Journal of the Korean Society for Industrial and Applied Mathematics

  • 제19권2호
  • /
  • Pages.137-170
  • /
  • 2015
  • /
  • 1226-9433(pISSN)
  • /
  • 1229-0645(eISSN)
한국산업응용수학회 (Korean Society for Industrial and Applied Mathematics)
권호 표지
DOI QR코드

DOI QR Code

GLOBAL THRESHOLD DYNAMICS IN HUMORAL IMMUNITY VIRAL INFECTION MODELS INCLUDING AN ECLIPSE STAGE OF INFECTED CELLS

  • ELAIW, A.M. (DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE, KING ABDULAZIZ UNIVERSITY)
  • 투고 : 2015.01.08
  • 심사 : 2015.05.06
  • 발행 : 2015.06.25
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, we propose and analyze three viral infection models with humoral immunity including an eclipse stage of infected cells. The incidence rate of infection is represented by bilinear incidence and saturated incidence in the first and second models, respectively, while it is given by a more general function in the third one. The neutralization rate of viruses is giv0en by bilinear form in the first two models, while it is given by a general function in the third one. For each model, we have derived two threshold parameters, the basic infection reproduction number which determines whether or not a chronic-infection can be established without humoral immunity and the humoral immune response activation number which determines whether or not a chronic-infection can be established with humoral immunity. By constructing suitable Lyapunov functions we have proven the global asymptotic stability of all equilibria of the models. For the third model, we have established a set of conditions on the threshold parameters and on the general functions which are sufficient for the global stability of the equilibria of the model. We have performed some numerical simulations for the third model with specific forms of the incidence and neutralization rates and have shown that the numerical results are consistent with the theoretical results.

키워드

참고문헌

  1. M. A. Nowak and R. M. May, "Virus dynamics: Mathematical Principles of Immunology and Virology," Oxford Uni., Oxford, 2000.
  2. M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. https://doi.org/10.1126/science.272.5258.74
  3. A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44. https://doi.org/10.1137/S0036144598335107
  4. L. Wang and M. Y. Li, Mathematical analysis of the global dynamics of a model for HIV infection of $CD4^+$ T cells, Math. Biosc., 200(1) (2006), 44-57. https://doi.org/10.1016/j.mbs.2005.12.026
  5. Y. Zhao, D. T. Dimitrov, H. Liu and Y. Kuang, Mathematical insights in evaluating state dependent effectiveness of HIV prevention interventions, Bull. Math. Biol., 75 (2013), 649-675. https://doi.org/10.1007/s11538-013-9824-7
  6. D. S. Callaway and A. S. Perelson, HIV-1 infection and low steady state viral loads, Bull. Math. Biol., 64 (2002), 29-64. https://doi.org/10.1006/bulm.2001.0266
  7. P. K. Roy, A. N. Chatterjee, D. Greenhalgh and Q. J. A. Khan, Long term dynamics in a mathematical model of HIV-1 infection with delay in different variants of the basic drug therapy model, Nonlinear Anal. Real World Appl., 14 ( 2013), 1621-1633. https://doi.org/10.1016/j.nonrwa.2012.10.021
  8. A. M. Elaiw, I. A. Hassanien and S. A. Azoz, Global stability of HIV infection models with intracellular delays, J. Korean Math. Soc., 49 (2012), 779-794. https://doi.org/10.4134/JKMS.2012.49.4.779
  9. A. M. Elaiw and S. A. Azoz, Global properties of a class of HIV infection models with Beddington-DeAngelis functional response, Math. Methods Appl. Sci., 36 (2013), 383-394. https://doi.org/10.1002/mma.2596
  10. A. M. Elaiw, Global properties of a class of virus infection models with multitarget cells, Nonlinear Dynam., 69 (2012), 423-435. https://doi.org/10.1007/s11071-011-0275-0
  11. A. M. Elaiw and X. Xia, HIV dynamics: Analysis and robust multirate MPC-based treatment schedules, J. Math. Anal. Appl., 356 (2009), 285-301.
  12. A. M. Elaiw, Global properties of a class of HIV models, Nonlinear Anal. Real World Appl., 11 (2010), 2253-2263. https://doi.org/10.1016/j.nonrwa.2009.07.001
  13. S. Eikenberry, S. Hews, J. D. Nagy and Y. Kuang, The dynamics of a delay model of HBV infection with logistic hepatocyte growth, Math. Biosc. Eng., 6 (2009), 283-299. https://doi.org/10.3934/mbe.2009.6.283
  14. S. A. Gourley, Y. Kuang and J. D. Nagy, Dynamics of a delay differential equation model of hepatitis B virus infection, J. Biol. Dyn., 2 (2008), 140-153. https://doi.org/10.1080/17513750701769873
  15. J. Li, K.Wang and Y. Yang, Dynamical behaviors of an HBV infection model with logistic hepatocyte growth, Math. Comput. Modelling, 54 (2011), 704-711. https://doi.org/10.1016/j.mcm.2011.03.013
  16. R. Qesmi, J. Wu, J. Wu and J. M. Heffernan, Influence of backward bifurcation in a model of hepatitis B and C viruses, Math. Biosci., 224 (2010), 118-125. https://doi.org/10.1016/j.mbs.2010.01.002
  17. R. Qesmi, S. ElSaadany, J. M. Heffernan and J. Wu, A hepatitis B and C virus model with age since infection that exhibit backward bifurcation, SIAM J. Appl. Math., 71 (4) (2011), 1509-1530. https://doi.org/10.1137/10079690X
  18. A. U. Neumann, N. P. Lam, H. Dahari, D. R. Gretch, T. E. Wiley, T. J, Layden and A. S. Perelson, Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon-alpha therapy, Science, 282 (1998), 103-107. https://doi.org/10.1126/science.282.5386.103
  19. M. Y. Li and H. Shu, Global dynamics of a mathematical model for HTLV-I infection of CD4+ T cells with delayed CTL response, Nonlinear Anal. Real World Appl., 13 (2012), 1080-1092. https://doi.org/10.1016/j.nonrwa.2011.02.026
  20. P. Tanvi, G. Gujarati and G. Ambika, Virus antibody dynamics in primary and secondary dengue infections, J. Math. Biol., 69 (2014), 1773-1800. https://doi.org/10.1007/s00285-013-0749-4
  21. J. A. Deans and S. Cohen, Immunology of malaria, Ann. Rev. Microbiol. 37 (1983), 25-49. https://doi.org/10.1146/annurev.mi.37.100183.000325
  22. A. Murase, T. Sasaki and T. Kajiwara, Stability analysis of pathogen-immune interaction dynamics, J. Math. Biol., 51 (2005), 247-267. https://doi.org/10.1007/s00285-005-0321-y
  23. A. M. Elaiw and N. H. AlShameani, Global analysis for a delay-distributed viral infection model with antibodies and general nonlinear incidence rate, J. Korean Soc. Ind. Appl. Math., 18(4) (2014), 317-335. https://doi.org/10.12941/jksiam.2014.18.317
  24. M. A. Obaid and A. M. Elaiw, Stability of virus infection models with antibodies and chronically infected cells, Abstr. Appl. Anal, 2014, Article ID 650371.
  25. A. M. Elaiw, A. Alhejelan and M. A. Alghamdi, Global dynamics of virus infection model with antibody immune response and distributed delays, Discrete Dyn. Nat. Soc., 2013 (2013), Article ID 781407.
  26. T. Wang, Z. Hu and F. Liao, Stability and Hopf bifurcation for a virus infection model with delayed humoral immunity response, J. Math. Anal. Appl., 411 (2014) 63-74. https://doi.org/10.1016/j.jmaa.2013.09.035
  27. T. Wang, Z. Hu, F. Liao and W. Ma, Global stability analysis for delayed virus infection model with general incidence rate and humoral immunity, Math. Comput. Simulation, 89 (2013), 13-22. https://doi.org/10.1016/j.matcom.2013.03.004
  28. S. Wang and D. Zou, Global stability of in host viral models with humoral immunity and intracellular delays, J. Appl. Math. Mod., 36 (2012), 1313-1322. https://doi.org/10.1016/j.apm.2011.07.086
  29. A. S. Perelson, D. Kirschner and R. De Boer, Dynamics of HIV infection of $CD4^+$ T cells, Math. Biosci., 114(1) (1993), 81-125. https://doi.org/10.1016/0025-5564(93)90043-A
  30. A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol. 66 (2004), 879-883 https://doi.org/10.1016/j.bulm.2004.02.001
  31. B. Buonomo and C. Vargas-De-Le, Global stability for an HIV-1 infection model including an eclipse stage of infected cells, J. Math. Anal. Appl. 385 (2012), 709-720. https://doi.org/10.1016/j.jmaa.2011.07.006
  32. J. K. Hale and S. Verduyn Lunel, "Introduction to functional differential equations," Springer-Verlag, New York, 1993.
  33. X. Song, A. U. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329 (2007), 281-297. https://doi.org/10.1016/j.jmaa.2006.06.064
  34. A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886. https://doi.org/10.1007/s11538-007-9196-y
  35. R. R. Regoes, D. Ebert, S. Bonhoeffer, Dose-dependent infection rates of parasites produce the Allee effect in epidemiology, Proc. R. Soc. Lond. Ser. B, 269 (2002), 271-279. https://doi.org/10.1098/rspb.2001.1816
  36. R. Xu, Global stability of an HIV-1 infection model with saturation infection and intracellular delay, J. Math. Anal. Appl., 375 (2011), 75-81. https://doi.org/10.1016/j.jmaa.2010.08.055
  37. G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infection, SIAM J. Appl. Math., 70 (2010), 2693-2708. https://doi.org/10.1137/090780821
  38. R. Larson and B. H. Edwards, "Calculus of a single variable," Cengage Learning, Inc., USA, (2010).