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Song, Xinyu; Zhou, Xueyong; Zhao, Xiang

Properties of stability and Hopf bifurcation for a HIV infection model with time delay. (English) Zbl 1193.34152

Appl. Math. Modelling 34, No. 6, 1511-1523 (2010).
Summary: We consider the classical mathematical model with saturation response of the infection rate and time delay. By stability analysis we obtain sufficient conditions for the global stability of the infection-free steady state and the permanence of the infected steady state. Numerical simulations are carried out to explain the mathematical conclusions.

Cited in 27 Documents

MSC:

34K20 Stability theory of functional-differential equations
34K18 Bifurcation theory of functional-differential equations
92D30 Epidemiology

Keywords:

time delay; HIV infection; Hopf bifurcation; global stability
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\textit{X. Song} et al., Appl. Math. Modelling 34, No. 6, 1511--1523 (2010; Zbl 1193.34152)
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References:

[1] Perelson, A. S.; Essunger, P.; Ho, D. D., Dynamics of HIV-1 and CD \(4^+\) lymphocytes in vivo, AIDS, 11, Suppl. A, S17-S24 (1997)
[2] Song, X.; Neumann, A. U., Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329, 281-297 (2007) · Zbl 1105.92011
[3] Wei, X.; Ghosh, S.; Taylor, M.; Johnson, V.; Emini, E.; Deutsch, P.; Lifson, J.; Bonhoeffer, S.; Nowak, M.; Hahn, B.; Saag, S.; Shaw, G., Viral dynamics in human immunodeficiency virus type 1 infection, Nature, 373, 117 (1995)
[4] Wang, X.; Tao, Y., Lyapunov function and global properties of virus dynamics with immune response, Int. J. Biomath., 1, 4, 443-448 (2008) · Zbl 1156.92322
[5] De Boer, R. J.; Perelson, A. S., Target cell limited and immune control models of HIV infection: a comparison, J. Theor. Biol., 190, 201-214 (1998)
[6] Perelson, A.; Nelson, P., Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41, 1, 3-44 (1999) · Zbl 1078.92502
[7] Culshaw, R. V.; Ruan, S., A delay-differential equation model of HIV infection of D4+ T-cells, Math. Biosci., 165, 27-39 (2000) · Zbl 0981.92009
[8] Culshaw, R. V.; Ruan, S.; Webb, G., A mathematical model of cell-to-cell HIV-1 that include a time delay, J. Math. Biol., 46, 425-444 (2003) · Zbl 1023.92011
[9] Nelson, P. W.; Perelson, A. S., Mathematical analysis of a delay differential equation models of HIV-1 infection, Math. Biosci., 179, 73-94 (2002) · Zbl 0992.92035
[10] Tam, J., Delay effect in a model for virus replication, IMA J. Math. Appl. Med. Biol., 16, 1, 29-37 (1999) · Zbl 0914.92012
[11] Song, X.; Cheng, S., A delay-differential equation model of HIV infection of \(\text{CD} 4^+\) T-cells, J. Koreal Math. Soc., 42, 5, 1071-1086 (2005) · Zbl 1078.92042
[12] Hale, J. K.; Waltman, P., Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20, 388-396 (1989) · Zbl 0692.34053
[13] Beretta, E.; Kuang, Y., Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33, 1144-1165 (2002) · Zbl 1013.92034
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