×

zbMATH — the first resource for mathematics

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.

MSC:
34K20 Stability theory of functional-differential equations
34K18 Bifurcation theory of functional-differential equations
92D30 Epidemiology
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Perelson, A.S.; Essunger, P.; Ho, D.D., Dynamics of HIV-1 and CD4^{+} 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.