2013, 10(2): 483-498. doi: 10.3934/mbe.2013.10.483

Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays

1. 

College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China

2. 

Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7

Received  January 2012 Revised  September 2012 Published  January 2013

We consider a mathematical model that describes the interactions of the HIV virus, CD4 cells and CTLs within host, which is a modification of some existing models by incorporating (i) two distributed kernels reflecting the variance of time for virus to invade into cells and the variance of time for invaded virions to reproduce within cells; (ii) a nonlinear incidence function $f$ for virus infections, and (iii) a nonlinear removal rate function $h$ for infected cells. By constructing Lyapunov functionals and subtle estimates of the derivatives of these Lyapunov functionals, we shown that the model has the threshold dynamics: if the basic reproduction number (BRN) is less than or equal to one, then the infection free equilibrium is globally asymptotically stable, meaning that HIV virus will be cleared; whereas if the BRN is larger than one, then there exist an infected equilibrium which is globally asymptotically stable, implying that the HIV-1 infection will persist in the host and the viral concentration will approach a positive constant level. This together with the dependence/independence of the BRN on $f$ and $h$ reveals the effect of the adoption of these nonlinear functions.
Citation: Zhaohui Yuan, Xingfu Zou. Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays. Mathematical Biosciences & Engineering, 2013, 10 (2) : 483-498. doi: 10.3934/mbe.2013.10.483
References:
[1]

L. K. Andrea and S. Ranjan, Evaluation of HIV-1 kinetic models using quantitative discrimination analysis, Bioinformatics, 21 (2005), 1668-1677.

[2]

R. Arnaout, M. Nowak and D. Wodarz, HIV-1 dynamics revisited: Biphasic decay by cytotoxic lymphocyte killing?, Proc. Roy. Soc. Lond. B, 265 (2000), 1347-1354.

[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]

S. Bonhoeffer, R. M. May, G. M. Shaw and M. A. Nowak, Virus dynamics and drug therapy, Proc. Natl. Acad. Sci. USA, 94 (1997), 6971-6976.

[5]

N. Burić, M. Mudrinic and N. Vasović, Time delay in a basic model of the immune response, Chaos, Solitons and Fractals, 12 (2001), 483-489

[6]

T. A. Burton, Volterra integral and differential equations, in "Mathematics In Science And Engineering" $2^{nd}$ edition, Elsevier, Amsterdam-Boston, 202 (2005).

[7]

L. Cai and J. Wu, Analysis of an HIV/AIDS treatment model with a nonlinear incidence, Chaos, Solitons and Fractals, 41 (2009), 175-182. doi: 10.1016/j.chaos.2007.11.023.

[8]

D. Callaway and A. S. Perelson, HIV-1 infection and low steady state viral loads, Bulletin of Mathematical Biology, 64 (2002), 29-64.

[9]

A. A. Canabarro, I. M. Gléria and M. L. Lyra, Periodic solutions and chaos in a non-linear model for the delayed cellular immune response, Physica A, 342 (2004), 234-241.

[10]

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.

[11]

R. Culshaw, S. Ruan and R. Spiteri, Optimal HIV treatment by maximising immune response, J. Math. Biol., 48 (2004), 545-562. doi: 10.1007/s00285-003-0245-3.

[12]

R. J. De Boer and A. S. Perelson, Towards a general function describing T cell proliferation, J. Theoret. Biol., 175 (1995), 567-576.

[13]

R. J. De Boer and A. S. Perelson, Target cell limited and immune control models of HIV infection: A comparison, J. Theoret. Biol., 190 (1998), 201-214.

[14]

P. Georgescu and Y. Hsieh, Global stability for a virus dynamics model with nonlinear incidence of infection and removal, SIAM J. Appl. Math., 67 (2006), 337-353. doi: 10.1137/060654876.

[15]

T. Kajiwara and T. Sasaki, A note on the stability analysis of pathogen-immune interaction dynamics, Discrete Continuous Dynam. Systems-B, 4 (2004), 615-622. doi: 10.3934/dcdsb.2004.4.615.

[16]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Academic Press, San Diego, 1993.

[17]

Y. Li, R. Xu, Z. Li and S. Mao, Global dynamics of a delayed HIV-1 infection model with CTL immune response, Discrete Dynamics in Nature and Society, 2011 (2011), Art. ID 673843, 13 pp. doi: 10.1155/2011/673843.

[18]

S. Liu and L. Wang, Global stability of an HIV-1 model with dstributed intracellular delays and a combination therapy, Math. Biosci. and Eng., 7 (2010), 675-685. doi: 10.3934/mbe.2010.7.675.

[19]

W. Liu, Nonlinear oscillation in models of immune response to persistent viruses, Theor. Popul. Biol., 52 (1997), 224-230.

[20]

C. Lv and Z. Yuan, Stability analysis of delay differential equation models of HIV-1 therapy for fighting a virus with another virus, J. Math. Anal. Appl., 352 (2009), 672-683. doi: 10.1016/j.jmaa.2008.11.026.

[21]

J. Mittler, B. Sulzer, A. Neumann and A. Perelson, Influence of delayed viral production on viral dynamics in HIV-1 infected patients, Math. Biosci., 152 (1998), 143-163.

[22]

Y. Nakata, Global dynamics of a cell mediated immunity in viral infection models with distributed delays, J. Math. Anal. Appl., 375 (2011), 14-27. doi: 10.1016/j.jmaa.2010.08.025.

[23]

P. 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.

[24]

P. Nelson and A. S. Perelson, Mathematica analysis of delay differential equation models of HIV-1 infection, Math. Biosci., 179 (2002), 73-94. doi: 10.1016/S0025-5564(02)00099-8.

[25]

M. Nowak and C. Bangham, Population dynamics of immune response to persistent viruses, Science, 272 (1996), 74-79.

[26]

M. A. Nowak, S. Bonhoeffer, G. M. Shaw and R. M. May, Anti-viral drug treatment: Dynamics of resistance in free virus and infected cell populations, J. Theor. Biol., 184 (1997), 203-217.

[27]

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.

[28]

A. S. Perelson and P. Nelson, Mathematical models of HIV dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107.

[29]

A. S. Perelson, A. Neumann, M. Markowitz, J. Leonard and D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time, Science, 271 (1996), 1582-1586.

[30]

A. N. Phillips, Reduction of HIV concentration during acute infection: Independence from a specific immune response, Science, 271 (1996), 497-499.

[31]

J. Wang, G. Huang and Y. Takeuchi, Global asymptotic stability for HIV-1 dynamics with two distributed delays,, Mathematical Medicine and Biology, ().  doi: 10.1093/imammb/dqr009.

[32]

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.

[33]

K. Wang, W. Wang, H. Pang and X. Liu, Complex dynamic behavior in a viral model with delayed immune response, Physica D, 226 (2007), 197-208. doi: 10.1016/j.physd.2006.12.001.

[34]

R. Xu, Global dynamics of an HIV-1 infection model with distributed intracellular delays, Comput. Math. Appl., 61 (2011), 2799-2805. doi: 10.1016/j.camwa.2011.03.050.

[35]

R. Xu, Global stability of an HIV-1 infection model with saturation infection and intracellular delay, J. Math. Anal. Appl., 375 (2011), 75-81. doi: 10.1016/j.jmaa.2010.08.055.

[36]

H. Zhu and X. Zou, Impact of delays in cell infection and virus production on HIV-1 dynamics, Mathematical Medicine and Biology, IMA, 25 (2008), 99-112.

[37]

H. Zhu and X. Zou, Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay, Discrete Continuous Dynam. Systems-B, 12 (2009), 511-524.

[38]

H. Zhu, Y. Luo and M. 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.

show all references

References:
[1]

L. K. Andrea and S. Ranjan, Evaluation of HIV-1 kinetic models using quantitative discrimination analysis, Bioinformatics, 21 (2005), 1668-1677.

[2]

R. Arnaout, M. Nowak and D. Wodarz, HIV-1 dynamics revisited: Biphasic decay by cytotoxic lymphocyte killing?, Proc. Roy. Soc. Lond. B, 265 (2000), 1347-1354.

[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]

S. Bonhoeffer, R. M. May, G. M. Shaw and M. A. Nowak, Virus dynamics and drug therapy, Proc. Natl. Acad. Sci. USA, 94 (1997), 6971-6976.

[5]

N. Burić, M. Mudrinic and N. Vasović, Time delay in a basic model of the immune response, Chaos, Solitons and Fractals, 12 (2001), 483-489

[6]

T. A. Burton, Volterra integral and differential equations, in "Mathematics In Science And Engineering" $2^{nd}$ edition, Elsevier, Amsterdam-Boston, 202 (2005).

[7]

L. Cai and J. Wu, Analysis of an HIV/AIDS treatment model with a nonlinear incidence, Chaos, Solitons and Fractals, 41 (2009), 175-182. doi: 10.1016/j.chaos.2007.11.023.

[8]

D. Callaway and A. S. Perelson, HIV-1 infection and low steady state viral loads, Bulletin of Mathematical Biology, 64 (2002), 29-64.

[9]

A. A. Canabarro, I. M. Gléria and M. L. Lyra, Periodic solutions and chaos in a non-linear model for the delayed cellular immune response, Physica A, 342 (2004), 234-241.

[10]

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.

[11]

R. Culshaw, S. Ruan and R. Spiteri, Optimal HIV treatment by maximising immune response, J. Math. Biol., 48 (2004), 545-562. doi: 10.1007/s00285-003-0245-3.

[12]

R. J. De Boer and A. S. Perelson, Towards a general function describing T cell proliferation, J. Theoret. Biol., 175 (1995), 567-576.

[13]

R. J. De Boer and A. S. Perelson, Target cell limited and immune control models of HIV infection: A comparison, J. Theoret. Biol., 190 (1998), 201-214.

[14]

P. Georgescu and Y. Hsieh, Global stability for a virus dynamics model with nonlinear incidence of infection and removal, SIAM J. Appl. Math., 67 (2006), 337-353. doi: 10.1137/060654876.

[15]

T. Kajiwara and T. Sasaki, A note on the stability analysis of pathogen-immune interaction dynamics, Discrete Continuous Dynam. Systems-B, 4 (2004), 615-622. doi: 10.3934/dcdsb.2004.4.615.

[16]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Academic Press, San Diego, 1993.

[17]

Y. Li, R. Xu, Z. Li and S. Mao, Global dynamics of a delayed HIV-1 infection model with CTL immune response, Discrete Dynamics in Nature and Society, 2011 (2011), Art. ID 673843, 13 pp. doi: 10.1155/2011/673843.

[18]

S. Liu and L. Wang, Global stability of an HIV-1 model with dstributed intracellular delays and a combination therapy, Math. Biosci. and Eng., 7 (2010), 675-685. doi: 10.3934/mbe.2010.7.675.

[19]

W. Liu, Nonlinear oscillation in models of immune response to persistent viruses, Theor. Popul. Biol., 52 (1997), 224-230.

[20]

C. Lv and Z. Yuan, Stability analysis of delay differential equation models of HIV-1 therapy for fighting a virus with another virus, J. Math. Anal. Appl., 352 (2009), 672-683. doi: 10.1016/j.jmaa.2008.11.026.

[21]

J. Mittler, B. Sulzer, A. Neumann and A. Perelson, Influence of delayed viral production on viral dynamics in HIV-1 infected patients, Math. Biosci., 152 (1998), 143-163.

[22]

Y. Nakata, Global dynamics of a cell mediated immunity in viral infection models with distributed delays, J. Math. Anal. Appl., 375 (2011), 14-27. doi: 10.1016/j.jmaa.2010.08.025.

[23]

P. 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.

[24]

P. Nelson and A. S. Perelson, Mathematica analysis of delay differential equation models of HIV-1 infection, Math. Biosci., 179 (2002), 73-94. doi: 10.1016/S0025-5564(02)00099-8.

[25]

M. Nowak and C. Bangham, Population dynamics of immune response to persistent viruses, Science, 272 (1996), 74-79.

[26]

M. A. Nowak, S. Bonhoeffer, G. M. Shaw and R. M. May, Anti-viral drug treatment: Dynamics of resistance in free virus and infected cell populations, J. Theor. Biol., 184 (1997), 203-217.

[27]

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.

[28]

A. S. Perelson and P. Nelson, Mathematical models of HIV dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107.

[29]

A. S. Perelson, A. Neumann, M. Markowitz, J. Leonard and D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time, Science, 271 (1996), 1582-1586.

[30]

A. N. Phillips, Reduction of HIV concentration during acute infection: Independence from a specific immune response, Science, 271 (1996), 497-499.

[31]

J. Wang, G. Huang and Y. Takeuchi, Global asymptotic stability for HIV-1 dynamics with two distributed delays,, Mathematical Medicine and Biology, ().  doi: 10.1093/imammb/dqr009.

[32]

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.

[33]

K. Wang, W. Wang, H. Pang and X. Liu, Complex dynamic behavior in a viral model with delayed immune response, Physica D, 226 (2007), 197-208. doi: 10.1016/j.physd.2006.12.001.

[34]

R. Xu, Global dynamics of an HIV-1 infection model with distributed intracellular delays, Comput. Math. Appl., 61 (2011), 2799-2805. doi: 10.1016/j.camwa.2011.03.050.

[35]

R. Xu, Global stability of an HIV-1 infection model with saturation infection and intracellular delay, J. Math. Anal. Appl., 375 (2011), 75-81. doi: 10.1016/j.jmaa.2010.08.055.

[36]

H. Zhu and X. Zou, Impact of delays in cell infection and virus production on HIV-1 dynamics, Mathematical Medicine and Biology, IMA, 25 (2008), 99-112.

[37]

H. Zhu and X. Zou, Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay, Discrete Continuous Dynam. Systems-B, 12 (2009), 511-524.

[38]

H. Zhu, Y. Luo and M. 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.

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