scholar

[16]

Y. Li and J. S. Muldowney, On Bendixson's criterion,, J. Differential Equations, 106 (1993), 27.  doi: 10.1006/jdeq.1993.1097.  Google Scholar

[17]

M. Y. Li and H. Shu, Impact of intercellar delays and target-cell dynamics on in vivo viral infections,, SIAM J. Appl. Math., 70 (2010), 2434.  doi: 10.1137/090779322.  Google Scholar

[18]

M. Y. Li and H. Shu, Joint effects of mitosis and intracellular delay on viral dynamics: two-parameter bifurcation analysis,, J. Math. Biol., 64 (2012), 1005.  doi: 10.1007/s00285-011-0436-2.  Google Scholar

[19]

R. H. Martin Jr., Logarithmic norms and projections applied to linear differential systems,, J. Math. Anal. Appl., 45 (1974), 432.  doi: 10.1016/0022-247X(74)90084-5.  Google Scholar

[20]

J. S. Muldowney, Compound matrices and ordinary differential equations,, Rocky Mountain J. Math., 20 (1990), 857.  doi: 10.1216/rmjm/1181073047.  Google Scholar

[21]

A. Murase, T. Sasaki and T. Kajiwara, Stability analysis of pathogen-immune interaction dynamics,, J. Math. Biol., 51 (2005), 247.  doi: 10.1007/s00285-005-0321-y.  Google Scholar

[22]

P. W. Nelson, J. E. Mittler and A. S. Perelson, Effect of drug efficacy and the eclipse phase of the viral life cycle on the estimates of HIV viral dynamic parameters,, Journal of Aids, 26 (2001), 405.   Google Scholar

[23]

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

[24]

M. A. Nowak and R. M. May, Virus Dynamics,, Cambridge University Press, (2000).   Google Scholar

[25]

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo,, SIAM Rev., 41 (1999), 3.  doi: 10.1137/S0036144598335107.  Google Scholar

[26]

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, (1996), 1582.  doi: 10.1126/science.271.5255.1582.  Google Scholar

[27]

R. Qesmi, J. Wu, J. Wu and J. M. Heffernan, Influence of backward bifurcation in a model of hepatitis B and C viruses,, Math. Biosci., 224 (2010), 118.  doi: 10.1016/j.mbs.2010.01.002.  Google Scholar

[28]

T. Revilla and G. García-Ramos, Fighting a virus with a virus: A dynamical model for HIV-1 therapy,, Math. Biosci., 185 (2003), 191.  doi: 10.1016/S0025-5564(03)00091-9.  Google Scholar

[29]

H. Shu and L. Wang, Role of CD$4^+$ T-cell proliferation in HIV infection under antiretroviral therapy,, J. Math. Anal. Appl., 394 (2012), 529.  doi: 10.1016/j.jmaa.2012.05.027.  Google Scholar

[30]

H. L. Smith, Monotone Dynamical Systems,, AMS, (1995).   Google Scholar

[31]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[32]

P. Waltman, A brief survey of persistence in dynamical systems,, in Delay Differential Equations and Dynamical Systems (eds. S. Busenberg, 1475 (1991), 31.  doi: 10.1007/BFb0083477.  Google Scholar

[33]

L. Wang and S. Ellermeyer, HIV infection and CD$4^+$ T cell dynamics,, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1417.  doi: 10.3934/dcdsb.2006.6.1417.  Google Scholar

[34]

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.  doi: 10.1016/j.mbs.2005.12.026.  Google Scholar

[35]

Y. Yan and W. Wang, Global stability of a five-dimensional model with immune responses and delay,, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 401.  doi: 10.3934/dcdsb.2012.17.401.  Google Scholar

[36]

H. Zhu and X. Zou, Dynamics of a HIV-1 Infection model with cell-mediated immune response and intracellular delay,, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 511.  doi: 10.3934/dcdsb.2009.12.511.  Google Scholar

show all references

References:
[1]

N. P. Bhatia and G. P. Szegö, Dynamical Systems: Stability Theory and Applications,, Lecture Notes in Math. 35, (1967).   Google Scholar

[2]

G. Butler and P. Waltman, Persistence in dynamical Systems,, J. Differential Equations, 63 (1986), 255.  doi: 10.1016/0022-0396(86)90049-5.  Google Scholar

[3]

W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations,, Health, (1995).   Google Scholar

[4]

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.   Google Scholar

[5]

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

[6]

N. M. Dixit and A. S. Perelson, Complex patterns of viral load decay under antiretroviral therapy: influence of pharmacokineticsand intracellular delay,, J. Theor. Biol., 226 (2004), 95.  doi: 10.1016/j.jtbi.2003.09.002.  Google Scholar

[7]

H. I. Freedman, S. G. Ruan and M. X. Tang, Uniform persistence and flows near a closed positively invariant set,, J. Dynam. Differen. Equat., 6 (1994), 583.  doi: 10.1007/BF02218848.  Google Scholar

[8]

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

[9]

J. M. Heffernan and L. M. Wahl, Monte Carlo estimates of natural variation in HIV infection,, J. Theor. Biol., 236 (2005), 137.  doi: 10.1016/j.jtbi.2005.03.002.  Google Scholar

[10]

T. B. Kepler and A. S. Perelson, Drug concentration heterogeneity facilitates the evolution of drug resistance,, Proc. Natl. Acad. Sci. USA, 95 (1998), 11514.  doi: 10.1073/pnas.95.20.11514.  Google Scholar

[11]

D. Kirschner, Using mathematics to understand HIV immune dynamics,, Notices of the AMS, 43 (1996), 191.   Google Scholar

[12]

A. Korobeinikov, Global properties of basic virus dynamics models,, Bull. Math. Biol., 66 (2004), 879.  doi: 10.1016/j.bulm.2004.02.001.  Google Scholar

[13]

J. P. LaSalle, The Stability of Dynamical Systems,, Regional Conference Series in Applied Mathematics, (1976).   Google Scholar

[14]

M. Y. Li, J. R. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size,, Math. Biosci., 160 (1999), 191.  doi: 10.1016/S0025-5564(99)00030-9.  Google Scholar

[15]

M. Y. Li and J. S. Muldowney, A geometric approach to the global-stability problems,, SIAM J. Math. Anal., 27 (1996), 1070.  doi: 10.1137/S0036141094266449.  Google Scholar

[16]

Y. Li and J. S. Muldowney, On Bendixson's criterion,, J. Differential Equations, 106 (1993), 27.  doi: 10.1006/jdeq.1993.1097.  Google Scholar

[17]

M. Y. Li and H. Shu, Impact of intercellar delays and target-cell dynamics on in vivo viral infections,, SIAM J. Appl. Math., 70 (2010), 2434.  doi: 10.1137/090779322.  Google Scholar

[18]

M. Y. Li and H. Shu, Joint effects of mitosis and intracellular delay on viral dynamics: two-parameter bifurcation analysis,, J. Math. Biol., 64 (2012), 1005.  doi: 10.1007/s00285-011-0436-2.  Google Scholar

[19]

R. H. Martin Jr., Logarithmic norms and projections applied to linear differential systems,, J. Math. Anal. Appl., 45 (1974), 432.  doi: 10.1016/0022-247X(74)90084-5.  Google Scholar

[20]

J. S. Muldowney, Compound matrices and ordinary differential equations,, Rocky Mountain J. Math., 20 (1990), 857.  doi: 10.1216/rmjm/1181073047.  Google Scholar

[21]

A. Murase, T. Sasaki and T. Kajiwara, Stability analysis of pathogen-immune interaction dynamics,, J. Math. Biol., 51 (2005), 247.  doi: 10.1007/s00285-005-0321-y.  Google Scholar

[22]

P. W. Nelson, J. E. Mittler and A. S. Perelson, Effect of drug efficacy and the eclipse phase of the viral life cycle on the estimates of HIV viral dynamic parameters,, Journal of Aids, 26 (2001), 405.   Google Scholar

[23]

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

[24]

M. A. Nowak and R. M. May, Virus Dynamics,, Cambridge University Press, (2000).   Google Scholar

[25]

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo,, SIAM Rev., 41 (1999), 3.  doi: 10.1137/S0036144598335107.  Google Scholar

[26]

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, (1996), 1582.  doi: 10.1126/science.271.5255.1582.  Google Scholar

[27]

R. Qesmi, J. Wu, J. Wu and J. M. Heffernan, Influence of backward bifurcation in a model of hepatitis B and C viruses,, Math. Biosci., 224 (2010), 118.  doi: 10.1016/j.mbs.2010.01.002.  Google Scholar

[28]

T. Revilla and G. García-Ramos, Fighting a virus with a virus: A dynamical model for HIV-1 therapy,, Math. Biosci., 185 (2003), 191.  doi: 10.1016/S0025-5564(03)00091-9.  Google Scholar

[29]

H. Shu and L. Wang, Role of CD$4^+$ T-cell proliferation in HIV infection under antiretroviral therapy,, J. Math. Anal. Appl., 394 (2012), 529.  doi: 10.1016/j.jmaa.2012.05.027.  Google Scholar

[30]

H. L. Smith, Monotone Dynamical Systems,, AMS, (1995).   Google Scholar

[31]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[32]

P. Waltman, A brief survey of persistence in dynamical systems,, in Delay Differential Equations and Dynamical Systems (eds. S. Busenberg, 1475 (1991), 31.  doi: 10.1007/BFb0083477.  Google Scholar

[33]

L. Wang and S. Ellermeyer, HIV infection and CD$4^+$ T cell dynamics,, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1417.  doi: 10.3934/dcdsb.2006.6.1417.  Google Scholar

[34]

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.  doi: 10.1016/j.mbs.2005.12.026.  Google Scholar

[35]

Y. Yan and W. Wang, Global stability of a five-dimensional model with immune responses and delay,, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 401.  doi: 10.3934/dcdsb.2012.17.401.  Google Scholar

[36]

H. Zhu and X. Zou, Dynamics of a HIV-1 Infection model with cell-mediated immune response and intracellular delay,, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 511.  doi: 10.3934/dcdsb.2009.12.511.  Google Scholar

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