Global stability and periodic solution of the viral dynamics. (English) Zbl 1105.92011

Summary: It is well known that mathematical models provide very important information for the research of human immunodeficiency virus-type 1 and hepatitis C virus (HCV). However, the infection rate of almost all mathematical models is linear. The linearity shows the simple interaction between the T cells and the viral particles.
We consider the classical mathematical model with saturation response of the infection rate. By stability analysis we obtain sufficient conditions on the parameters for the global stability of the infected steady state and the infection-free steady state. We also obtain the conditions for the existence of an orbitally asymptotically stable periodic solution. Numerical simulations are presented to illustrate the results.


92C60 Medical epidemiology
34C25 Periodic solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
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[1] Neumann, A.; Lam, N.; Dahari, H.; Gretch, D.; Wiley, T.; Layden, T.; Perelson, A., Hepatitis C viral dynamics in vivo and antiviral efficacy of the interferon-\(α\) therapy, Science, 282, 103-107 (1998)
[2] Wei, X.; Ghosh, S.; Taylor, M.; Johnson, V.; Emini, E.; Deutsch, P.; Lifson, J.; Bonhoeffer, S.; Nowak, M.; Hahn, B.; Saag, S.; Shaw, G., Viral dynamics in human immunodeficiency virus type 1 infection, Nature, 373, 117 (1995)
[3] Perelson, A.; Neumann, A.; Markowitz, M.; Leonard, J.; Ho, D., HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time, Science, 271, 1582 (1996)
[4] Perelson, A.; Essunger, P.; Cao, Y.; Vesanen, M.; Hurley, A.; Saksela, K.; Markowitz, M.; Ho, D., Decay characteristics of HIV-1-infected compartments during combination therapy, Nature, 387, 188 (1997)
[5] Ho, D.; Neumann, A.; Perelson, A.; Chen, W.; Leonard, J.; Markowitz, M., Rapid turnover of plasma virions and \(CD 4^+\) lymphocytes in HIV-1 infection, Nature, 373, 123-126 (1995)
[6] Perelson, A.; Nelson, P., Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41, 1, 3-44 (1999) · Zbl 1078.92502
[7] Nowak, M. A.; Bonhoeffer, S.; Hill, A. M.; Boehme, R.; Thomas, H. C.; McDade, H., Viral dynamics in hepatitis B virus infection, Proc. Natl. Acad. Sci. USA, 93, 4398-4402 (1996)
[8] Nowak, M. A.; Lloyd, A. L.; Vadquez, G. M., Viral dynamics of primary viremia and antitroviral therapy in simian immunodeficiency virus infection, J. Virology, 71, 7518-7525 (1997)
[9] Hirsch, M. W., System of differential equations which are competitive or cooperative, IV, SIAM J. Math. Anal., 21, 1225-1234 (1990) · Zbl 0734.34042
[10] Butler, G.; Freedman, H. I.; Waltman, P., Uniform persistence system, Proc. Amer. Math. Soc., 96, 425-430 (1986)
[11] Zhu, H. R.; Smith, H. L., Stable periodic orbits for a class of three-dimensional competitive systems, J. Differential Equations, 110, 143-156 (1994) · Zbl 0802.34064
[12] Smith, H. L.; Thieme, H., Convergence for strongly ordered preserving semiflows, SIAM J. Math. Anal., 22, 1081-1101 (1991) · Zbl 0739.34040
[13] Muldowney, J. S., Compound matrices and ordinary differential equations, Rocky Mountain J. Math., 20, 857-872 (1990) · Zbl 0725.34049
[14] Li, Y.; Muldowney, J. S., Global stability for the SEIR model in epidemiology, Math. Biosci., 125, 155-164 (1995) · Zbl 0821.92022
[15] Perelson, A. S.; Essunger, P.; Ho, D. D., Dynamics of HIV-1 and \(CD 4^+\) lymphocytes in vivo, AIDS, 11, Suppl. A, S17-S24 (1997)
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