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Global Properties for an HIV-1 Infection Model Including an Eclipse Stage of Infected Cells and Saturation Infection


Mahmoud Hesaaraki

Department of mathematics,
Sharif University of Technology,
P.O.Box: 11155-9414,
Tehran Iran

Mahtab Sabzevari

Department of mathematics,
Sharif University of Technology,
P.O.Box: 11155-9414,
Tehran Iran


In this paper we study a fourth-dimensional human immunodeficiency virus (HIV) model including an eclipse stage of infected cells and saturation infection. One feature of this model is that an eclipse stage for the infected cells is included and cells in this stage may revert to the uninfected class . The other feature is that system has nonlinear incidence of infection of health CD4^+T cells. For the analysis of nonlinear autonomous differential equations with or without time delay, the stability of equilibria is important. We will obtain sufficient conditions for the global stability of the equilibria system by using Lyapunov direct method and the geometric approach to stability, based on the generalization of the Poincare-Bendixson criterion for system of n ordinary differential equations.


  1. S. Bonhoeffer, R. M. May, G. M. Shaw, and M. A. Nowak, Virus dynamics and drug therapy, Proceedings of the National Academy of Sciences, vol. 94, no. 13, pp. 6971-6976, 1997
  2. D. D. Ho, A. U. Neumann, A. S. Perelson, W. Chen, J. M. Leonard, M. Markowitz, et al., Rapid turnover of plasma virions and cd4 lymphocytes in hiv-1 infection, Nature, vol. 373, no. 6510, pp. 123-126, 1995
  3. A. S. Perelson, A. U. Neumann, M. Markowitz, J. M. Leonard, and D. D. Ho, Hiv-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time, Science, vol. 271, no. 5255, pp. 1582-1586, 196
  4. A. S. Perelson and P. W. Nelson, Mathematical analysis of hiv-1 dynamics in vivo, SIAM review, vol. 41, no. 1, pp. 3-44, 1999
  5. M. Nowak and R. M. May, Virus dynamics: mathematical principles of immunology and virology. Oxford university press, 2000
  6. L. Rong, M. A. Gilchrist, Z. Feng, and A. S. Perelson, Modeling within host hiv-1 dynamics and the evolution of drug resistance: trade-trade-offs between viral enzyme function and drug susceptibility, Journal of Theoretical biology, vol. 247, no. 4, pp. 804-818, 2007
  7. B. Buonomo and C. Vargas-De-Leon, Global stability for an hiv-1 infection model including an eclipse stage of infected cells, Journal of Mathematical Analysis and Applications, vol. 385, no. 2, pp. 709-720, 2012
  8. C. L. Althaus, A. S. De Vos, and R. J. De Boer, Reassessing the human immunodeficiency virus type 1 life cycle through age-structured modeling: life span of infected cells, viral generation time, and basic reproductive number, r0, Journal of virology, vol. 83, no. 15, pp. 7659-7667, 2009
  9. D. Ebert, C. D. Zschokke-Rohringer, and H. J. Carius, Dose effects and density-dependent regulation of two microparasites of daphnia magna, Oecologia, vol. 122, no. 2, pp. 200-209, 2000
  10. X. Song and A. U. Neumann, Global stability and periodic solution of the viral dynamics, Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 281-297, 2007
  11. Y. Li and J. S. Muldowney, On Bendixson's criterion, Journal of Differential Equations, vol. 106, no. 1, pp. 27-39, 1993
  12. M. Y. Li and J. S. Muldowney, On Smith's autonomous convergence theorem, JOURNAL OF MATHEMATICS, vol. 25, no. 1, 1995
  13. M. Y. Li and J. S. Muldowney, A geometric approach to global-stability problems, SIAM Journal on Mathematical Analysis, vol. 27, no. 4, pp. 1070-1083, 1996
  14. M. M. Ballyk, C. C. McCluskey, and G. S. Wolkowicz, Global analysis of competition for perfectly substitutable resources with linear response, Journal of mathematical biology, vol. 51, no. 4, pp. 458-490, 2005
  15. B. Buonomo and D. Lacitignola, Analysis of a tuberculosis model with a case study in Uganda, Journal of Biological Dynamics, vol. 4, no. 6, pp. 571-593, 2010
  16. H. Freedman, S. Ruan, and M. Tang, Uniform persistence and flows near a closed positively invariant set, Journal of Dynamics and Differential Equations, vol. 6, no. 4, pp. 583-600, 1994
  17. V. Hutson and K. Schmitt, Permanence and the dynamics of biological systems, Mathematical Biosciences, vol. 111, no. 1, pp. 1-71, 1992
  18. R. H. Martin, Logarithmic norms and projections applied to linear differential systems, Journal of Mathematical Analysis and Applications, vol. 45, no. 2, pp. 432-454, 1974
  19. B. Buonomo and D. Lacitignola, Global stability for a four dimensional epidemic model, Note di Matematica, vol. 30, no. 2, pp. 83-96, 2011

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