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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

 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.
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
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