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Global stability for an HIV-1 infection model including an eclipse stage of infected cells. (English) Zbl 1223.92024
Summary: We consider the mathematical model for the viral dynamics of HIV-1 introduced by L. Rong et al. [J. Theor. Biol. 247, 804–818 (2007); see also Bull. Math. Biol. 69, No. 6, 2027–2060 (2007; Zbl 1298.92053)]. One main feature of this model is that an eclipse stage for the infected cells is included and cells in this stage may revert to the uninfected class. The viral dynamics is described by four nonlinear ordinary differential equations. Rong et al. have analyzed the stability of the infected equilibrium locally. We perform a global stability analysis using two techniques, the Lyapunov direct method and the geometric approach to stability, based on a higher-order generalization of Bendixson’s criterion. We obtain sufficient conditions in terms of the system parameters. Numerical simulations are also provided to give a more complete representation of the system dynamics.

92C50 Medical applications (general)
34D23 Global stability of solutions to ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
65C20 Probabilistic models, generic numerical methods in probability and statistics
Full Text: DOI
[1] Althaus, C.L.; De Vos, A.S.; De Boer, R.J., Reassessing the human immunodeficiency virus type 1 life cycle through age-structured modeling: life span of infected cells, viral generation time, and basic reproductive number, \(R_0\), J. virol., 83, 7659-7667, (2009)
[2] Althaus, C.L.; De Boer, R.J., Implications of CTL-mediated Killing of HIV-infected cells during the non-productive stage of infection, Plos one, 6, 2, e16468, (2011)
[3] Anderson, R.M.; May, R.M., Infectious diseases in humans: dynamics and control, (1991), Oxford University Press Oxford
[4] 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
[5] Ballyk, M.M.; McCluskey, C.C.; Wolkowicz, G.K.S., Global analysis of competition for perfectly substitutable resources with linear response, J. math. biol., 51, 458-490, (2005) · Zbl 1090.92045
[6] Beretta, E.; Solimano, F.; Takeuchi, Y., Negative criteria for the existence of periodic solutions in a class of delay-differential equations, Nonlinear anal., 50, 941-966, (2002) · Zbl 1087.34542
[7] Buonomo, B.; dʼOnofrio, A.; Lacitignola, D., Global stability of an SIR epidemic model with information dependent vaccination, Math. biosci., 216, 9-16, (2008) · Zbl 1152.92019
[8] Buonomo, B.; Lacitignola, D., General conditions for global stability in a single species population-toxicant model, Nonlinear anal. real world appl., 5, 749-762, (2004) · Zbl 1074.92036
[9] Buonomo, B.; Lacitignola, D., On the use of the geometric approach to global stability for three-dimensional ODE systems: a bilinear case, J. math. anal. appl., 348, 255-266, (2008) · Zbl 1158.34033
[10] Buonomo, B.; Lacitignola, D., On the dynamics of an SEIR epidemic model with a convex incidence rate, Ric. mat., 57, 261-281, (2008) · Zbl 1232.34061
[11] Buonomo, B.; Lacitignola, D., Analysis of a tuberculosis model with a case study in uganda, J. biol. dyn., 4, 571-593, (2010) · Zbl 1403.92286
[12] Buonomo, B.; Lacitignola, D., Global stability for a four dimensional epidemic model, Note mat., 30, 81-93, (2010)
[13] Dieckmann, O.; Heesterbeek, J.A.P., Mathematical epidemiology of infectious diseases, Wiley series in mathematical and computational biology, (2000), Wiley West Sussex · Zbl 0997.92505
[14] Freedman, H.I.; Ruan, S.; Tang, M., Uniform persistence and flows near a closed positively invariant set, J. differential equations, 6, 583-600, (1994) · Zbl 0811.34033
[15] Funk, G.A.; Fischer, M.; Joos, B.; Opravil, M.; Guenthard, H.F.; Ledergerber, B.; Bonhoeffer, S., Quantification of in vivo replicative capacity of HIV-1 in different compartments of infected cells, J. acquir. immune defic. syndr., 26, 397-404, (2001)
[16] Gumel, A.B.; McCluskey, C.C.; Watmough, J., Modelling the potential impact of a SARS vaccine, Math. biosci. eng., 3, 485-512, (2006) · Zbl 1092.92039
[17] Hutson, V.; Schmitt, K., Permanence and the dynamics of biological systems, Math. biosci., 111, 1-71, (1992) · Zbl 0783.92002
[18] Korobeinikov, A., Global properties of basic virus dynamics models, Bull. math. biol., 66, 879-883, (2004) · Zbl 1334.92409
[19] Korobeinikov, A., Lyapunov functions and global properties for SEIR and SEIS epidemic models, IMA J. math. appl. med. biol., 21, 75-83, (2004) · Zbl 1055.92051
[20] Korobeinikov, A.; Wake, G.C., Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Appl. math. lett., 15, 955-960, (2002) · Zbl 1022.34044
[21] Li, M.Y.; Graef, J.R.; Wang, L.; Karsai, J., Global stability of a SEIR model with varying total population size, Math. biosci., 160, 191-213, (1999) · Zbl 0974.92029
[22] Li, M.Y.; Muldowney, J.S., On bendixsonʼs criterion, J. differential equations, 106, 27-39, (1993)
[23] Li, M.Y.; Muldowney, J.S., On R.A. smithʼs autonomous convergence theorem, Rocky mountain J. math., 25, 365-379, (1995) · Zbl 0841.34052
[24] Li, M.Y.; Muldowney, J.S., Global stability for the SEIR model in epidemiology, Math. biosci., 125, 155-164, (1995) · Zbl 0821.92022
[25] Li, M.Y.; Muldowney, J.S., A geometric approach to global-stability problems, SIAM J. math. anal., 27, 1070-1083, (1996) · Zbl 0873.34041
[26] Li, M.Y.; Smith, H.L.; Wang, L., Global dynamics of an SEIR epidemic model with vertical transmission, SIAM J. math. anal., 62, 58-69, (2001) · Zbl 0991.92029
[27] Lyapunov, A.M., The general problem of the stability of motion, (1992), Taylor & Francis London · Zbl 0786.70001
[28] Margheri, A.; Rebelo, C., Some examples of persistence in epidemiological models, J. math. biol., 46, 564-570, (2003) · Zbl 1023.92027
[29] Martin, R.H., Logarithmic norms and projections applied to linear differential systems, J. math. anal. appl., 45, 432-454, (1974) · Zbl 0293.34018
[30] McCluskey, C.C., A strategy for constructing Lyapunov functions for non-autonomous linear differential equations, Linear algebra appl., 409, 100-110, (2005) · Zbl 1076.37012
[31] McCluskey, C.C., Global stability for a class of mass action systems allowing for latency in tuberculosis, J. math. anal. appl., 338, 518-535, (2008) · Zbl 1131.92042
[32] Nelson, P.W.; Gilchrist, M.A.; Coombs, D.; Hyman, J.M.; Perelson, A.S., An age structured model of HIV infection that allows for variations in the production rate of viral particles and the death rate of productively infected cells, Math. biosci. eng., 1, 267-288, (2004) · Zbl 1060.92038
[33] Nowak, M.A.; May, R.M., Virus dynamics: mathematical principles of immunology and virology, (2000), Oxford University Press Oxford · Zbl 1101.92028
[34] 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)
[35] Perelson, A.S.; Nelson, P.W., Mathematical analysis of HIV-1 dynamics in vivo, SIAM rev., 41, 3-44, (1999) · Zbl 1078.92502
[36] Perelson, A.S., Modelling viral and immune system dynamics, Nat. rev. immunol., 2, 28-36, (2002)
[37] Rong, L.; Gilchrist, M.A.; Feng, Z.; Perelson, A.S., Modeling within-host HIV-1 dynamics and the evolution of drug resistance: trade-offs between viral enzyme function and drug susceptibility, J. theoret. biol., 247, 804-818, (2007)
[38] Rong, L.; Feng, Z.; Perelson, A.S., Mathematical analysis of age-structured HIV-1 dynamics with combination antiretroviral therapy, SIAM J. appl. math., 67, 731-756, (2007) · Zbl 1121.92043
[39] Tumwiine, J.; Mugisha, J.Y.T.; Luboobi, L.S., A host-vector model for malaria with infective immigrants, J. math. anal. appl., 361, 139-149, (2010) · Zbl 1176.92045
[40] 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
[41] Vargas-De-León, C., Constructions of Lyapunov functions for classic SIS, SIR and SIRS epidemic models with variable population size, Foro-red-mat: revista electrónica de contenido matemático, 26, (2009)
[42] C. Vargas-De-León, Analysis of a model for the dynamics of hepatitis B with noncytolytic loss of infected cells, World J. Model. Simul., submitted for publication.
[43] Wang, L.; Li, M.Y., Mathematical analysis of the global dynamics of a model for HIV infection of \(\operatorname{C} \operatorname{D} 4^+ \operatorname{T}\) cells, Math. biosci., 200, 44-57, (2006)
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