GLOBAL MATHEMATICAL ANALYSIS OF AN HIV-1 INFECTION MODEL WITH HOLLING TYPE-II INCIDENCE

TitleGLOBAL MATHEMATICAL ANALYSIS OF AN HIV-1 INFECTION MODEL WITH HOLLING TYPE-II INCIDENCE
Publication TypeJournal Article
Year of Publication2011
AuthorsWANG, LIANCHENG
Secondary TitleCommunications in Applied Analysis
Volume15
Issue1
Start Page47
Pagination56
Date Published01/2011
Type of Workscientific: mathematics
ISSN1083–2564
AMS34A34, 34D23
Abstract
In this paper, we study an HIV-1 infection mathematical model with Holling typeII incidence. Both local and global mathematical analysis is carried out. By identifying a basic reproduction number R 0 , we show that if R0 ≤ 1, the uninfected steady state P0 is the only equilibrium in the feasible region, and P0 is globally asymptotically stable. Therefore, no HIV-1 infection persists and infected T cells and HIV-1 virus are cleared over time. However, if R0 > 1, a unique infected steady state P∗ emerges in the interior of the feasible region and P0 becomes unstable. We show that the system is uniformly persistent and the unique infected steady state P∗ is globally asymptotically stable in the interior of the feasible region. Therefore, HIV-1 infection persists and the concentrations of the healthy cells, infected cells, and HIV-1 virus will settle at the level of P∗.
URLhttp://www.acadsol.eu/en/articles/15/1/4.pdf
Short TitleGLOBAL ANALYSIS OF AN HIV MODEL
Refereed DesignationRefereed
Full Text

REFERENCES

[1] A. S. Perelson, A. Neumann, M. Markowitz, J. Leonard, and D. Ho, HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time, Science 271 (1996), 1582-1586.
[2] A. S. Perelson, D. E. Kirschner and R. D. Boer, Dynamics of HIV infection of CD4 + T cells, Math. Biosci. 114 (1993), 81-125.
[3] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-I dynamics in vivo, SIAM Review 41 (1999), 3-44.
[4] P. W. Nelson and A. S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection, Math. Biosci. 179 (2002), 73-94.
[5] D. Li and W. Ma, Asymptotic properties of a HIV-1 infection model with time delay, J. Math. Anal. Appl. 335 (2007), 683-691.
[6] M. Nowark, R. Anderson, M. Boerlijst, R. May and A. McMichael, HIV-1 evolution and disease progression, Science 274 (1996), 1008-1011.
[7] M. A. Nowark and R. M. May, Virus Dynamics, Oxford University Press, New York, 2000.
[8] D. Kirschner, Using mathematics to understand HIV immune dynamics, Notices, Amer. Math. Soc. 43 (1996), 191-202.
[9] R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4 + T-cells, Math. Biosci. 165 (2000), 27-39.
[10] A. Perelson and P. Nelson, Mathematical models of HIV dynamics in vivo, SIAM Review 41 (1999), 3-44.
[11] P. D. Leenheer and H. L. Smith, Virus dynamics: a global analysis, SIAM J. Appl. Math.63 (2003), 1313-1327.
[12] L. Wang and M. Y. Li, Mathematical analysis of the global dynamics of a model for HIV infection of CD4 + T cells, Math. Biosci. 200 (2006), 44-57.
[13] L. Wang and S. Ellermeyer, HIV infection and CD4 + T cell dynamics, Discrete Contin. Dyn. Syst. Ser. B 6 (2006), 1417-1430.
[14] M. Y. Li and J. S. Muldowney, A geometric approach to the global-stability problems, SIAM J. Math. Anal. 27 (1996), 1070-1083.
[15] H. I. Freedman, M. X. Tang, and S. G. Ruan, Uniform persistence and flows near a closed positively invariant set, J. Dynam. Diff. Equat. 6 (1994), 583-600.
[16] M. Y. Li, J. R. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with a varying total population size, Math. Biosci. 160 (1999), 191-213.
[17] G. J. Butler and P. Waltman, Persistence in dynamical systems, Proc. Amer. Math. Soc. 96 (1986), 425-430.
[18] P. Waltman, A brief survey of persistence, in Delay Differential Equations and Dynamical Systems, (Ed: S. Busenberg and M. Martelli), Springer-Verlag, New York, 1991, pp. 31-40.
[19] F. Brauer and J. A. Nohel, The Qualitative Theory of Ordinary Differential Equations: An Introduction, Dover Publ, Inc., New York, 1989.
[20] J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976.
[21] W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, Health, Boston, 1995.
[22] M. Fiedler, Additive compound matrices and inequalities for eigenvalues of stochastic matrices, Czech. Math. J. 99 (1974), 392-402.
[23] J. S. Muldowney, Compound matrices and ordinary differential equations, Rocky Mount. J. Math. 20 (1990), 857-872.
[24] R. H. Martin, Jr., Logarithmic norms and projections applied to linear differential systems, J. Math. Anal. Appl. 45 (1974), 432-454.