# American Institute of Mathematical Sciences

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. Google Scholar [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. 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] 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. Google Scholar [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 Google Scholar [6] T. A. 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Spiteri, Optimal HIV treatment by maximising immune response, J. Math. Biol., 48 (2004), 545-562. doi: 10.1007/s00285-003-0245-3.  Google Scholar [12] R. J. De Boer and A. S. Perelson, Towards a general function describing T cell proliferation, J. Theoret. Biol., 175 (1995), 567-576. Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [16] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Academic Press, San Diego, 1993.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] W. Liu, Nonlinear oscillation in models of immune response to persistent viruses, Theor. Popul. Biol., 52 (1997), 224-230. Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] M. Nowak and C. Bangham, Population dynamics of immune response to persistent viruses, Science, 272 (1996), 74-79. Google Scholar [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. Google Scholar [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.  Google Scholar [28] A. S. Perelson and P. Nelson, Mathematical models of HIV dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107.  Google Scholar [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. Google Scholar [30] A. N. Phillips, Reduction of HIV concentration during acute infection: Independence from a specific immune response, Science, 271 (1996), 497-499. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar

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. Google Scholar [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. 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] 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. Google Scholar [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 Google Scholar [6] T. A. Burton, Volterra integral and differential equations, in "Mathematics In Science And Engineering" $2^{nd}$ edition, Elsevier, Amsterdam-Boston, 202 (2005).  Google Scholar [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.  Google Scholar [8] D. Callaway and A. S. Perelson, HIV-1 infection and low steady state viral loads, Bulletin of Mathematical Biology, 64 (2002), 29-64. Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [12] R. J. De Boer and A. S. Perelson, Towards a general function describing T cell proliferation, J. Theoret. Biol., 175 (1995), 567-576. Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [16] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Academic Press, San Diego, 1993.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] W. Liu, Nonlinear oscillation in models of immune response to persistent viruses, Theor. Popul. Biol., 52 (1997), 224-230. Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] M. Nowak and C. Bangham, Population dynamics of immune response to persistent viruses, Science, 272 (1996), 74-79. Google Scholar [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. Google Scholar [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.  Google Scholar [28] A. S. Perelson and P. Nelson, Mathematical models of HIV dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107.  Google Scholar [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. Google Scholar [30] A. N. Phillips, Reduction of HIV concentration during acute infection: Independence from a specific immune response, Science, 271 (1996), 497-499. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar
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