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Analysis of the dynamics of a delayed HIV pathogenesis model. (English) Zbl 1185.92062
Summary: Considering full logistic proliferation of CD4$^{+}$ T cells, we study an HIV pathogenesis model with antiretroviral therapy and HIV replication time. We first analyze the existence and stability of the equilibrium, and then investigate the effect of the time delay on the stability of the infected steady state. Sufficient conditions are given to ensure that the infected steady state is asymptotically stable for all delays. Furthermore, we apply the Nyquist criterion to estimate the length of delay for which stability continues to hold, and investigate the existence of Hopf bifurcation by using a delay $\tau $ as a bifurcation parameter. Finally, numerical simulations are presented to illustrate the main results.

MSC:
92C50Medical applications of mathematical biology
34K20Stability theory of functional-differential equations
34K18Bifurcation theory of functional differential equations
34K60Qualitative investigation and simulation of models
65C20Models (numerical methods)
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Full Text: DOI
References:
[1] Perelson, A. S.; Nelson, P. W.: Mathematical analysis of HIV-1 dynamics in vivo, SIAM rev. 41, 3-44 (1999) · Zbl 1078.92502 · doi:10.1137/S0036144598335107
[2] Bonhoeffer, S.; May, R. M.; Shaw, G. M.: Virus dynamics and drug therapy, Proc. natl. Acad. sci. USA 94, 6971-6976 (1997)
[3] Nowak, M. A.; Bonhoeffer, S.; Shaw, G. M.; May, R. M.: Anti-viral drug treatment: dynamics of resistance in free virus and infected cell populations, J. theoret. Biol. 184, 203-217 (1997)
[4] Nowak, M. A.; May, R. M.: Virus dynamics: mathematical principles of immunology and virology, (2000) · Zbl 1101.92028
[5] Perelson, A. S.; Neumann, A. U.; Markowitz, M.: HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time, Science 271, 1582-1586 (1996)
[6] Stafford, M. A.; Coreya, L.; Cao, Y. Z.: Modeling plasma virus concentration during primary HIV infection, J. theoret. Biol. 203, 285-293 (2000)
[7] De Leenheer, P.; Smith, H. L.: Virus dynamics: A global analysis, SIAM J. Appl. math. 63, 1313-1327 (2003) · Zbl 1035.34045 · doi:10.1137/S0036139902406905
[8] Nelson, P. W.; Murray, J. D.; Perelson, A. S.: A model of HIV-1 pathogenesis that includes an intracellular delay, Math. biosci. 163, 201-216 (2000) · Zbl 0942.92017 · doi:10.1016/S0025-5564(99)00055-3
[9] Nelson, P. W.; Perelson, A. S.: Mathematical analysis of delay differential equation models of HIV-1 infection, Math. biosci. 179, 73-94 (2002) · Zbl 0992.92035 · doi:10.1016/S0025-5564(02)00099-8
[10] Song, X. Y.; Neumann, Avidan U.: Global stability and periodic solution of the viral dynamics, J. math. Anal. appl. 329, No. 1, 281-297 (2007) · Zbl 1105.92011 · doi:10.1016/j.jmaa.2006.06.064
[11] Zhou, X. Y.; Song, X. Y.; Shi, X. Y.: A differential equation model of HIV infection of CD4+ T-cells with cure rate, J. math. Anal. appl. 342, 1342-1355 (2008) · Zbl 1188.34062 · doi:10.1016/j.jmaa.2008.01.008
[12] Yang, J. Y.; Wang, X. Y.; Zhang, F. Q.: A differential equation model of HIV infection of CD4+ T-cells with delay, Discrete dyn. Nat. soc. 2008 (2009) · Zbl 1157.92029
[13] Wang, L. C.; Li, M. Y.: Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells, Math. biosci. 200, 44-57 (2006) · Zbl 1086.92035 · doi:10.1016/j.mbs.2005.12.026
[14] Wang, X.; Song, X. Y.: Global stability and periodic solution of a model for HIV infection of CD4+ T cells, Appl. math. Comput. 189, 1331-1340 (2007) · Zbl 1117.92040 · doi:10.1016/j.amc.2006.12.044
[15] Culshaw, R. V.; Ruan, S. G.: A delay-differential equation model of HIV infection of CD4+ T-cells, Math. biosci. 165, 27-39 (2000) · Zbl 0981.92009 · doi:10.1016/S0025-5564(00)00006-7
[16] Herz, A. V. M.; Bonhoer, S.; Anderson, R. M.; May, R. M.; Nowak, M. A.: Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay, Proc. natl. Acad. sci. USA 93, 7247-7251 (1996)
[17] Song, X. Y.; Cheng, S. H.: Delay-differential equation model of HIV infection of CD4+ T-cells, Korean math. Soc. 42, 1071-1086 (2005) · Zbl 1078.92042 · doi:10.4134/JKMS.2005.42.5.1071
[18] Jiang, X. W.; Zhou, X. Y.; Shi, X. Y.; Song, X. Y.: Analysis of stability and Hopf bifurcation for a delay-differential equation model of HIV infection of CD4+ T cells, Chaos solitons fractals 38, 447-460 (2008) · Zbl 1146.34313 · doi:10.1016/j.chaos.2006.11.026
[19] Zhou, X. Y.; Song, X. Y.; Shi, X. Y.: Analysis of stability and Hopf bifurcation for an HIV infection model with time delay, Appl. math. Comput. 199, 23-38 (2008) · Zbl 1136.92027 · doi:10.1016/j.amc.2007.09.030
[20] Wang, Y.; Zhou, Y. C.; Wu, J. H.; Heffernan, J.: Oscillatory viral dynamics in a delayed HIV pathogenesis model, Math. biosci. 3, 3-12 (2009) · Zbl 1168.92031
[21] Leonard, R.; Zagury, D.: Cytopathic effect of human immunodeficiency virus in T4 cells is linked to the last stage of virus infection, Proc. natl. Acad. sci. USA 85, 3570-3574 (1988)
[22] Wodarz, D.; Hamer, D. H.: Infection dynamic in HIV-specific CD4+ T cells, Math. biosci. 209, 14-29 (2007) · Zbl 1120.92026
[23] Cai, L. M.; Li, X. Z.: Stability of Hopf bifurcation in a delayed model for HIV infection of CD4+ T-cells, Chaos solitons fractals 42, 1-11 (2009) · Zbl 1198.37119 · doi:10.1016/j.chaos.2008.04.048
[24] Gopalsamy, K.: Stability and oscillation in delay different equations of population dynamics, (1992) · Zbl 0752.34039
[25] Freedman, H. I.; Rao, V. S. H: The trade-off between mutual interference and time lags in predator--prey systems, Bull. math. Biol. 45, 991-1004 (1983) · Zbl 0535.92024
[26] Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H.: Theory and application of Hopf bifurcation, (1981) · Zbl 0474.34002