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Global stability of an HIV-1 infection model with saturation infection and intracellular delay. (English) Zbl 1222.34101
Author’s abstract: An HIV-1 infection model with a saturation infection rate and an intracellular delay accounting for the time between viral entry into a target cell and the production of new virus particles is investigated. By analyzing the characteristic equations, the local stability of an infection-free equilibrium and a chronic-infection equilibrium of the model is established. By using suitable Lyapunov functionals and the LaSalle invariant principle, it is proved that if the basic reproduction ratio is less than one, the infection-free equilibrium is globally asymptotically stable; if the basic reproduction ratio is greater than one, the chronic-infection equilibrium is globally asymptotically stable.
MSC:
34K60Qualitative investigation and simulation of models
92C50Medical applications of mathematical biology
92D30Epidemiology
34K20Stability theory of functional-differential equations
References:
[1]Bonhoeffer, S.; May, R. M.; Shaw, G. M.; Nowak, M. A.: Virus dynamics and drug therapy, Proc. natl. Acad. sci. USA 94, 6971-6976 (1997)
[2]Culshaw, R. V.; 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
[3]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
[4]Ebert, D.; Zschokke-Rohringer, C. D.; Carius, H. J.: Dose effects and density-dependent regulation of two microparasites of daphnia magna, Oecologia 122, 200-209 (2000)
[5]Hale, J. K.; Lunel, S. Verduyn: Introduction to functional differential equations, (1993)
[6]Herz, A. V. M.; Bonhoeffer, 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)
[7]Ho, D.; Neumann, A.; Perelson, A.; Chen, W.; Leonard, J.; Markowitz, M.: Rapid turnover of plasma virions and CD4+ lymphocytes in HIV-1 infection, Nature 373, 123-126 (1995)
[8]Kirschner, D.: Using mathematics to understand HIV immune dynamics, Notices amer. Math. soc. 43, 191-202 (1996) · Zbl 1044.92503
[9]Kuang, Y.: Delay differential equations with applications in population dynamics, (1993) · Zbl 0777.34002
[10]Mccluskey, C. C.: Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. biosci. Eng. 6, No. 3, 603-610 (2009) · Zbl 1190.34108 · doi:10.3934/mbe.2009.6.603
[11]Mccluskey, C. C.: Complete global stability for an SIR epidemic model with delay-distributed or discrete, Nonlinear anal. Real world appl. 11, No. 1, 55-59 (2010) · Zbl 1185.37209 · doi:10.1016/j.nonrwa.2008.10.014
[12]Mittler, J. E.; Markowitz, B.; Ho, D. D.; Perelson, A. S.: Improved estimates for HIV-1 clearance rate and intracellular delay, Aids 13, 1415-1417 (1999)
[13]Mittler, J. E.; Sulzer, B.; Neumann, A. U.; Perelson, A. S.: Influence of delayed viral production on viral dynamics in HIV-1 infected patients, Math. biosci. 152, 143-163 (1998) · Zbl 0946.92011 · doi:10.1016/S0025-5564(98)10027-5
[14]Nelson, P. W.; Murray, J. D.; Perelson, A. S.: A model of HIV-1 pathogenesis that includes an intracellular delay, Math. biosci. 163, 201-215 (2000) · Zbl 0942.92017 · doi:10.1016/S0025-5564(99)00055-3
[15]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
[16]Nowak, M.; Anderson, R.; Boerlijst, M.; Bonhoeffer, S.; May, R.; Mcmichael, A.: HIV-1 evolution and disease progression, Science 274, 1008-1010 (1996)
[17]Nowak, M.; Bonhoeffer, S.; Shaw, G.; May, R.: Anti-viral drug treatment: dynamics of resistance in free virus and infected cell populations, J. theoret. Biol. 184, 203-217 (1997)
[18]Perelson, A.; Kirschner, D.; De Boer, R.: Dynamics of HIV infection of CD4+ T cells, Math. biosci. 114, 81-125 (1993) · Zbl 0796.92016 · doi:10.1016/0025-5564(93)90043-A
[19]Perelson, A.; Nelson, P.: Mathematical analysis of HIV-1 dynamics in vivo, SIAM rev. 41, 3-44 (1999) · Zbl 1078.92502 · doi:10.1137/S0036144598335107
[20]Perelson, A.; Neumann, A.; Markowitz, M.; Leonard, J.; Ho, D.: HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time, Science 271, 1582-1586 (1996)
[21]Regoes, R. R.; Ebert, D.; Bonhoeffer, S.: Dose-dependent infection rates of parasites produce the allee effect in epidemiology, Proc. R. Soc. lond. Ser. B 269, 271-279 (2002)
[22]Song, X.; Neumann, Avidan 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
[23]Tam, J.: Delay effect in a model for virus replication, IMA J. Math. appl. Med. biol. 16, 29-37 (1999) · Zbl 0914.92012
[24]Wang, K.; Wang, W.; Pang, H.; Liu, X.: Complex dynamic behavior in a viral model with delayed immune response, Phys. D 226, 197-208 (2007) · Zbl 1117.34081 · doi:10.1016/j.physd.2006.12.001