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Mathematical analysis of the global dynamics of a model for HIV infection of CD4$^{+}$ T cells. (English) Zbl 1086.92035
Summary: A mathematical model that describes HIV infection of CD4$^{+}$ T cells is analyzed. Global dynamics of the model is rigorously established. We prove that, if the basic reproduction number $R_{0}\le 1$, the HIV infection is cleared from the T-cell population; if $R_{0} > 1$, the HIV infection persists. For an open set of parameter values, the chronic-infection equilibrium $P^*$ can be unstable and periodic solutions may exist. We establish parameter regions for which $P^*$ is globally stable.

MSC:
 92C60 Medical epidemiology 34D05 Asymptotic stability of ODE 34D23 Global stability of ODE 37N25 Dynamical systems in biology
Full Text:
References:
 [1] Perelson, A. S.; Kirschner, D. E.; De Boer, R.: Dynamics of HIV infection of CD4+ T cells. Math. biosci. 114, 81 (1993) · Zbl 0796.92016 [2] Perelson, A. S.; Nelson, P. W.: Mathematical analysis of HIV-I dynamics in vivo. SIAM rev. 41, 3 (1999) · Zbl 1078.92502 [3] Nelson, P. W.; Perelson, A. S.: Mathematical analysis of delay differential equation models of HIV-1 infection. Math. biosci. 179, 73 (2002) · Zbl 0992.92035 [4] Perelson, A. S.: Modeling the interaction of the immune system with HIV. Lecture notes in biomathematics 83, 350 (1989) · Zbl 0683.92001 [5] Kirschner, D.: Using mathematics to understand HIV immune dynamics. Notices AMS 43, 191 (1996) · Zbl 1044.92503 [6] De Leenheer, P.; Smith, H. L.: Virus dynamics: a global analysis. SIAM J. Appl. math. 63, 1313 (2003) · Zbl 1035.34045 [7] Li, M. Y.; Muldowney, J. S.: A geometric approach to the global-stability problems. SIAM J. Math. anal. 27, 1070 (1996) · Zbl 0873.34041 [8] Li, M. Y.; Smith, H. L.; Wang, L.: Global dynamics of an SEIR model with vertical transmission. SIAM J. Appl. math. 62, 58 (2001) · Zbl 0991.92029 [9] Wang, L.; Li, M. Y.; Kirschner, D.: Mathematical analysis of the global dynamics of a model for HTLV-I infection and ATL progression. Math. biosci. 179, 207 (2002) · Zbl 1008.92026 [10] Lasalle, J. P.: The stability of dynamical systems, regional conference series in applied mathematics. (1976) · Zbl 0364.93002 [11] Freedman, H. I.; Tang, M. X.; Ruan, S. G.: Uniform persistence and flows near a closed positively invariant set. J. dynam. Differen. equat. 6, 583 (1994) · Zbl 0811.34033 [12] 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 (1999) · Zbl 0974.92029 [13] Ruan, S.; Wei, J.: On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion. IMA J. Math. appl. Med. bio. 18, 41 (2001) · Zbl 0982.92008 [14] Fiedler, M.: Additive compound matrices and inequality for eigenvalues of stochastic matrices. Czech math. J. 99, 392 (1974) · Zbl 0345.15013 [15] Muldowney, J. S.: Compound matrices and ordinary differential equations. Rocky mount. J. math. 20, 857 (1990) · Zbl 0725.34049 [16] Coppel, W. A.: Stability and asymptotic behavior of differential equations. (1995) · Zbl 0838.52014 [17] Butler, G. J.; Waltman, P.: Persistence in dynamical systems. Proc. am. Math. soc. 96, 425 (1986) · Zbl 0603.58033 [18] Waltman, P.: A brief survey of persistence. Delay differential equations and dynamical systems, 31 (1991) · Zbl 0756.34054 [19] Jr., R. H. Martin: Logarithmic norms and projections applied to linear differential systems. J. math. Anal. appl. 45, 432 (1974) · Zbl 0293.34018