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Stability and Hopf bifurcation of a HIV infection model with CTL-response delay. (English) Zbl 1232.37045
Summary: We consider a HIV infection model with CTL-response delay and analyze the effect of time delay on stability of equilibria. We obtain the global stability of the infection-free equilibrium and give sufficient conditions for the local stability of the CTL-absent equilibrium and CTL-present equilibrium. By choosing the CTL-response delay τ as a bifurcation parameter, we prove that the CTL-present equilibrium is locally asymptotically stable in a range of delays and a Hopf bifurcation occurs as τ crosses a critical value. Numerical simulations are given to support the theoretical results.
MSC:
37N25Dynamical systems in biology
92D30Epidemiology
34K18Bifurcation theory of functional differential equations
34K20Stability theory of functional-differential equations
References:
[1]Nelson, P.; Murray, J.; Perelson, A.: A model of HIV-1 pathogenesis that includes and intracellular delay, Math. biosci. 163, 201-215 (2000) · Zbl 0942.92017 · doi:10.1016/S0025-5564(99)00055-3
[2]Perelson, A.; Nelson, P.: Mathematical models of HIV dynamics in vivo, SIAM rev. 41, 3-44 (1999) · Zbl 1078.92502 · doi:10.1137/S0036144598335107
[3]Deboer, R. J.; Perelson, A. S.: Towards a general function describing T cell proliferation, J. theoret. Biol. 175, 567-576 (1995)
[4]Nowak, M.; May, R.: Virus dynamics, (2000)
[5]Nowak, M.; Bangham, C.: Population dynamics of immune responses to persistent viruses, Science (NS) 272, 74-79 (1996)
[6]Wang, K.; Wang, W.; Liu, X.: Global stability in a viral infection model with lytic and nonlytic immune response, Comput. math. Appl. 51, 1593-1610 (2006) · Zbl 1141.34034 · doi:10.1016/j.camwa.2005.07.020
[7]Zhu, H.; Zou, X.: Dynamics of a HIV-1 infection model with cell-mediated immume response and intracellular delay, Discrete contin. Dyn. syst. Ser. B 12, 513-526 (2009) · Zbl 1169.92033 · doi:10.3934/dcdsb.2009.12.511
[8]Arnaout, R.; Owak, M. N.; Wodarz, D.: HIV-1 dynamics revisited: biphasic decay by cytotoxic lymphocyte Killing, Proc. R. Soc. lond. Ser. B 265, 1347-1354 (2000)
[9]Culshaw, R.; Ruan, S.: 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
[10]Culshaw, R.; 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
[11]Nelson, P. W.; Perelson, A.: 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
[12]Nelson, P.; Murray, J.; Perelson, A.: A model of HIV-1 pathogenesis that includes an intracelluar delay, Math. biosci. 163, 201-215 (2000) · Zbl 0942.92017 · doi:10.1016/S0025-5564(99)00055-3
[13]Zhu, H.; Zou, X.: Impact of delays in cell infection and virus production on HIV-1 dynamics, Math. med. Biol. 25, 99-112 (2008) · Zbl 1155.92031 · doi:10.1093/imammb/dqm010
[14]Canabarro, A. A.; Gléria, I. M.; Lyra, M. L.: Periodic solutions and chos in a non-linear model for the delayed cellular immune response, Physica A 342, 234-241 (2004)
[15]Wang, K.; Wang, W.; Pang, H.; Liu, X.: Complex dynamic behavior in a viral model with delayed immune response, Physica D 226, 197-208 (2007) · Zbl 1117.34081 · doi:10.1016/j.physd.2006.12.001
[16]Culshaw, R.; Ruan, S.; Spiteri, R.: Optimal HIV treament by maximising immune response, J. math. Biol. 48, 545-562 (2004) · Zbl 1057.92035 · doi:10.1007/s00285-003-0245-3
[17]Hale, J. K.: Theoory of functional differential equations, (1997)
[18]Kuang, Y.: Delay differential equations with applications in population dynamics, (1993) · Zbl 0777.34002