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.


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
Full Text: DOI


[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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