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http://dx.doi.org/10.4134/BKMS.2011.48.3.555

GLOBAL STABILITY OF THE VIRAL DYNAMICS WITH CROWLEY-MARTIN FUNCTIONAL RESPONSE  

Zhou, Xueyong (School of Mathematical Sciences Nanjing Normal University, College of Mathematics and Information Science Xinyang Normal University)
Cui, Jingan (School of Science Beijing University of Civil Engineering and Architecture)
Publication Information
Bulletin of the Korean Mathematical Society / v.48, no.3, 2011 , pp. 555-574 More about this Journal
Abstract
It is well known that the mathematical models provide very important information for the research of human immunodeciency virus type. 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. In this paper, a differential equation model of HIV infection of $CD4^+$ T-cells with Crowley-Martin function response is studied. We prove that if the basic reproduction number $R_0$ < 1, the HIV infection is cleared from the T-cell population and the disease dies out; if $R_0$ > 1, the HIV infection persists in the host. We find that the chronic disease steady state is globally asymptotically stable if $R_0$ > 1. Numerical simulations are presented to illustrate the results.
Keywords
HIV infection; permanence; globally asymptotical stability;
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1 R. V. Culshaw, S. G. Ruan, and R. J. Spiteri, Optimal HIV treatment by maximising immune response, J. Math. Biol. 48 (2004), no. 5, 545-562.   DOI
2 F. R. Gantmacher, The Theory of Matrices, Chelsea Publ. Co., New York, 1959.
3 M. W. Hirsch, Systems of differential equations which are competitive or cooperative IV, SIAM J. Math. Anal. 21 (1990), 1225-1234.   DOI
4 A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-I dynamics in vivo, SIAM Rev. 41 (1999), 3-44.   DOI   ScienceOn
5 A. S. Perelson, A. U. Neumann, M. Markowitz, et al., HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time, Science 271 (1996), 1582-1586.   DOI   ScienceOn
6 H. L. Smith, Monotone dynamical systems: An Introduction to the theory of competitive and cooperative systems, Trans. Amer. Math. Soc., vol. 41, 1995.
7 H. L. Smith , Systems of ordinary differential equations which generate an order preserving flow, SIAM Rev. 30 (1998), 87-98.
8 X. Y. Song and A. U. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl. 329 (2007), no. 1, 281-297.   DOI   ScienceOn
9 H. R. Thieme, Persistence under relaxed point-dissipativity (with applications to an endemic model), SIAMJ. Math. Anal. 24 (1993), 407-435.   DOI   ScienceOn
10 X. Y. Zhou, X. Y. Song, and X. Y. Shi, A differential equation model of HIV infection of CD4+CD4+ T-cells with cure rate, J. Math. Anal. Appl. 342 (2008), no. 2, 1342-1355.   DOI   ScienceOn
11 X. Y. Zhou, X. Y. Song, and X. Y. Shi, Analysis of stability and Hopf bifurcation for an HIV infection model with time delay, Appl. Math. Comput. 199 (2008), no. 1, 23-38.   DOI   ScienceOn
12 H. R. Zhu and H. L. Smith, Stable periodic orbits for a class of three-dimensional competitive systems, J. Differential Equations 110 (1994), no. 1, 143-156.   DOI   ScienceOn
13 A. R. McLean and T. B. L. Kirkwood, A model of human immunode ciency virus infection in T helper cell clones, J. Theor. Biol. 147 (1990), 177-203.   DOI
14 P. De Leenheer and H. L. Smith, Virus dynamics: a global analysis, SIAM J. Appl. Math. 63 (2003), no. 4, 1313-1327.   DOI   ScienceOn
15 D. Li and W. Ma, Asymptotic properties of a HIV-1 infection model with time delay, J. Math. Anal. Appl. 335 (2007), no. 1, 683-691.   DOI   ScienceOn
16 A. L. Lloyd, The dependence of viral parameter estimates on the assumed viral life cycle: limitations of studies of viral load data, Proc. R. Soc. Lond. B 268 (2001), 847-854.   DOI   ScienceOn
17 A. R. McLean, M. M. Rosado, F. Agenes, R. Vasconcellos, and A. A. Freitas, Resource competition as a mechanism for B cell homeostasis, Proc. Natl Acad. Sci. USA 94 (1997), 5792-5797.   DOI
18 L. Q. Min, Y. M. Su, and Y. Kuang, Mathematical analysis of a basic virus infection model with application to HBV infection, Rocky Mountain J. Math. 38 (2008), no. 5, 1573-1585.   DOI   ScienceOn
19 J. E. Mittler, B. Sulzer, A. U. Neumann, and A. S. Perelson, In uence of delayed viral production on viral dynamics in HIV-1 infected patients, Math. Biosci. 152 (1998), 143-163.   DOI   ScienceOn
20 J. S. Muldowney, Compound matrices and ordinary differential equations, Rocky Mountain J. Math. 20 (1990), no. 4, 857-872.   DOI
21 N. Nagumo, Uber die lage der integralkurven gewohnlicher differential gleichungen, Proc. Phys-Math. Soc. Japan 24 (1942), 551-559.
22 S. M. Ciupe, R. M. Ribeiro, P. W. Nelson, and A. S. Perelson, Modeling the mechanisms of acute hepatitis B virus infection, J. Theor. Biol. 247 (2007), no. 1, 23-35.   DOI   ScienceOn
23 P. W. Nelson, J. D. Murray, and A. S. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay, Math. Biosci. 163 (2000), no. 2, 201-215.   DOI   ScienceOn
24 A. S. Perelson, D. E. Kirschner, and R. de Boer, Dynamics of HIV infection of CD4+ T cells, Math. Biosci. 114 (1993), 81-125.   DOI   ScienceOn
25 S. Bonhoeffer, R. M. May, G. M. Shaw, and M. A. Nowak, Virus dynamics and drug therapy, Proc. Natl. Acad. Sci. USA 94 (1997), 6971-6976.   DOI
26 P. H. Crowley and E. K. Martin, Functional responses and interference within and between year classes of a dragon y population, J. North. Am. Benth. Soc. 8 (1989), 211-221.   DOI   ScienceOn
27 R. V. Culshaw and S. G. Ruan, A delay-differential equation model of HIV infection of CD4+ T-cells, Math. Biosci. 165 (2000), 27-39.   DOI   ScienceOn