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Global stability of an HIV pathogenesis model with cure rate. (English) Zbl 1231.34094
Summary: We consider an HIV pathogenesis model including cure rate and the full logistic proliferation term of \(CD4^{+}\) T cells in healthy and infected populations. Let \(N\) be the number of virus released by each productive infected \(CD4^{+}\) T cell. The critical number that ensures the existence of the positive equilibrium is obtained. We further show that if , then there exists a unique uninfected equilibrium point \(E_{0}\) that is locally asymptotically stable. If , then the system is persistent and the only infected steady state \(E^{\ast }\) is globally asymptotically stable in the feasible region. Numerical simulations are presented to illustrate the obtained main results. Moreover, we find that there exist periodic solutions when the infected steady state \(E^{\ast }\) is unstable.

MSC:
34D23 Global stability of solutions to ordinary differential equations
92D30 Epidemiology
34C11 Growth and boundedness of solutions to ordinary differential equations
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[1] Gao, S.J.; Teng, Z.D.; Nieto, J.J.; Torres, A., Analysis of an SIR epidemic model with pulse vaccination and distributed time delay, Journal of biomedicine and biotechnology, 2007, (2007), Article ID 64870, 10 pages
[2] Cai, L.M.; Li, X.Z.; Ghoshc, M.; Guo, B.Z., Stability analysis of an HIV/AIDS epidemic model with treatment, Journal of computational and applied mathematics, 229, 313-323, (2009) · Zbl 1162.92035
[3] Samanta, G.P., Permanence and extinction of a nonautonomous HIV/AIDS epidemic model with distributed time delay, Nonlinear analysis: real world applications, 12, 1163-1177, (2011) · Zbl 1203.92051
[4] 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)
[5] 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, Journal of theoretical biology, 184, 203-217, (1997)
[6] Bonhoeffer, S.; May, R.M.; Shaw, G.M.; Nowak, M.A., Virus dynamics and drug therapy, Proceedings of national Academy of sciences of the united states of amercia, 94, 6971-6976, (1997)
[7] Stafford, M.A.; Corey, L.; Cao, Y.Z.; Daar, E.S.; Ho, D.D.; Perelson, A.S., Modeling plasma virus concentration during primary HIV infection, Journal of theoretical biology, 203, 285-293, (2000)
[8] Yu, Y.M.; Nieto, J.J.; Torres, A.; Wang, K.F., A viral infection model with a nonlinear infection rate, Boundary value problems, 2009, (2009), Article ID 958016, 19 pages · Zbl 1187.34062
[9] Xu, R., Global stability of an HIV-1 infection model with saturation infection and intracellular delay, Journal of mathematical analysis and applications, 375, 75-81, (2011) · Zbl 1222.34101
[10] Elaiw, A.M., Global properties of a class of HIV models, Nonlinear analysis: real world applications, 11, 2253-2263, (2010) · Zbl 1197.34073
[11] Ho, D.D.; Neumann, A.U.; Perelson, A.S.; Chen, W.; Leonard, J.M.; Markowitz, M., Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection, Nature, 373, 117-122, (1995)
[12] Sachsenberg, N.; Perelson, A.S.; Yerly, S.; Schockmel, G.A.; Leduc, D.; Hirschel, B.; Perrin, L., Turnover of CD4^+ and CD8+ T lymphocytes in HIV-1 infection as measured by ki-67 antigen, The journal of experimental medicine, 187, 1295-1303, (1998)
[13] Perelson, A.S.; Nelson, P.W., Mathematical analysis of HIV-1 dynamics in vivo, SIAM review, 41, 3-44, (1999) · Zbl 1078.92502
[14] Leenheer, P.D.; Smith, H.L., Virus dynamics: a global analysis, SIAM journal on applied mathematics, 63, 1313-1327, (2003) · Zbl 1035.34045
[15] Song, X.Y.; Neumann, A.U., Global stability and periodic solution of the viral dynamics, Journal of mathematical analysis and applications, 329, 281-297, (2007) · Zbl 1105.92011
[16] Wang, L.C.; Li, M.Y., Mathematical analysis of the global dynamics of a model for HIV infection of CD4^+ T cells, Mathematical biosciences, 200, 44-57, (2006) · Zbl 1086.92035
[17] Wang, X.; Song, X.Y., Global stability and periodic solution of a model for HIV infection of CD4^+ T cells, Applied mathematics and computation, 189, 1331-1340, (2007) · Zbl 1117.92040
[18] Culshaw, R.V.; Ruan, S.G., A delay-differential equation model of HIV infection of CD4^+ T-cells, Mathematical biosciences, 165, 27-39, (2000) · Zbl 0981.92009
[19] Leonard, R.; Zagury, D.; Bernard, J.; Zagury, J.F.; Gallo, R.C., Cytopathic effect of human immunodeficiency virus in T4 cells is linked to the last stage of virus infection, Proceedings of national Academy of sciences of the united states of amercia, 85, 3570-3574, (1988)
[20] Wodarz, D.; Hamer, D.H., Infection dynamic in HIV-specific CD4^+ T cells, Mathematical biosciences, 209, 14-29, (2007) · Zbl 1120.92026
[21] Wang, L.C.; Ellermeyer, S., HIV infection and CD4^+ T cell dynamics, Discrete and continuous dynanmical system, 6, 1417-1430, (2006) · Zbl 1123.92033
[22] Cai, L.M.; Li, X.Z., Stability of Hopf bifurcation in a delayed model for HIV infection of CD4^+ T-cells, Chaos, solitons & fractals, 42, 1-11, (2009) · Zbl 1198.37119
[23] Hu, Z.X.; Liu, X.D.; Wang, H.; Ma, W.B., Analysis of the dynamics of a delayed HIV pathogenesis model, Journal of computational and applied mathematics, 234, 461-476, (2010) · Zbl 1185.92062
[24] Guidotti, L.G.; Rochford, R.; Chung, J.; Shapiro, M.; Purcell, R.; Chisari, F.V., Viral clearance without destruction of infected cells during acute HBV infection, Science, 284, 825-829, (1999)
[25] Lewin, S.; Walters, T.; Locarnini, S., Hepatitis B treatment: rational combination chemotherapy based on viral kinetic and animal model studies, Antiviral research, 35, 381-396, (2002)
[26] Wang, K.F.; Fan, A.J.; Torres, A., Global properties of an improved hepatitis B virus model, Nonlinear analysis: real world applications, 11, 3131-3138, (2010) · Zbl 1197.34081
[27] Zhou, X.Y.; Song, X.Y.; Shi, X.Y., A differential equation model of HIV infection of CD4^+ T-cells with cure rate, Journal of mathematical analysis and applications, 342, 1342-1355, (2008) · Zbl 1188.34062
[28] Zack, J.A.; Arrigo, S.J.; Weitsman, S.R.; Go, A.S.; Haislip, A.; Chen, I.S., HIV-1 entry into quiescent primary lymphocytes: molecular analysis reveals a labile latent viral structure, Cell, 61, 213-222, (1990)
[29] Zack, J.A.; Haislip, A.M.; Krogstad, P.; Chen, I.S., Incompletely reverse-transcribed human immunodeficiency virus type 1 genomes in quiescent cells can function as intermediates in the retroviral cycle, Journal of virology, 66, 1717-1725, (1992)
[30] Essunger, P.; Perelson, A.S., Modeling HIV infection of CD4^+ T-cell subpopulations, Journal of theoretical biology, 170, 367-391, (1994)
[31] Srivastava, P.K.; Chandra, P., Modeling the dynamics of HIV and CD4^+ T cells during primary infection, Nonlinear analysis: real world applications, 11, 612-618, (2010) · Zbl 1181.37122
[32] Hale, J.K.; Waltman, P., Persistence in infinite-dimensional systems, SIAM journal on mathematical analysis, 20, 388-395, (1989) · Zbl 0692.34053
[33] Li, M.Y.; Muldowney, J.S., A geometric approach to the global stability problems, SIAM journal on mathematical analysis, 27, 1070-1083, (1996) · Zbl 0873.34041
[34] Muldowney, J.S., Compound matrices and ordinary differential equations, Rocky mountain journal of mathematics, 20, 857-872, (1990) · Zbl 0725.34049
[35] Fiedler, M., Additive compound matrices and inequality for eigenvalues of stochastic matrices, Czechoslovak mathematical journal, 99, 392-402, (1974) · Zbl 0345.15013
[36] Coppel, W.A., Stability and asymptotic behavior of differential equations, (1965), Health Boston · Zbl 0154.09301
[37] Butler, G.J.; Waltman, P., Persistence in dynamical system, Journal of differential equations, 63, 255-263, (1986) · Zbl 0603.58033
[38] Waltman, P., A brief survey of persistence, Delay differential equations and dynamical systems, 1475, 31-40, (1991) · Zbl 0756.34054
[39] Martin, R.H., Logarithmic norms and projections applied to linear differential systems, Journal of mathematical analysis and applications, 45, 432-442, (1974) · Zbl 0293.34018
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