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

34K20Stability theory of functional-differential equations
34K18Bifurcation theory of functional differential equations
Full Text: DOI
[1] Perelson, A. S.; Essunger, P.; Ho, D. D.: Dynamics of HIV-1 and CD4+ lymphocytes in vivo, Aids 11, No. 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 · doi:10.1016/j.jmaa.2006.06.064
[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, No. 4, 443-448 (2008) · Zbl 1156.92322 · doi:10.1142/S1793524508000382
[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, No. 1, 3-44 (1999) · Zbl 1078.92502 · doi:10.1137/S0036144598335107
[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 · doi:10.1016/S0025-5564(00)00006-7
[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 · doi:10.1007/s00285-002-0191-5
[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 · doi:10.1016/S0025-5564(02)00099-8
[10] Tam, J.: Delay effect in a model for virus replication, IMA J. Math. appl. Med. biol. 16, No. 1, 29-37 (1999) · Zbl 0914.92012
[11] Song, X.; Cheng, S.: A delay-differential equation model of HIV infection of CD4+ T-cells, J. koreal math. Soc. 42, No. 5, 1071-1086 (2005) · Zbl 1078.92042 · doi:10.4134/JKMS.2005.42.5.1071
[12] Hale, J. K.; Waltman, P.: Persistence in infinite-dimensional systems, SIAM J. Math. anal. 20, 388-396 (1989) · Zbl 0692.34053 · doi:10.1137/0520025
[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 · doi:10.1137/S0036141000376086