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August  2017, 22(6): 2365-2387. doi: 10.3934/dcdsb.2017121

## Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays

 a. College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang 830046, China b. The Basic Science Department, Xinjiang Institute of Engineering, Urumqi, Xinjiang 830091, China

Received  April 2015 Revised  March 2017 Published  March 2017

Fund Project: This research was supported by the Natural Science Foundation of Xinjiang (Grant No. 2016D03022) and the Doctorial Subjects Foundation of the Ministry of Education of China (Grant No. 2013651110001), the National Natural Science Foundation of China (Grant No. 11661076).

In this paper, the dynamical behaviors of a viral infection model with cytotoxic T-lymphocyte (CTL) immune response, immune response delay and production delay are investigated. The threshold values for virus infection and immune response are established. By means of Lyapunov functionals methods and LaSalle's invariance principle, sufficient conditions for the global stability of the infection-free and CTL-absent equilibria are established. Global stability of the CTL-present infection equilibrium is also studied when there is no immune delay in the model. Furthermore, to deal with the local stability of the CTL-present infection equilibrium in a general case with two delays being positive, we extend an existing geometric method to treat the associated characteristic equation. When the two delays are positive, we show some conditions for Hopf bifurcation at the CTL-present infection equilibrium by using the immune delay as a bifurcation parameter. Numerical simulations are performed in order to illustrate the dynamical behaviors of the model.

Citation: Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121
##### References:
 [1] E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent paramaters, SIAM J. Math. Anal., 33 (2002), 1144-1165. Google Scholar [2] A. A. Canabarro, I. M. Gleria and M. L. Lyra, Periodic solutions and chaos in a nonlinear model for the delayed cellular immune response, Physica A, 342 (2004), 234-241. Google Scholar [3] S. Chen, C. Cheng and Y. Takeuchi, Stability analysis in delayed within-host viral dynamics with both viral and cellular infections, J. Math. Anal. Appl., 442 (2016), 642-672. Google Scholar [4] R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4+Tcell, Math. Biosci., 165 (2000), 27-39. Google Scholar [5] R. Culshaw, S. Ruan and G. Webb, A mathematical model of cell-to-cell HIV-1 that include a time delay, J. Math. Biol., 46 (2003), 425-444. Google Scholar [6] D. Ebert, C. D. Z. Rohringer and H. J. Carius, Dose effects and density-dependent regulation of two microparasites of Daphnia magna, Oecologia, 122 (2000), 200-209. Google Scholar [7] H. W. Hethcote, M. A. Lewis and P. Driessche, An epidemiological model with a delay and a nonlinear incidence rate, J. Math. Biol., 27 (1989), 49-64. Google Scholar [8] G. Huang, W. Ma and Y. Takeuchi, Global properties for virus dynamics model with Beddington-DeAngelis functional response, Appl. Math. Lett., 22 (2009), 1690-1693. Google Scholar [9] G. Huang, H. Yokoi, Y. Takeuchi, T. Kajiwara and T. Sasaki, Impact of intracellular delay, immune activation delay and nonlinear incidence on viral dynamics, Japan J. Indust. Appl. Math., 28 (2011), 383-411. Google Scholar [10] R. A. Koup, J. T. Safrit, Y. Cao, C. A. Andrews, G. McLeod, W. Borkowsky, C. Farthing and D. D. Ho, Temporal association of cellular immune responses with the initial control of viremia in primary human immunodeficiency virus type 1 syndrome, J. Virol., 68 (1994), 4650-4655. Google Scholar [11] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, 191, Academic Press, Inc. , Boston, MA, 1993. Google Scholar [12] M. Y. Li and H. Shu, Global dynamics of a mathematical model for HTLV-1 infection of CD4+T cells with delayed CTL response, Nonlinear Anal.: RWA., 13 (2012), 1080-1092. Google Scholar [13] Z. Ma, Y. Zhou, W. Wang and Z. Jin, Mathematical Models and Dynamics of Infectious Diseases, China Science Press, Beijing, 2004.   Google Scholar [14] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. Google Scholar [15] P. W. Nelson and A. Perelson, Mathematical analysis of a delay-differential equation model of HIV-1 infection, Math. Biosci., 179 (2002), 73-94. Google Scholar [16] R. Ouifki and G. Witten, Stability analysis of a model for HIV infection with RTI and three intracellular delays, Biosys., 95 (2009), 1-6. Google Scholar [17] 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. Google Scholar [18] R. R. Regoes, D. Ebert and S. Bonhoeffer, Dose-dependent infection rates of parasites produce the Allee effect in epidemiology, Proc. R. Soc. Lond. Ser. B., 269 (2002), 271-279. Google Scholar [19] H. Shu, L. Wang and J. Watmough, Global stability of a nonlinear viral infection model with in infinitely distributed intracellular delays and CTL immune response, SIAM J. Appl. Math., 73 (2013), 1280-1302. Google Scholar [20] X. Song and A. U. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329 (2007), 281-297. Google Scholar [21] X. Song, X. Zhou and X. Zhao, Properties of stability and Hopf bifurcation for a HIV infection model with time delay, Appl. Math. Modelling, 34 (2010), 1511-1523. Google Scholar [22] M. Stafford, L. Corey, Y. Cao, E. Daar, D. Ho and A. Perelson, Modelling plasma virus concentration during primary HIV infection, J. Theor. Biol., 203 (2000), 285-301. Google Scholar [23] X. Tian and Rui. Xu, Global stability ang Hopf bifurcation of an HIV-1 infection model with saturation incidence and delayed CTL immune response, Appl. Math. Comput., 237 (2014), 146-154. Google Scholar [24] X. Wang, A. Elaiw and X. Song, Global properties of a delayed HIV infection model with CTL immune response, Appl. Math. Comput., 218 (2012), 9405-9414. Google Scholar [25] 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. Google Scholar [26] Y. Wang, Y. Zhou, F. Brauer and J. M. Heffernan, Viral dynamics model with CTL immune response incorporating antiretroviral therapy, J. Math. Biol., 67 (2013), 901-934. Google Scholar [27] D. Wodarz, Hepatitis C virus dynamics and pathology: The role of CTL and antibody responses, J. Gen. Virol., 84 (2003), 1743-1750. Google Scholar [28] R. Xu, Global stability of an HIV-1 infection model with saturation infection and intracellular delay, J. Math. Anal. Appl., 375 (2011), 75-81. Google Scholar [29] Y. Yang, L. Zou and S. Ruan, Global dynamics of a delayed within-host viral infection model with both virus-to-cell and cell-to-cell transmissions, Math. Biosci., 270 (2015), 183-191. Google Scholar [30] X. Zhou, X. Song and X. Shi, A differential equation model of HIV infection of CD4+ T-cells with cure rate, J. Math. Anal. Appl., 342 (2008), 1342-1355. Google Scholar [31] 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. Google Scholar [32] 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. Google Scholar

show all references

##### References:
 [1] E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent paramaters, SIAM J. Math. Anal., 33 (2002), 1144-1165. Google Scholar [2] A. A. Canabarro, I. M. Gleria and M. L. Lyra, Periodic solutions and chaos in a nonlinear model for the delayed cellular immune response, Physica A, 342 (2004), 234-241. Google Scholar [3] S. Chen, C. Cheng and Y. Takeuchi, Stability analysis in delayed within-host viral dynamics with both viral and cellular infections, J. Math. Anal. Appl., 442 (2016), 642-672. Google Scholar [4] R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4+Tcell, Math. Biosci., 165 (2000), 27-39. Google Scholar [5] R. Culshaw, S. Ruan and G. Webb, A mathematical model of cell-to-cell HIV-1 that include a time delay, J. Math. Biol., 46 (2003), 425-444. Google Scholar [6] D. Ebert, C. D. Z. Rohringer and H. J. Carius, Dose effects and density-dependent regulation of two microparasites of Daphnia magna, Oecologia, 122 (2000), 200-209. Google Scholar [7] H. W. Hethcote, M. A. Lewis and P. Driessche, An epidemiological model with a delay and a nonlinear incidence rate, J. Math. Biol., 27 (1989), 49-64. Google Scholar [8] G. Huang, W. Ma and Y. Takeuchi, Global properties for virus dynamics model with Beddington-DeAngelis functional response, Appl. Math. Lett., 22 (2009), 1690-1693. Google Scholar [9] G. Huang, H. Yokoi, Y. Takeuchi, T. Kajiwara and T. Sasaki, Impact of intracellular delay, immune activation delay and nonlinear incidence on viral dynamics, Japan J. Indust. Appl. Math., 28 (2011), 383-411. Google Scholar [10] R. A. Koup, J. T. Safrit, Y. Cao, C. A. Andrews, G. McLeod, W. Borkowsky, C. Farthing and D. D. Ho, Temporal association of cellular immune responses with the initial control of viremia in primary human immunodeficiency virus type 1 syndrome, J. Virol., 68 (1994), 4650-4655. Google Scholar [11] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, 191, Academic Press, Inc. , Boston, MA, 1993. Google Scholar [12] M. Y. Li and H. Shu, Global dynamics of a mathematical model for HTLV-1 infection of CD4+T cells with delayed CTL response, Nonlinear Anal.: RWA., 13 (2012), 1080-1092. Google Scholar [13] Z. Ma, Y. Zhou, W. Wang and Z. Jin, Mathematical Models and Dynamics of Infectious Diseases, China Science Press, Beijing, 2004.   Google Scholar [14] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. Google Scholar [15] P. W. Nelson and A. Perelson, Mathematical analysis of a delay-differential equation model of HIV-1 infection, Math. Biosci., 179 (2002), 73-94. Google Scholar [16] R. Ouifki and G. Witten, Stability analysis of a model for HIV infection with RTI and three intracellular delays, Biosys., 95 (2009), 1-6. Google Scholar [17] 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. Google Scholar [18] R. R. Regoes, D. Ebert and S. Bonhoeffer, Dose-dependent infection rates of parasites produce the Allee effect in epidemiology, Proc. R. Soc. Lond. Ser. B., 269 (2002), 271-279. Google Scholar [19] H. Shu, L. Wang and J. Watmough, Global stability of a nonlinear viral infection model with in infinitely distributed intracellular delays and CTL immune response, SIAM J. Appl. Math., 73 (2013), 1280-1302. Google Scholar [20] X. Song and A. U. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329 (2007), 281-297. Google Scholar [21] X. Song, X. Zhou and X. Zhao, Properties of stability and Hopf bifurcation for a HIV infection model with time delay, Appl. Math. Modelling, 34 (2010), 1511-1523. Google Scholar [22] M. Stafford, L. Corey, Y. Cao, E. Daar, D. Ho and A. Perelson, Modelling plasma virus concentration during primary HIV infection, J. Theor. Biol., 203 (2000), 285-301. Google Scholar [23] X. Tian and Rui. Xu, Global stability ang Hopf bifurcation of an HIV-1 infection model with saturation incidence and delayed CTL immune response, Appl. Math. Comput., 237 (2014), 146-154. Google Scholar [24] X. Wang, A. Elaiw and X. Song, Global properties of a delayed HIV infection model with CTL immune response, Appl. Math. Comput., 218 (2012), 9405-9414. Google Scholar [25] 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. Google Scholar [26] Y. Wang, Y. Zhou, F. Brauer and J. M. Heffernan, Viral dynamics model with CTL immune response incorporating antiretroviral therapy, J. Math. Biol., 67 (2013), 901-934. Google Scholar [27] D. Wodarz, Hepatitis C virus dynamics and pathology: The role of CTL and antibody responses, J. Gen. Virol., 84 (2003), 1743-1750. Google Scholar [28] R. Xu, Global stability of an HIV-1 infection model with saturation infection and intracellular delay, J. Math. Anal. Appl., 375 (2011), 75-81. Google Scholar [29] Y. Yang, L. Zou and S. Ruan, Global dynamics of a delayed within-host viral infection model with both virus-to-cell and cell-to-cell transmissions, Math. Biosci., 270 (2015), 183-191. Google Scholar [30] X. Zhou, X. Song and X. Shi, A differential equation model of HIV infection of CD4+ T-cells with cure rate, J. Math. Anal. Appl., 342 (2008), 1342-1355. Google Scholar [31] 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. Google Scholar [32] 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. Google Scholar
The time series of model (2) before Hopf bifurcation occurs for $\tau_2=0.0516$
The time series and the phase trajectories of model (2) when Hopf bifurcation occurs for $\tau_2=1.3215$
The time series of model (2) after Hopf bifurcation occurs for $\tau_2=4.3215$
The time series and the phase trajectories of model (2) when Hopf bifurcation occurs again for $\tau_2=6.8000$
List of parameters
 Parameter Interpretation Value Source $s$ production rate of uninfected cells 10 $\mu l^{-1}day^{-1}$ [22,26] $d$ death rate of uninfected cells 0.01 $day^{-1}$ [22,26] $a$ death rate of infected cells 0.5 $day^{-1}$ [17,26] $p$ CTL effectiveness 1 $\mu l day^{-1}$ [17,26] $\beta$ the infection rate 0.45 $\mu l day^{-1}$ [26] $\alpha$ Saturation coefficient 0.01 Assumed $k$ production rate of free virus 0.4 $cell^{-1}day^{-1}$ [17,26] $u$ clearance rate of free virus 3 $day^{-1}$ [27,17] $c$ proliferation rate of CTL response 0.1 $\mu l day^{-1}$ [27,17] $b$ death rate of CTL 0.15 $day^{-1}$ [27,17] $m$ death rate for infected cells during $[t-\tau_1, t]$ 0.01 Assumed
 Parameter Interpretation Value Source $s$ production rate of uninfected cells 10 $\mu l^{-1}day^{-1}$ [22,26] $d$ death rate of uninfected cells 0.01 $day^{-1}$ [22,26] $a$ death rate of infected cells 0.5 $day^{-1}$ [17,26] $p$ CTL effectiveness 1 $\mu l day^{-1}$ [17,26] $\beta$ the infection rate 0.45 $\mu l day^{-1}$ [26] $\alpha$ Saturation coefficient 0.01 Assumed $k$ production rate of free virus 0.4 $cell^{-1}day^{-1}$ [17,26] $u$ clearance rate of free virus 3 $day^{-1}$ [27,17] $c$ proliferation rate of CTL response 0.1 $\mu l day^{-1}$ [27,17] $b$ death rate of CTL 0.15 $day^{-1}$ [27,17] $m$ death rate for infected cells during $[t-\tau_1, t]$ 0.01 Assumed
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