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Abstract and Applied Analysis
Volume 2014, Article ID 650371, 12 pages
http://dx.doi.org/10.1155/2014/650371
Research Article

Stability of Virus Infection Models with Antibodies and Chronically Infected Cells

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 9 December 2013; Revised 18 February 2014; Accepted 6 March 2014; Published 3 April 2014

Academic Editor: Malay Banerjee

Copyright © 2014 Mustafa A. Obaid and A. M. Elaiw. 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 Virology, University of Oxford, Oxford, UK, 2000.
  2. M. A. Nowak and C. R. M. Bangham, “Population dynamics of immune responses to persistent viruses,” Science, vol. 272, no. 5258, pp. 74–79, 1996. View at Google Scholar · View at Scopus
  3. 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 Google Scholar · View at Scopus
  4. 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, 1996. View at Google Scholar · View at Scopus
  5. A. U. Neumann, N. P. Lam, H. Dahari et al., “Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon-α therapy,” Science, vol. 282, no. 5386, pp. 103–107, 1998. View at Publisher · View at Google Scholar · View at Scopus
  6. A. S. Perelson, D. E. Kirschner, and R. De Boer, “Dynamics of HIV infection of CD4+ T cells,” Mathematical Biosciences, vol. 114, no. 1, pp. 81–125, 1993. View at Publisher · View at Google Scholar · View at Scopus
  7. 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 Scopus
  8. A. M. Elaiw, “Global properties of a class of virus infection models with multitarget cells,” Nonlinear Dynamics, pp. 1–13, 2011. View at Publisher · View at Google Scholar · View at Scopus
  9. A. M. Elaiw and S. A. Azoz, “Global properties of a class of HIV infection models with Beddington- DeAngelis functional response,” Mathematical Methods in the Applied Sciences, vol. 36, no. 4, pp. 383–394, 2013. View at Google Scholar
  10. A. M. Elaiw and A. S. Alsheri, “Global dynamics of HIV infection of CD4+ T cells and macrophages,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 264759, 8 pages, 2013. View at Publisher · View at Google Scholar
  11. 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, vol. 49, no. 4, pp. 779–794, 2012. View at Google Scholar
  12. A. M. Elaiw and M. A. Alghamdi, “Global properties of virus dynamics models with multitarget cells and discrete-time delays,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 201274, 19 pages, 2011. View at Publisher · View at Google Scholar · View at Scopus
  13. A. M. Elaiw, “Global dynamics of an HIV infection model with two classes of target cells and distributed delays,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 253703, 13 pages, 2012. View at Publisher · View at Google Scholar
  14. M. A. Obaid, “Global analysis of a virus infection model with multitarget cells and distributed intracellular delays,” Life Science Journal, vol. 9, pp. 1500–1508, 2012. View at Google Scholar
  15. K. Wang, A. Fan, and A. Torres, “Global properties of an improved hepatitis B virus model,” Nonlinear Analysis: Real World Applications, vol. 11, no. 4, pp. 3131–3138, 2010. View at Publisher · View at Google Scholar · View at Scopus
  16. X. Wang, A. Elaiw, and X. Song, “Global properties of a delayed HIV infection model with CTL immune response,” Applied Mathematics and Computation, vol. 218, no. 18, pp. 9405–9414, 2012. View at Publisher · View at Google Scholar · View at Scopus
  17. J. Li, K. Wang, and Y. Yang, “Dynamical behaviors of an HBV infection model with logistic hepatocyte growth,” Mathematical and Computer Modelling, vol. 54, no. 1-2, pp. 704–711, 2011. View at Publisher · View at Google Scholar · View at Scopus
  18. 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 Scopus
  19. R. M. Anderson, R. M. May, and S. Gupta, “Non-linear phenomena in host-parasite interactions,” Parasitology, vol. 99, pp. S59–S79, 1989. View at Google Scholar · View at Scopus
  20. A. Murase, T. Sasaki, and T. Kajiwara, “Stability analysis of pathogen-immune interaction dynamics,” Journal of Mathematical Biology, vol. 51, no. 3, pp. 247–267, 2005. View at Publisher · View at Google Scholar · View at Scopus
  21. D. Wodarz, R. M. May, and M. A. Nowak, “The role of antigen-independent persistence of memory cytotoxic T lymphocytes,” International Immunology, vol. 12, no. 4, pp. 467–477, 2000. View at Google Scholar · View at Scopus
  22. C. Chiyaka, W. Garira, and S. Dube, “Modelling immune response and drug therapy in human malaria infection,” Computational and Mathematical Methods in Medicine, vol. 9, no. 2, pp. 143–163, 2008. View at Publisher · View at Google Scholar · View at Scopus
  23. A. S. Perelson, “Modelling viral and immune system dynamics,” Nature Reviews Immunology, vol. 2, no. 1, pp. 28–36, 2002. View at Google Scholar · View at Scopus
  24. S. Wang and D. Zou, “Global stability of in-host viral models with humoral immunity and intracellular delays,” Applied Mathematical Modelling, vol. 36, no. 3, pp. 1313–1322, 2012. View at Publisher · View at Google Scholar · View at Scopus
  25. H. F. Huo, Y. L. Tang, and L. X. Feng, “A virus dynamics model with saturation infection and humoral immunity,” Journal of Mathematical Analysis and Applications, vol. 6, no. 40, pp. 1977–1983, 2012. View at Google Scholar
  26. A. M. Elaiw, A. Alhejelan, and M. A. Alghamdi, “Global dynamics of virus infection model with antibody immune response and distributed delays,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 781407, 9 pages, 2013. View at Publisher · View at Google Scholar
  27. T. Wang, Z. Hu, and F. Liao, “Stability and Hopf bifurcation for a virus infection model with delayed humoral immunity response,” Journal of Mathematical Analysis and Applications, vol. 411, no. 1, pp. 63–74, 2014. View at Google Scholar
  28. X. Wang and S. Liu, “A class of delayed viral models with saturation infection rate and immune response,” Mathematical Methods in the Applied Sciences, vol. 36, no. 2, pp. 125–142, 2013. View at Google Scholar
  29. 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. View at Publisher · View at Google Scholar · View at Scopus
  30. P. Georgescu and Y.-H. Hsieh, “Global stability for a virus dynamics model with nonlinear incidence of infection and removal,” SIAM Journal on Applied Mathematics, vol. 67, no. 2, pp. 337–353, 2006. View at Publisher · View at Google Scholar · View at Scopus
  31. A. Korobeinikov, “Global asymptotic properties of virus dynamics models with dose-dependent parasite reproduction and virulence and non-linear incidence rate,” Mathematical Medicine and Biology, vol. 26, no. 3, pp. 225–239, 2009. View at Publisher · View at Google Scholar · View at Scopus
  32. 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. View at Google Scholar · View at Scopus
  33. R. R. Regoes, D. Ebert, and S. Bonhoeffer, “Dose-dependent infection rates of parasites produce the Allee effect in epidemiology,” Proceedings of the Royal Society B: Biological Sciences, vol. 269, no. 1488, pp. 271–279, 2002. View at Publisher · View at Google Scholar · View at Scopus