American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Existence of the global attractor for the plate equation with nonlocal nonlinearity in $ \mathbb{R} ^{n}$
  • DCDS-B Home
  • This Issue
  • Next Article
    Global phase portraits of uniform isochronous centers with quartic homogeneous polynomial nonlinearities
January  2016, 21(1): 133-149. doi: 10.3934/dcdsb.2016.21.133

Global stability of a delayed viral infection model with nonlinear immune response and general incidence rate

Yu Ji 1, and  Lan Liu 2,

1. 

Department of Mathematics, Beijing Technology and Business University, Beijing, 100048
2. 

Department of Mathematics, Beijing Technology and Business University, Beijing 100048, China

Received  February 2015 Revised  April 2015 Published  November 2015

In this paper, a delayed viral infection model with nonlinear immune response and general incidence rate is discussed. We prove the existence and uniqueness of the equilibria. We study the effect of three kinds of time delays on the dynamics of the model. By using the Lyapunov functional and LaSalle invariance principle, we obtain the conditions of global stabilities of the infection-free equilibrium, the immune-exhausted equilibrium and the endemic equilibrium. It is shown that an increase of the viral-infection delay and the virus-production delay may stabilize the infection-free equilibrium, but the immune response delay can destabilize the equilibrium, leading to Hopf bifurcations. Numerical simulations are given to verify the analytical results. This can provide a possible interpretation for the viral oscillation observed in chronic hepatitis B virus (HBV) and human immunodeficiency virus (HIV) infected patients.
Keywords: virus infection, general incidence rate., immune response, stability, Time delay.
Mathematics Subject Classification: Primary: 34K20, 34K25; Secondary: 92D3.
Citation: 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
References:
[1]

A. Bertoletti and M. K. Maini, Protection or damage: a dual role for the virus-specific cytotoxic T lymphocyte response in hepatitis B and C infection?,, Curr. Opin. Microbiol., 3 (2000), 387.   Google Scholar

[2]

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

[3]

S. Bonhoeffer, R. M. May and G. M. Shaw, et al., Virus dynamics and drug therapy,, P. Natl. Acad. Sci. USA., 94 (1997), 6971.  doi: 10.1073/pnas.94.13.6971.  Google Scholar

[4]

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.  doi: 10.1016/j.physa.2004.04.083.  Google Scholar

[5]

X. Chen, L. Q. Min and Y. Zheng, et al., Dynamics of acute hepatitis B virus infection in chimpanzees,, Math. Comput. Simulat., 96 (2014), 157.  doi: 10.1016/j.matcom.2013.05.003.  Google Scholar

[6]

Y. K. Chun, J. Y. Kim and H. J. Woo, et al., No significant correlation exists between core promoter mutations, viral replication, and liver damage in chronic hepatitis B infection,, Hepatology, 32 (2000), 1154.   Google Scholar

[7]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations,, J. Math. Biol., 28 (1990), 365.  doi: 10.1007/BF00178324.  Google Scholar

[8]

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

[9]

S. Eikenberry, S. Hews and J. D. Nagy, et al., The dynamics of a delay model of HBV infection with logistic hepatocyte growth,, Math. Biosci. Eng., 6 (2009), 283.  doi: 10.3934/mbe.2009.6.283.  Google Scholar

[10]

S. A. Gourley, Y. Kuang and J. D. Nagy, Dynamics of a delay differential model of hepatitis B virus,, J. Biol. Dynam., 2 (2008), 140.  doi: 10.1080/17513750701769873.  Google Scholar

[11]

S. J. Hadziyannis, N. C. Tassopoulos and E. J. Heathcote, et al., Long-term therapy with adefovir dipivoxil for HBeAg-negative chronic hepatitis B,, New Engl. J. Med., 352 (2005), 2673.  doi: 10.1056/NEJMoa042957.  Google Scholar

[12]

K. Hattaf, N. Yousfi and A. Tridane, Mathematical analysis of a virus dynamics model with general incidence rate and cure rate,, Nonlinear Anal-Real, 13 (2012), 1866.  doi: 10.1016/j.nonrwa.2011.12.015.  Google Scholar

[13]

Z. X. Hu, J. J. Zhang and H. Wang, et al., Dynamics analysis of a delayed viral infection model with logistic growth and immune impairment,, Appl. Math. Model., 38 (2014), 524.  doi: 10.1016/j.apm.2013.06.041.  Google Scholar

[14]

G. Huang, W. B. Ma and Y. Takeuchi, Global analysis for delay virus dynamics model with Beddington-DeAngelis functional response,, Appl. Math. Lett., 24 (2011), 1199.  doi: 10.1016/j.aml.2011.02.007.  Google Scholar

[15]

Y. Ji, L. Q. Min and Y. A. Ye, Global analysis of a viral infection model with application to HBV infection,, J. Biol. Syst., 18 (2010), 325.  doi: 10.1142/S0218339010003299.  Google Scholar

[16]

Y. Ji, Global stability of a multiple delayed viral infection model with general incidence rate and an application to HIV infection,, Math. Biosci. Eng., 12 (2015), 525.  doi: 10.3934/mbe.2015.12.525.  Google Scholar

[17]

M. Y. Li and H. Y. Shu, Global dynamics of a mathematical model for HTLV-I infection of CD$4^+$ T cells with delayed CTL response,, Nonlinear Anal-Real, 13 (2012), 1080.  doi: 10.1016/j.nonrwa.2011.02.026.  Google Scholar

[18]

L. Q. Min, Y. M. Su and Y. Kuang, Mathematical analysis of a basic virus infection model with application to HBV infection,, Rocky Mt. J. Math., 38 (2008), 1573.  doi: 10.1216/RMJ-2008-38-5-1573.  Google Scholar

[19]

Y. Nakata, Global dynamics of a viral infection model with a latent period and Beddington-DeAngelis response,, Nonlinear Anal-Theor., 74 (2011), 2929.  doi: 10.1016/j.na.2010.12.030.  Google Scholar

[20]

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

[21]

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

[22]

A. Penna, F. V. Chisari and A. Bertoletti, et al., Cytotoxic T lymphocytes recognize an HLA-A2-restricted epitope within the hepatitis B virus nucleocapsid antigen,, J. Exp. Med., 174 (1991), 1565.   Google Scholar

[23]

X. Y. Song and A. U. Neumann, Global stability and periodic solution of the viral dynamics,, J. Math. Anal. Appl., 329 (2007), 281.  doi: 10.1016/j.jmaa.2006.06.064.  Google Scholar

[24]

X. Y. Song, S. L. Wang and J. Dong, Stability properties and Hopf bifurcation of a delayed viral infection model with lytic immune response,, J. Math. Anal. Appl., 373 (2011), 345.  doi: 10.1016/j.jmaa.2010.04.010.  Google Scholar

[25]

X. Y. Song, X. Y. Zhou and X. Zhao, Properties of stability and Hopf bifurcaion for a HIV infection model with time delay,, Appl. Math. Model., 34 (2010), 1511.  doi: 10.1016/j.apm.2009.09.006.  Google Scholar

[26]

M. A. Stafford, L. Corey and Y. Z. Cao, et al., Modeling plasma virus concentration during primary HIV infection,, J. Theor. Biol., 203 (2000), 285.  doi: 10.1006/jtbi.2000.1076.  Google Scholar

[27]

Y. N. Tian and X. N. Liu, Global dynamics of a virus dynamical model with general incidence rate and cure rate,, Nonlinear Anal-Real, 16 (2014), 17.  doi: 10.1016/j.nonrwa.2013.09.002.  Google Scholar

[28]

Z. P. Wang and R. Xu, Stability and Hopf bifurcation in a viral infection model with nonlinear incidence rate and delayed immune response,, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 964.  doi: 10.1016/j.cnsns.2011.06.024.  Google Scholar

[29]

Y. Zheng, L. Q. Min and Y. Ji, et al., Global stability of endemic equilibrium point of basic virus infection model with application to HBV infection,, J. Syst. Sci. Complex., 23 (2010), 1221.  doi: 10.1007/s11424-010-8467-0.  Google Scholar

[30]

X. Zhou and J. Cui, Global stability of the viral dynamics with Crowley-Martin functional response,, B. Korean Math. Soc., 48 (2011), 555.  doi: 10.4134/BKMS.2011.48.3.555.  Google Scholar

[31]

H. Y. Zhu, Y. Luo and M. L. Chen, Stability and Hopf bifurcation of a HIV infection model with CTL-response delay,, Comput. Math. Appl., 62 (2011), 3091.  doi: 10.1016/j.camwa.2011.08.022.  Google Scholar

show all references

References:
[1]

A. Bertoletti and M. K. Maini, Protection or damage: a dual role for the virus-specific cytotoxic T lymphocyte response in hepatitis B and C infection?,, Curr. Opin. Microbiol., 3 (2000), 387.   Google Scholar

[2]

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

[3]

S. Bonhoeffer, R. M. May and G. M. Shaw, et al., Virus dynamics and drug therapy,, P. Natl. Acad. Sci. USA., 94 (1997), 6971.  doi: 10.1073/pnas.94.13.6971.  Google Scholar

[4]

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.  doi: 10.1016/j.physa.2004.04.083.  Google Scholar

[5]

X. Chen, L. Q. Min and Y. Zheng, et al., Dynamics of acute hepatitis B virus infection in chimpanzees,, Math. Comput. Simulat., 96 (2014), 157.  doi: 10.1016/j.matcom.2013.05.003.  Google Scholar

[6]

Y. K. Chun, J. Y. Kim and H. J. Woo, et al., No significant correlation exists between core promoter mutations, viral replication, and liver damage in chronic hepatitis B infection,, Hepatology, 32 (2000), 1154.   Google Scholar

[7]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations,, J. Math. Biol., 28 (1990), 365.  doi: 10.1007/BF00178324.  Google Scholar

[8]

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

[9]

S. Eikenberry, S. Hews and J. D. Nagy, et al., The dynamics of a delay model of HBV infection with logistic hepatocyte growth,, Math. Biosci. Eng., 6 (2009), 283.  doi: 10.3934/mbe.2009.6.283.  Google Scholar

[10]

S. A. Gourley, Y. Kuang and J. D. Nagy, Dynamics of a delay differential model of hepatitis B virus,, J. Biol. Dynam., 2 (2008), 140.  doi: 10.1080/17513750701769873.  Google Scholar

[11]

S. J. Hadziyannis, N. C. Tassopoulos and E. J. Heathcote, et al., Long-term therapy with adefovir dipivoxil for HBeAg-negative chronic hepatitis B,, New Engl. J. Med., 352 (2005), 2673.  doi: 10.1056/NEJMoa042957.  Google Scholar

[12]

K. Hattaf, N. Yousfi and A. Tridane, Mathematical analysis of a virus dynamics model with general incidence rate and cure rate,, Nonlinear Anal-Real, 13 (2012), 1866.  doi: 10.1016/j.nonrwa.2011.12.015.  Google Scholar

[13]

Z. X. Hu, J. J. Zhang and H. Wang, et al., Dynamics analysis of a delayed viral infection model with logistic growth and immune impairment,, Appl. Math. Model., 38 (2014), 524.  doi: 10.1016/j.apm.2013.06.041.  Google Scholar

[14]

G. Huang, W. B. Ma and Y. Takeuchi, Global analysis for delay virus dynamics model with Beddington-DeAngelis functional response,, Appl. Math. Lett., 24 (2011), 1199.  doi: 10.1016/j.aml.2011.02.007.  Google Scholar

[15]

Y. Ji, L. Q. Min and Y. A. Ye, Global analysis of a viral infection model with application to HBV infection,, J. Biol. Syst., 18 (2010), 325.  doi: 10.1142/S0218339010003299.  Google Scholar

[16]

Y. Ji, Global stability of a multiple delayed viral infection model with general incidence rate and an application to HIV infection,, Math. Biosci. Eng., 12 (2015), 525.  doi: 10.3934/mbe.2015.12.525.  Google Scholar

[17]

M. Y. Li and H. Y. Shu, Global dynamics of a mathematical model for HTLV-I infection of CD$4^+$ T cells with delayed CTL response,, Nonlinear Anal-Real, 13 (2012), 1080.  doi: 10.1016/j.nonrwa.2011.02.026.  Google Scholar

[18]

L. Q. Min, Y. M. Su and Y. Kuang, Mathematical analysis of a basic virus infection model with application to HBV infection,, Rocky Mt. J. Math., 38 (2008), 1573.  doi: 10.1216/RMJ-2008-38-5-1573.  Google Scholar

[19]

Y. Nakata, Global dynamics of a viral infection model with a latent period and Beddington-DeAngelis response,, Nonlinear Anal-Theor., 74 (2011), 2929.  doi: 10.1016/j.na.2010.12.030.  Google Scholar

[20]

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

[21]

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

[22]

A. Penna, F. V. Chisari and A. Bertoletti, et al., Cytotoxic T lymphocytes recognize an HLA-A2-restricted epitope within the hepatitis B virus nucleocapsid antigen,, J. Exp. Med., 174 (1991), 1565.   Google Scholar

[23]

X. Y. Song and A. U. Neumann, Global stability and periodic solution of the viral dynamics,, J. Math. Anal. Appl., 329 (2007), 281.  doi: 10.1016/j.jmaa.2006.06.064.  Google Scholar

[24]

X. Y. Song, S. L. Wang and J. Dong, Stability properties and Hopf bifurcation of a delayed viral infection model with lytic immune response,, J. Math. Anal. Appl., 373 (2011), 345.  doi: 10.1016/j.jmaa.2010.04.010.  Google Scholar

[25]

X. Y. Song, X. Y. Zhou and X. Zhao, Properties of stability and Hopf bifurcaion for a HIV infection model with time delay,, Appl. Math. Model., 34 (2010), 1511.  doi: 10.1016/j.apm.2009.09.006.  Google Scholar

[26]

M. A. Stafford, L. Corey and Y. Z. Cao, et al., Modeling plasma virus concentration during primary HIV infection,, J. Theor. Biol., 203 (2000), 285.  doi: 10.1006/jtbi.2000.1076.  Google Scholar

[27]

Y. N. Tian and X. N. Liu, Global dynamics of a virus dynamical model with general incidence rate and cure rate,, Nonlinear Anal-Real, 16 (2014), 17.  doi: 10.1016/j.nonrwa.2013.09.002.  Google Scholar

[28]

Z. P. Wang and R. Xu, Stability and Hopf bifurcation in a viral infection model with nonlinear incidence rate and delayed immune response,, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 964.  doi: 10.1016/j.cnsns.2011.06.024.  Google Scholar

[29]

Y. Zheng, L. Q. Min and Y. Ji, et al., Global stability of endemic equilibrium point of basic virus infection model with application to HBV infection,, J. Syst. Sci. Complex., 23 (2010), 1221.  doi: 10.1007/s11424-010-8467-0.  Google Scholar

[30]

X. Zhou and J. Cui, Global stability of the viral dynamics with Crowley-Martin functional response,, B. Korean Math. Soc., 48 (2011), 555.  doi: 10.4134/BKMS.2011.48.3.555.  Google Scholar

[31]

H. Y. Zhu, Y. Luo and M. L. Chen, Stability and Hopf bifurcation of a HIV infection model with CTL-response delay,, Comput. Math. Appl., 62 (2011), 3091.  doi: 10.1016/j.camwa.2011.08.022.  Google Scholar

[1]

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
[2]

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
[3]

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
[4]

Ting Guo, Haihong Liu, Chenglin Xu, Fang Yan. Global stability of a diffusive and delayed HBV infection model with HBV DNA-containing capsids and general incidence rate. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4223-4242. doi: 10.3934/dcdsb.2018134
[5]

Shouying Huang, Jifa Jiang. Epidemic dynamics on complex networks with general infection rate and immune strategies. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2071-2090. doi: 10.3934/dcdsb.2018226
[6]

Bao-Zhu Guo, Li-Ming Cai. A note for the global stability of a delay differential equation of hepatitis B virus infection. Mathematical Biosciences & Engineering, 2011, 8 (3) : 689-694. doi: 10.3934/mbe.2011.8.689
[7]

Haitao Song, Weihua Jiang, Shengqiang Liu. Virus dynamics model with intracellular delays and immune response. Mathematical Biosciences & Engineering, 2015, 12 (1) : 185-208. doi: 10.3934/mbe.2015.12.185
[8]

Lianwen Wang, Zhijun Liu, Yong Li, Dashun Xu. Complete dynamical analysis for a nonlinear HTLV-I infection model with distributed delay, CTL response and immune impairment. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 917-933. doi: 10.3934/dcdsb.2019196
[9]

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
[10]

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
[11]

C. Connell McCluskey. Global stability of an $SIR$ epidemic model with delay and general nonlinear incidence. Mathematical Biosciences & Engineering, 2010, 7 (4) : 837-850. doi: 10.3934/mbe.2010.7.837
[12]

Jinling Zhou, Yu Yang. Traveling waves for a nonlocal dispersal SIR model with general nonlinear incidence rate and spatio-temporal delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1719-1741. doi: 10.3934/dcdsb.2017082
[13]

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
[14]

Shouying Huang, Jifa Jiang. Global stability of a network-based SIS epidemic model with a general nonlinear incidence rate. Mathematical Biosciences & Engineering, 2016, 13 (4) : 723-739. doi: 10.3934/mbe.2016016
[15]

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
[16]

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
[17]

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
[18]

Gesham Magombedze, Winston Garira, Eddie Mwenje. Modelling the human immune response mechanisms to mycobacterium tuberculosis infection in the lungs. Mathematical Biosciences & Engineering, 2006, 3 (4) : 661-682. doi: 10.3934/mbe.2006.3.661
[19]

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
[20]

Yuri Nechepurenko, Michael Khristichenko, Dmitry Grebennikov, Gennady Bocharov. Bistability analysis of virus infection models with time delays. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2385-2401. doi: 10.3934/dcdss.2020166

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (53)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences