• Previous Article
    Oscillation theorems for impulsive parabolic differential system of neutral type
  • DCDS-B Home
  • This Issue
  • Next Article
    Numerical solutions of viscoelastic bending wave equations with two term time kernels by Runge-Kutta convolution quadrature
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.

[2]

A. A. CanabarroI. 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.

[3]

S. ChenC. 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.

[4]

R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4+Tcell, Math. Biosci., 165 (2000), 27-39.

[5]

R. CulshawS. 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.

[6]

D. EbertC. D. Z. Rohringer and H. J. Carius, Dose effects and density-dependent regulation of two microparasites of Daphnia magna, Oecologia, 122 (2000), 200-209.

[7]

H. W. HethcoteM. A. Lewis and P. Driessche, An epidemiological model with a delay and a nonlinear incidence rate, J. Math. Biol., 27 (1989), 49-64.

[8]

G. HuangW. Ma and Y. Takeuchi, Global properties for virus dynamics model with Beddington-DeAngelis functional response, Appl. Math. Lett., 22 (2009), 1690-1693.

[9]

G. HuangH. YokoiY. TakeuchiT. 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.

[10]

R. A. KoupJ. T. SafritY. CaoC. A. AndrewsG. McLeodW. BorkowskyC. 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.

[11]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, 191, Academic Press, Inc. , Boston, MA, 1993.

[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.

[13] Z. MaY. ZhouW. Wang and Z. Jin, Mathematical Models and Dynamics of Infectious Diseases, China Science Press, Beijing, 2004.
[14]

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

[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.

[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.

[17]

K. A. PawelekS. LiuF. 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.

[18]

R. R. RegoesD. 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.

[19]

H. ShuL. 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.

[20]

X. Song and A. U. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329 (2007), 281-297.

[21]

X. SongX. 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.

[22]

M. StaffordL. CoreyY. CaoE. DaarD. Ho and A. Perelson, Modelling plasma virus concentration during primary HIV infection, J. Theor. Biol., 203 (2000), 285-301.

[23]

X. Tian and Xu Rui., 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.

[24]

X. WangA. Elaiw and X. Song, Global properties of a delayed HIV infection model with CTL immune response, Appl. Math. Comput., 218 (2012), 9405-9414.

[25]

K. WangW. WangH. Pang and X. Liu, Complex dynamic behavior in a viral model with delayed immune response, Physica D, 226 (2007), 197-208.

[26]

Y. WangY. ZhouF. Brauer and J. M. Heffernan, Viral dynamics model with CTL immune response incorporating antiretroviral therapy, J. Math. Biol., 67 (2013), 901-934.

[27]

D. Wodarz, Hepatitis C virus dynamics and pathology: The role of CTL and antibody responses, J. Gen. Virol., 84 (2003), 1743-1750.

[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.

[29]

Y. YangL. 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.

[30]

X. ZhouX. 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.

[31]

H. ZhuY. Luo and M. Chen, Stability and Hopf bifurcation of a HIV infection model with CTL-response delay, Comput. Math. Appl., 62 (2011), 3091-3102.

[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.

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.

[2]

A. A. CanabarroI. 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.

[3]

S. ChenC. 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.

[4]

R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4+Tcell, Math. Biosci., 165 (2000), 27-39.

[5]

R. CulshawS. 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.

[6]

D. EbertC. D. Z. Rohringer and H. J. Carius, Dose effects and density-dependent regulation of two microparasites of Daphnia magna, Oecologia, 122 (2000), 200-209.

[7]

H. W. HethcoteM. A. Lewis and P. Driessche, An epidemiological model with a delay and a nonlinear incidence rate, J. Math. Biol., 27 (1989), 49-64.

[8]

G. HuangW. Ma and Y. Takeuchi, Global properties for virus dynamics model with Beddington-DeAngelis functional response, Appl. Math. Lett., 22 (2009), 1690-1693.

[9]

G. HuangH. YokoiY. TakeuchiT. 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.

[10]

R. A. KoupJ. T. SafritY. CaoC. A. AndrewsG. McLeodW. BorkowskyC. 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.

[11]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, 191, Academic Press, Inc. , Boston, MA, 1993.

[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.

[13] Z. MaY. ZhouW. Wang and Z. Jin, Mathematical Models and Dynamics of Infectious Diseases, China Science Press, Beijing, 2004.
[14]

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

[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.

[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.

[17]

K. A. PawelekS. LiuF. 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.

[18]

R. R. RegoesD. 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.

[19]

H. ShuL. 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.

[20]

X. Song and A. U. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329 (2007), 281-297.

[21]

X. SongX. 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.

[22]

M. StaffordL. CoreyY. CaoE. DaarD. Ho and A. Perelson, Modelling plasma virus concentration during primary HIV infection, J. Theor. Biol., 203 (2000), 285-301.

[23]

X. Tian and Xu Rui., 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.

[24]

X. WangA. Elaiw and X. Song, Global properties of a delayed HIV infection model with CTL immune response, Appl. Math. Comput., 218 (2012), 9405-9414.

[25]

K. WangW. WangH. Pang and X. Liu, Complex dynamic behavior in a viral model with delayed immune response, Physica D, 226 (2007), 197-208.

[26]

Y. WangY. ZhouF. Brauer and J. M. Heffernan, Viral dynamics model with CTL immune response incorporating antiretroviral therapy, J. Math. Biol., 67 (2013), 901-934.

[27]

D. Wodarz, Hepatitis C virus dynamics and pathology: The role of CTL and antibody responses, J. Gen. Virol., 84 (2003), 1743-1750.

[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.

[29]

Y. YangL. 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.

[30]

X. ZhouX. 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.

[31]

H. ZhuY. Luo and M. Chen, Stability and Hopf bifurcation of a HIV infection model with CTL-response delay, Comput. Math. Appl., 62 (2011), 3091-3102.

[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.

Figure 1.  The time series of model (2) before Hopf bifurcation occurs for $\tau_2=0.0516$
Figure 2.  The time series and the phase trajectories of model (2) when Hopf bifurcation occurs for $\tau_2=1.3215$
Figure 3.  The time series of model (2) after Hopf bifurcation occurs for $\tau_2=4.3215$
Figure 4.  The time series and the phase trajectories of model (2) when Hopf bifurcation occurs again for $\tau_2=6.8000$
Table 1.  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
[1]

Alexander Rezounenko. Stability of a viral infection model with state-dependent delay, CTL and antibody immune responses. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1547-1563. doi: 10.3934/dcdsb.2017074

[2]

Yu Ji, Lan Liu. Global stability of a delayed viral infection model with nonlinear immune response and general incidence rate. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 133-149. doi: 10.3934/dcdsb.2016.21.133

[3]

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

[4]

Jinliang Wang, Lijuan Guan. Global stability for a HIV-1 infection model with cell-mediated immune response and intracellular delay. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 297-302. doi: 10.3934/dcdsb.2012.17.297

[5]

Jinliang Wang, Jiying Lang, Xianning Liu. Global dynamics for viral infection model with Beddington-DeAngelis functional response and an eclipse stage of infected cells. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3215-3233. doi: 10.3934/dcdsb.2015.20.3215

[6]

Huiyan Zhu, Xingfu Zou. Dynamics of a HIV-1 Infection model with cell-mediated immune response and intracellular delay. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 511-524. doi: 10.3934/dcdsb.2009.12.511

[7]

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

[8]

Alexander Rezounenko. Viral infection model with diffusion and state-dependent delay: Stability of classical solutions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1091-1105. doi: 10.3934/dcdsb.2018143

[9]

Cameron Browne. Immune response in virus model structured by cell infection-age. Mathematical Biosciences & Engineering, 2016, 13 (5) : 887-909. doi: 10.3934/mbe.2016022

[10]

Fabrizio Clarelli, Roberto Natalini. A pressure model of immune response to mycobacterium tuberculosis infection in several space dimensions. Mathematical Biosciences & Engineering, 2010, 7 (2) : 277-300. doi: 10.3934/mbe.2010.7.277

[11]

Jeng-Huei Chen. An analysis of functional curability on HIV infection models with Michaelis-Menten-type immune response and its generalization. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2089-2120. doi: 10.3934/dcdsb.2017086

[12]

Sumei Li, Yicang Zhou. Backward bifurcation of an HTLV-I model with immune response. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 863-881. doi: 10.3934/dcdsb.2016.21.863

[13]

Hossein Mohebbi, Azim Aminataei, Cameron J. Browne, Mohammad Reza Razvan. Hopf bifurcation of an age-structured virus infection model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 861-885. doi: 10.3934/dcdsb.2018046

[14]

Yu Ji. Global stability of a multiple delayed viral infection model with general incidence rate and an application to HIV infection. Mathematical Biosciences & Engineering, 2015, 12 (3) : 525-536. doi: 10.3934/mbe.2015.12.525

[15]

Hee-Dae Kwon, Jeehyun Lee, Myoungho Yoon. An age-structured model with immune response of HIV infection: Modeling and optimal control approach. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 153-172. doi: 10.3934/dcdsb.2014.19.153

[16]

Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026

[17]

Yincui Yan, Wendi Wang. Global stability of a five-dimensional model with immune responses and delay. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 401-416. doi: 10.3934/dcdsb.2012.17.401

[18]

Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325

[19]

Mudassar Imran, Hal L. Smith. The dynamics of bacterial infection, innate immune response, and antibiotic treatment. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 127-143. doi: 10.3934/dcdsb.2007.8.127

[20]

Yilong Li, Shigui Ruan, Dongmei Xiao. The Within-Host dynamics of malaria infection with immune response. Mathematical Biosciences & Engineering, 2011, 8 (4) : 999-1018. doi: 10.3934/mbe.2011.8.999

2017 Impact Factor: 0.972

Metrics

  • PDF downloads (25)
  • HTML views (5)
  • Cited by (0)

Other articles
by authors

[Back to Top]