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Role of \(CD4^{+}\) T-cell proliferation in HIV infection under antiretroviral therapy. (English) Zbl 1275.92033
The authors study a mathematical model describing the interaction between HIV viruses and CD4+ T cells. Both RT and protease inhibitor treatments are incorporated in the model. They assume that the CD4+ T cells proliferate according to a saturation form which is different from the commonly used logistic proliferation form in the literature. Both local and global analyses are carried out. A backward bifurcation has been observed. The global stability of the unique infected equilibrium has been established by using a new criterion developed by M.Y. Li and J.S. Muldowney which has been successfully utilized by some researchers for some high dimensional systems.

MSC:
92C50 Medical applications (general)
92C60 Medical epidemiology
34C60 Qualitative investigation and simulation of ordinary differential equation models
37N25 Dynamical systems in biology
34C23 Bifurcation theory for ordinary differential equations
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[1] Weiss, R.A., How does HIV cause AIDS?, Science, 260, 1273-1279, (1993)
[2] UNAIDS, 2010 report on the global AIDS epidemic.
[3] Gray, R.T.; Zhang, L.; Lupiwa, T.; Wilson, D.P., Forecasting the population-level impact of reductions in HIV antiretroviral therapy in papua new guinea, AIDS res. treatment, 2011, 8, (2011), Article ID 891593
[4] Magnus, C.; Regoes, R.R., Restricted occupancy models for neutralization of HIV virions and populations, J. theoret. biol., 283, 192-202, (2011) · Zbl 1397.92655
[5] Wodarz, D.; Hamer, D.H., Infection dynamics in HIV-specific CD4 T cells: does a CD4 T cell boost benefit the host or the virus?, Math. biosci., 209, 14-29, (2007) · Zbl 1120.92026
[6] Bonhoeffer, S.; Coffin, J.M.; Nowak, M.A., Human immunodeficiency virus drug therapy and virus load, J. virol., 71, 3275-3278, (1997)
[7] 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)
[8] Nowak, M.A.; Bonhoeffer, S.; Shaw, G.M.; May, R.M., Anti-viral drug treatment: dynamics of resistance in free virus and infected cell populations, J. theoret. biol., 184, 203-217, (1997)
[9] Perelson, A.S.; Nelson, P.W., Mathematical analysis of HIV-1 dynamics in vivo, SIAM rev., 41, 3-44, (1999) · Zbl 1078.92502
[10] Perelson, A.S.; Neumann, A.U.; Markowitz, M.; Leonard, J.M.; Ho, D.D., HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time, Science, 271, 1582-1586, (1996)
[11] Kepler, T.B.; Perelson, A.S., Drug concentration heterogeneity facilitates the evolution of drug resistance, Proc. natl. acad. sci. USA, 95, 11514-11519, (1998) · Zbl 0919.92023
[12] Nelson, P.W.; Mittler, J.E.; Perelson, A.S., Effect of drug efficacy and the eclipse phase of the viral life cycle on the estimates of HIV viral dynamic parameters, Journal of aids, 26, 405-412, (2001)
[13] Revilla, T.; García-Ramos, G., Fighting a virus with a virus: a dynamic model for HIV-1 therapy, Math. biosci., 185, 191-203, (2003) · Zbl 1021.92015
[14] Nowak, M.A.; May, R.M., Virus dynamics, (2000), Cambridge University Press Cambridge · Zbl 1101.92028
[15] De Leenheer, P.; Smith, H.L., Virus dynamics: a global analysis, SIAM J. appl. math., 63, 1313-1327, (2003) · Zbl 1035.34045
[16] Dixit, N.M.; Perelson, A.S., Complex patternsof viral load decay under antiretroviral therapy: influence of pharmacokineticsand intracellular delay, J. theoret. biol., 226, 95-109, (2004)
[17] De Boer, R.J.; Perelson, A.S., Target cell limited and immune control models of HIV infection: a comparison, J. theoret. biol., 190, 201-214, (1998)
[18] Li, M.Y.; Shu, H., Global dynamics of a mathematical model for HTLV-I infection of CD4^+ T cells with delayed CTL response, Nonlinear anal. RWA, 13, 1080-1092, (2012) · Zbl 1239.34086
[19] Nowak, M.A.; Bangham, C.R.M., Population dynamics of immune responses to persistent viruses, Science, 272, 74-79, (1996)
[20] Van Gulck, E.; Vlieghe, E.; Vekemans, M.; Van Tendeloo, V.F.; Van De Velde, A.; Smits, E.; Anguille, S.; Cools, N.; Goossens, H.; Mertens, L.; De Haes, W.; Wong, J.; Florence, E.; Vanham, G.; Berneman, Z.N., MRNA-based dendritic cell vaccination induces potent antiviral T-cell responses in HIV-1- infected patients, Aids, 26, F1-F12, (2012)
[21] Korobeinikov, A., Global properties of basic virus dynamics models, Bull. math. biol., 66, 879-883, (2004) · Zbl 1334.92409
[22] Li, M.Y.; Shu, H., Joint effects of mitosis and intracellular delay on viral dynamics: two-parameter bifurcation analysis, J. math. biol., 64, 1005-1020, (2012) · Zbl 1303.92060
[23] Wang, L.; Li, M.Y., Mathematical analysis of the global dynamics of a model for HIV infection of CD4^+ T cells, Math. biosci., 200, 44-57, (2006) · Zbl 1086.92035
[24] Arino, J.; McCluskey, C.C.; van den Driessche, P., Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J. appl. math., 64, 260-276, (2003) · Zbl 1034.92025
[25] Dushoff, J.; Huang, W.; Castillo-Chavez, C., Backwards bifurcations and catastrophe in simple models of fatal diseases, J. math. biol., 36, 227-248, (1998) · Zbl 0917.92022
[26] Gömez-Acevedo, H.; Li, M.Y., Backward bifurcation in a model for HTLV-I infection of CD4^+ T cells, Bull. math. biol., 67, 101-114, (2005) · Zbl 1334.92231
[27] Qesmi, R.; Wu, J.; Wu, J.; Heffernan, J.M., Influence of backward bifurcation in a model of hepatitis B and C viruses, Math. biosci., 224, 118-125, (2010) · Zbl 1188.92017
[28] Sharomi, O.; Podder, C.N.; Gumel, A.B.; Elbasha, E.H.; Watmough, J., Role of incidence function in vaccine-induced backward bifurcation in some HIV models, Math. biosci., 210, 436-463, (2007) · Zbl 1134.92026
[29] van den Driessche, P.; Watmough, J., A simple SIS epidemic model with a backward bifurcation, J. math. biol., 40, 525-540, (2000) · Zbl 0961.92029
[30] Buonomo, B.; Varga-De-León, C., Global stability for an HIV-1 infection model including an eclipse stage of infected cells, J. math. anal. appl., 385, 709-720, (2012) · Zbl 1223.92024
[31] Liu, S.; Wang, L., Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy, Math. biosci. eng., 7, 675-685, (2010) · Zbl 1260.92065
[32] Liu, X.; Wang, H.; Hu, Z.; Ma, W., Global stability of an HIV pathogenesis model with cure rate, Nonlinear anal. RWA, 12, 2947-2961, (2011) · Zbl 1231.34094
[33] Samanta, G.P., Permanence and extinction of a nonautonomous HIV/AIDS epidemic model with distributed time delay, Nonlinear anal. RWA, 12, 1163-1177, (2011) · Zbl 1203.92051
[34] Xu, R., Global stability of an HIV-1 infection model with saturation infection and intracellular delay, J. math. anal. appl., 375, 75-81, (2011) · Zbl 1222.34101
[35] Février, M.; Dorgham, K.; Rebollo, A., CD4^+ T cell depletion in human immunodeficiency virus (HIV) infection: role of apoptosis, Viruses, 3, 586-612, (2011)
[36] Kirschner, D., Using mathematics to understand HIV immune dynamics, Notices amer. math. soc., 43, 191-202, (1996) · Zbl 1044.92503
[37] van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. biosci., 180, 29-48, (2002) · Zbl 1015.92036
[38] LaSalle, J.P., ()
[39] Freedman, H.I.; Ruan, S.G.; Tang, M.X., Uniform persistence and flows near a closed positively invariant set, J. dynam. differential equations, 6, 583-600, (1994) · Zbl 0811.34033
[40] Li, M.Y.; Graef, J.R.; Wang, L.; Karsai, J., Global dynamics of a SEIR model with a varying total population size, Math. biosci., 160, 191-213, (1999) · Zbl 0974.92029
[41] Li, M.Y.; Muldowney, J.S., A geometric approach to the global-stability problems, SIAM J. math. anal., 27, 1070-1083, (1996) · Zbl 0873.34041
[42] Li, Y.; Muldowney, J.S., On bendixson’s criterion, J. differential equations, 106, 27-39, (1993) · Zbl 0786.34033
[43] Muldowney, J.S., Compound matrices and ordinary differential equations, Rocky mountain J. math., 20, 857-871, (1990) · Zbl 0725.34049
[44] Coppel, W.A., Stability and asymptotic behavior of differential equations, (1995), Health Boston · Zbl 0838.52014
[45] Butler, G.; Waltman, P., Persistence in dynamical systems, J. differential equations, 63, 255-263, (1986) · Zbl 0603.58033
[46] Waltman, P., A brief survey of persistence, () · Zbl 0756.34054
[47] Martin, R.H., Logarithmic norms and projections applied to linear differential systems, J. math. anal. appl., 45, 432-454, (1974) · Zbl 0293.34018
[48] Heffernan, J.M.; Wahl, L.M., Monte Carlo estimates of natural variation in HIV infection, J. theoret. biol., 236, 137-153, (2005)
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