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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 |
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), 387-392. |
[2] |
R. J. De Boer and A. S. Perelson, Towards a general function describing T cell proliferation, J. Theor. Biol., 175 (1995), 567-576. |
[3] |
S. Bonhoeffer, R. M. May and G. M. Shaw, et al., Virus dynamics and drug therapy, P. Natl. Acad. Sci. USA., 94 (1997), 6971-6976.
doi: 10.1073/pnas.94.13.6971. |
[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), 234-241.
doi: 10.1016/j.physa.2004.04.083. |
[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), 157-170.
doi: 10.1016/j.matcom.2013.05.003. |
[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), 1154-1162. |
[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), 365-382.
doi: 10.1007/BF00178324. |
[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), 29-48.
doi: 10.1016/S0025-5564(02)00108-6. |
[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), 283-299.
doi: 10.3934/mbe.2009.6.283. |
[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), 140-153.
doi: 10.1080/17513750701769873. |
[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), 2673-2681.
doi: 10.1056/NEJMoa042957. |
[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), 1866-1872.
doi: 10.1016/j.nonrwa.2011.12.015. |
[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), 524-534.
doi: 10.1016/j.apm.2013.06.041. |
[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), 1199-1203.
doi: 10.1016/j.aml.2011.02.007. |
[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), 325-337.
doi: 10.1142/S0218339010003299. |
[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), 525-536.
doi: 10.3934/mbe.2015.12.525. |
[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), 1080-1092.
doi: 10.1016/j.nonrwa.2011.02.026. |
[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), 1573-1585.
doi: 10.1216/RMJ-2008-38-5-1573. |
[19] |
Y. Nakata, Global dynamics of a viral infection model with a latent period and Beddington-DeAngelis response, Nonlinear Anal-Theor., 74 (2011), 2929-2940.
doi: 10.1016/j.na.2010.12.030. |
[20] |
M. A. Nowak and C. R. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79.
doi: 10.1126/science.272.5258.74. |
[21] |
M. A. Nowak and R. M. May, Virus Dynamics, Oxford University Press, New York, 2000. |
[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), 1565-1570. |
[23] |
X. Y. Song and A. U. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329 (2007), 281-297.
doi: 10.1016/j.jmaa.2006.06.064. |
[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), 345-355.
doi: 10.1016/j.jmaa.2010.04.010. |
[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), 1511-1523.
doi: 10.1016/j.apm.2009.09.006. |
[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), 285-301.
doi: 10.1006/jtbi.2000.1076. |
[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), 17-26.
doi: 10.1016/j.nonrwa.2013.09.002. |
[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), 964-978.
doi: 10.1016/j.cnsns.2011.06.024. |
[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), 1221-1230.
doi: 10.1007/s11424-010-8467-0. |
[30] |
X. Zhou and J. Cui, Global stability of the viral dynamics with Crowley-Martin functional response, B. Korean Math. Soc., 48 (2011), 555-574.
doi: 10.4134/BKMS.2011.48.3.555. |
[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), 3091-3102.
doi: 10.1016/j.camwa.2011.08.022. |
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), 387-392. |
[2] |
R. J. De Boer and A. S. Perelson, Towards a general function describing T cell proliferation, J. Theor. Biol., 175 (1995), 567-576. |
[3] |
S. Bonhoeffer, R. M. May and G. M. Shaw, et al., Virus dynamics and drug therapy, P. Natl. Acad. Sci. USA., 94 (1997), 6971-6976.
doi: 10.1073/pnas.94.13.6971. |
[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), 234-241.
doi: 10.1016/j.physa.2004.04.083. |
[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), 157-170.
doi: 10.1016/j.matcom.2013.05.003. |
[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), 1154-1162. |
[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), 365-382.
doi: 10.1007/BF00178324. |
[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), 29-48.
doi: 10.1016/S0025-5564(02)00108-6. |
[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), 283-299.
doi: 10.3934/mbe.2009.6.283. |
[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), 140-153.
doi: 10.1080/17513750701769873. |
[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), 2673-2681.
doi: 10.1056/NEJMoa042957. |
[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), 1866-1872.
doi: 10.1016/j.nonrwa.2011.12.015. |
[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), 524-534.
doi: 10.1016/j.apm.2013.06.041. |
[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), 1199-1203.
doi: 10.1016/j.aml.2011.02.007. |
[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), 325-337.
doi: 10.1142/S0218339010003299. |
[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), 525-536.
doi: 10.3934/mbe.2015.12.525. |
[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), 1080-1092.
doi: 10.1016/j.nonrwa.2011.02.026. |
[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), 1573-1585.
doi: 10.1216/RMJ-2008-38-5-1573. |
[19] |
Y. Nakata, Global dynamics of a viral infection model with a latent period and Beddington-DeAngelis response, Nonlinear Anal-Theor., 74 (2011), 2929-2940.
doi: 10.1016/j.na.2010.12.030. |
[20] |
M. A. Nowak and C. R. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79.
doi: 10.1126/science.272.5258.74. |
[21] |
M. A. Nowak and R. M. May, Virus Dynamics, Oxford University Press, New York, 2000. |
[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), 1565-1570. |
[23] |
X. Y. Song and A. U. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329 (2007), 281-297.
doi: 10.1016/j.jmaa.2006.06.064. |
[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), 345-355.
doi: 10.1016/j.jmaa.2010.04.010. |
[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), 1511-1523.
doi: 10.1016/j.apm.2009.09.006. |
[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), 285-301.
doi: 10.1006/jtbi.2000.1076. |
[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), 17-26.
doi: 10.1016/j.nonrwa.2013.09.002. |
[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), 964-978.
doi: 10.1016/j.cnsns.2011.06.024. |
[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), 1221-1230.
doi: 10.1007/s11424-010-8467-0. |
[30] |
X. Zhou and J. Cui, Global stability of the viral dynamics with Crowley-Martin functional response, B. Korean Math. Soc., 48 (2011), 555-574.
doi: 10.4134/BKMS.2011.48.3.555. |
[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), 3091-3102.
doi: 10.1016/j.camwa.2011.08.022. |
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