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Global stability for a class of virus models with cytotoxic T Lymphocyte immune response and antigenic variation. (English) Zbl 1225.92022
Summary: We study the global stability of a class of models for in-vivo virus dynamics that take into account the cytotoxic T Lymphocyte immune response and display antigenic variation. This class includes a number of models that have been extensively used to model HIV dynamics. We show that models in this class are globally asymptotically stable, under mild hypothesis, by using appropriate Lyapunov functions. We also characterise the stable equilibrium points for the entire biologically relevant parameter range. As a by-product, we are able to determine what is the diversity of the persistent strains.

92C50 Medical applications (general)
92C60 Medical epidemiology
34D23 Global stability of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
37N25 Dynamical systems in biology
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[1] Asquith, B., Bangham, C.R.M. (2003). An introduction to lymphocyte and viral dynamics: the power and limitations of mathematical analysis. Proc. R. Soc. Lond. Ser. B: Biol. Sci., 270(1525), 1651–1657. · doi:10.1098/rspb.2003.2386
[2] Bocharov, G.A., Romanyukha, A.A. (1994). Mathematical-model of antiviral immune-response-iii–influenza-A Virus-infection. J. Theor. Biol., 167(4), 323–360. · doi:10.1006/jtbi.1994.1074
[3] Bonhoeffer, S., et al. (1997). Virus dynamics and drug therapy. Proc. Natl. Acad. Sci. USA, 94, 6971–6974. · doi:10.1073/pnas.94.13.6971
[4] de Leenheer, P., Smith, H.L. (2003). Virus dynamics: a global analysis. SIAM J. Appl. Math., 63, 1313–1327. · Zbl 1035.34045 · doi:10.1137/S0036139902406905
[5] Janeway, C., et al. (2004). Immunobiology (6th ed.). New York: Garland Science.
[6] Kooi, B., Hanegraaf, P. (2001). Bi-trophic food chain dynamics with multiple component populations. Bull. Math. Biol., 63(2), 271–299. · Zbl 1323.92175 · doi:10.1006/bulm.2000.0219
[7] Kooi, B., et al. (1998). On the use of the logistic equation in models of food chains. Bull. Math. Biol., 60, 231–246. · Zbl 1053.92520 · doi:10.1006/bulm.1997.0016
[8] Korobeinikov, A. (2004a). Global properties of basic virus dynamics models. Bull. Math. Biol., 66, 879–883. · Zbl 1334.92409 · doi:10.1016/j.bulm.2004.02.001
[9] Korobeinikov, A. (2004b). Lyapunov functions and global properties for SEIR and SEIS epidemic models. Math. Med. Biol., 21(2), 75–83. · Zbl 1055.92051 · doi:10.1093/imammb/21.2.75
[10] Korobeinikov, A. (2009). Global asymptotic properties of virus dynamics models with dose-dependent parasite reproduction and virulence and non-linear incidence rate. Math. Med. Biol., 26, 225–239. · Zbl 1171.92034 · doi:10.1093/imammb/dqp006
[11] Korebeinikov, A., Wake, G.C. (1999). Global properties of three-dimensional predator-prey models. J. Appl. Math. Decis. Sci., 3(2), 155–162. · Zbl 0951.92031 · doi:10.1155/S1173912699000085
[12] LaSalle, J.P. (1964). Recent advances in Liapunov stability theory. SIAM Rev., 6, 1–11. · Zbl 0126.29809 · doi:10.1137/1006001
[13] Li, M.Y., Muldowney, J.S. (1995). Global stability for the seir model in epidemiology. Math. Biosci., 125(2), 155–164. · Zbl 0821.92022 · doi:10.1016/0025-5564(95)92756-5
[14] Marchuk, G.I., et al. (1991). Mathematical-model of antiviral immune-response. 2. Parameters identification for acute viral Hepatitis-B. J. Theor. Biol., 151(1), 41–70. · doi:10.1016/S0022-5193(05)80143-2
[15] Neumann, A.U., et al. (1998). Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon-alpha therapy. Science, 282(5386), 103–107. · doi:10.1126/science.282.5386.103
[16] Nowak, M.A., Bangham, C.R.M. (1996). Population dynamics of immune responses to persitent viruses. Science, 272, 74–79. · doi:10.1126/science.272.5258.74
[17] Nowak, M.A., May, R.M. (2000). Virus dynamics: mathematical principles of immunology and virology. Oxford: Oxford University Press. · Zbl 1101.92028
[18] Pastore, D.H. (2005). A Dinâmica no sistema imunológico na presença de mutação. Ph.D. thesis, IMPA.
[19] Perelson, A.S., Nelson, P.W. (1999). Mathematical analysis of HIV-1 dynamics in vivo. SIAM Rev., 41, 3–44. · Zbl 1078.92502 · doi:10.1137/S0036144598335107
[20] Perelson, A.S., et al. (1993). Dynamics of HIV-infection of Cd4+ T-cells. Math. Biosci., 114(1), 81–125. · Zbl 0796.92016 · doi:10.1016/0025-5564(93)90043-A
[21] Perelson, A.S., et al. (1996). HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time. Science, 271(5255), 1582–1586. · doi:10.1126/science.271.5255.1582
[22] Roy, A.B., Solimano, F. (1986). Global stability of partially closed food-chains with resources. Bull. Math. Biol., 48(5–6), 455–468. · Zbl 0613.92026
[23] Smith, H.L. (1995). Monotone dynamical systems. Providence: AMS. · Zbl 0821.34003
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