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Global stability and periodic solution of a model for HIV infection of CD4$^{+}$ T cells. (English) Zbl 1117.92040
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 non-linear infection rate. The global dynamics of this model is rigorously established. We prove that, if the basic reproduction number $R_{0}\leqslant 1$, the HIV infection is cleared from the T-cells population; if $R_{0}>1$, the HIV infection persists. Further, the existence of a non-trivial periodic solution is also studied by means of numerical simulations.

##### MSC:
 92C60 Medical epidemiology 37N25 Dynamical systems in biology 34D23 Global stability of ODE 34K20 Stability theory of functional-differential equations
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##### References:
 [1] Ho, D.; Neumann, A.; Perelson, A.; Chen, W.; Leonard, J.; Markowitz, M.: Rapid turnover of plasma virions and CD4+ lymphocytes in HIV-1 infection. Nature 373, 123-126 (1995) [2] De Boer, R. J.; Perelson, A. S.: Target cell limited and immune control models of HIV infection: a comparison. J. theor. Biol. 190, 201 (1998) [3] Perelson, A. S.; Nelson, P. W.: Mathematical analysis of HIV-I dynamics in vivo. SIAM rev. 41, 3 (1999) · Zbl 1078.92502 [4] Perelson, A.; Nelson, P.: Mathematical analysis of HIV-1 dynamics in vivo. SIAM rev. 41, No. 1, 3-44 (1999) · Zbl 1078.92502 [5] 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) [6] Ho, D.; Neumann, A.; Perelson, A.; Chen, W.; Leonard, J.; Markowitz, M.: Rapid turnover of plasma virions and CD4+ lymphocytes in HIV-1 infection. Nature 373, 123-126 (1995) [7] 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) [8] Lasalle, J. P.: The stability of dynamical system, regional conference series in applied mathematics. (1976) · Zbl 0364.93002 [9] Freedman, H. I.; Tang, M. X.; Ruan, S. G.: Uniform persistence and flows near a closed positively invariant set. J. dyn. Differen. equat. 6, 583 (1994) · Zbl 0811.34033 [10] 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 [11] De Leenheer, P.; Smith, H. L.: Virus dynamics: a global analysis. SIAM J. Appl. math. 63, No. 4, 1313-1327 (2003) · Zbl 1035.34045 [12] Li, M. Y.; Muldowney, J. S.: A geometric approach to the global stability problems. SIAM J. Math. anal. 27, 1070 (1996) · Zbl 0873.34041 [13] Fiedler, M.: Additive compound matrices and inequality for eigenvalues of stochastic matrices. Czech math. J. 99, 392 (1974) · Zbl 0345.15013 [14] Muldowney, J. S.: Compound matrices and ordinary differential equations. Rocky mountain J. Math. 20, 857-872 (1990) · Zbl 0725.34049 [15] Coppel, W. A.: Stability and asymptotic behavior of differential equations. (1995) · Zbl 0838.52014 [16] Butler, G. J.; Waltman, P.: Persistence in dynamical system. Proc. am. Math. soc. 96, 425 (1986) · Zbl 0603.58033 [17] Waltman, P.: A brief survey of persistence. Delay differential equations and dynamical systems, 31 (1991) · Zbl 0756.34054 [18] Jr., R. H. Martin: Logarithmic norms and projections applied to linear differential system. J. math. Anal. appl. 45, 432 (1974)