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Discrete Dynamics in Nature and Society
Volume 2011, Article ID 201274, 19 pages
http://dx.doi.org/10.1155/2011/201274
Research Article

Global Properties of Virus Dynamics Models with Multitarget Cells and Discrete-Time Delays

1Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, Egypt

Received 8 July 2011; Accepted 16 October 2011

Academic Editor: Yong Zhou

Copyright © 2011 A. M. Elaiw and M. A. Alghamdi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. M. A. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virolog, Oxford University Press, Oxford, UK, 2000.
  2. A. V. M. Herz, S. Bonhoeffer, R. M. Anderson, R. M. May, and M. A. Nowak, “Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay,” Proceedings of the National Academy of Sciences of the United States of America, vol. 93, no. 14, pp. 7247–7251, 1996. View at Publisher · View at Google Scholar
  3. R. V. Culshaw and S. Ruan, “A delay-differential equation model of HIV infection of CD4+ T-cells,” Mathematical Biosciences, vol. 165, no. 1, pp. 27–39, 2000. View at Publisher · View at Google Scholar
  4. N. M. Dixit and A. S. Perelson, “Complex patterns of viral load decay under antiretroviral therapy: influence of pharmacokinetics and intracellular delay,” Journal of Theoretical Biology, vol. 226, no. 1, pp. 95–109, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  5. J. E. Mittler, B. Sulzer, A. U. Neumann, and A. S. Perelson, “Influence of delayed viral production on viral dynamics in HIV-1 infected patients,” Mathematical Biosciences, vol. 152, no. 2, pp. 143–163, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. P. W. Nelson, J. D. Murray, and A. S. Perelson, “A model of HIV-1 pathogenesis that includes an intracellular delay,” Mathematical Biosciences, vol. 163, no. 2, pp. 201–215, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. P. W. Nelson and A. S. Perelson, “Mathematical analysis of delay differential equation models of HIV-1 infection,” Mathematical Biosciences, vol. 179, no. 1, pp. 73–94, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. D. Li and W. Ma, “Asymptotic properties of a HIV-1 infection model with time delay,” Journal of Mathematical Analysis and Applications, vol. 335, no. 1, pp. 683–691, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. R. Xu, “Global stability of an HIV-1 infection model with saturation infection and intracellular delay,” Journal of Mathematical Analysis and Applications, vol. 375, no. 1, pp. 75–81, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. Y. Nakata, “Global dynamics of a viral infection model with a latent period and Beddington-DeAngelis response,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 9, pp. 2929–2940, 2011. View at Publisher · View at Google Scholar
  11. M. Y. Li and H. Shu, “Global dynamics of an in-host viral model with intracellular delay,” Bulletin of Mathematical Biology, vol. 72, no. 6, pp. 1492–1505, 2010. View at Publisher · View at Google Scholar · View at PubMed · View at Zentralblatt MATH
  12. G. Huang, W. Ma, and Y. Takeuchi, “Global analysis for delay virus dynamics model with Beddington-DeAngelis functional response,” Applied Mathematics Letters, vol. 24, no. 7, pp. 1199–1203, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. Y. Wang, Y. Zhou, J. Wu, and J. Heffernan, “Oscillatory viral dynamics in a delayed HIV pathogenesis model,” Mathematical Biosciences, vol. 219, no. 2, pp. 104–112, 2009. View at Publisher · View at Google Scholar · View at PubMed · View at Zentralblatt MATH
  14. X. Wang, Y. Tao, and X. Song, “A delayed HIV-1 infection model with Beddington-DeAngelis functional response,” Nonlinear Dynamics, vol. 62, no. 1-2, pp. 67–72, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. X. Song, X. Zhou, and X. Zhao, “Properties of stability and Hopf bifurcation for a HIV infection model with time delay,” Applied Mathematical Modelling, vol. 34, no. 6, pp. 1511–1523, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. X. Song, S. Wang, and J. Dong, “Stability properties and Hopf bifurcation of a delayed viral infection model with lytic immune response,” Journal of Mathematical Analysis and Applications, vol. 373, no. 2, pp. 345–355, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. X. Shi, X. Zhou, and X. Song, “Dynamical behavior of a delay virus dynamics model with CTL immune response,” Nonlinear Analysis. Real World Applications, vol. 11, no. 3, pp. 1795–1809, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. Z. Hu, X. Liu, H. Wang, and W. Ma, “Analysis of the dynamics of a delayed HIV pathogenesis model,” Journal of Computational and Applied Mathematics, vol. 234, no. 2, pp. 461–476, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. Q. Xie, D. Huang, S. Zhang, and J. Cao, “Analysis of a viral infection model with delayed immune response,” Applied Mathematical Modelling, vol. 34, no. 9, pp. 2388–2395, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. G. Huang, Y. Takeuchi, and W. Ma, “Lyapunov functionals for delay differential equations model of viral infections,” SIAM Journal on Applied Mathematics, vol. 70, no. 7, pp. 2693–2708, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. H. Zhu and X. Zou, “Impact of delays in cell infection and virus production on HIV-1 dynamics,” Mathematical Medicine and Biology, vol. 25, no. 2, pp. 99–112, 2008. View at Publisher · View at Google Scholar · View at PubMed · View at Zentralblatt MATH
  22. R. Ouifki and G. Witten, “Stability analysis of a model for HIV infection with RTI and three intracellular delays,” BioSystems, vol. 95, no. 1, pp. 1–6, 2009. View at Publisher · View at Google Scholar · View at PubMed
  23. Y. Nakata, “Global dynamics of a cell mediated immunity in viral infection models with distributed delays,” Journal of Mathematical Analysis and Applications, vol. 375, no. 1, pp. 14–27, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. R. Xu, “Global dynamics of an HIV-1 infection model with distributed intracellular delays,” Computers & Mathematics with Applications, vol. 61, no. 9, pp. 2799–2805, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. A. S. Perelson, P. Essunger, Y. Cao et al., “Decay characteristics of HIV-1-infected compartments during combination therapy,” Nature, vol. 387, no. 6629, pp. 188–191, 1997. View at Publisher · View at Google Scholar · View at PubMed
  26. D. S. Callaway and A. S. Perelson, “HIV-1 infection and low steady state viral loads,” Bulletin of Mathematical Biology, vol. 64, no. 1, pp. 29–64, 2002. View at Publisher · View at Google Scholar · View at PubMed
  27. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. A. M. Elaiw, “Global properties of a class of HIV models,” Nonlinear Analysis. Real World Applications, vol. 11, no. 4, pp. 2253–2263, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. A. M. Elaiw, I. A. Hassanien, and S. A. Azoz, “Global properties of a class of HIV infection models with Beddington-DeAngelis functional response,” Submitted to Mathematical Methods in the Applied Sciences.
  30. A. M. Elaiw and X. Xia, “HIV dynamics: analysis and robust multirate MPC-based treatment schedules,” Journal of Mathematical Analysis and Applications, vol. 359, no. 1, pp. 285–301, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. A. M. Elaiw, “Global properties of a class of virus infection models with multitarget cells,” Nonlinear Dynamics, In Press. View at Publisher · View at Google Scholar
  32. J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, Springer, New York, NY, USA, 1993.
  33. M. A. Nowak, R. Anderson, M. Boerlijst, S. Bonhoeffer, R. May, and A. McMichael, “HIV-1 evolution and disease progression,” Science, vol. 274, pp. 1008–1010, 1996. View at Google Scholar
  34. A. M. Elaiw, I. A. Hassanien, and S. A. Azoz, “Global stability of HIV infection models with intracellular delays,” Journal of the Korean Mathematical Society.