Browse > Article
http://dx.doi.org/10.12941/jksiam.2015.19.137

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)
Publication Information
Journal of the Korean Society for Industrial and Applied Mathematics / v.19, no.2, 2015 , pp. 137-170 More about this Journal
Abstract
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.
Keywords
Viral infection; Global stability; Immune response; Lyapunov function;
Citations & Related Records
Times Cited By KSCI : 2  (Citation Analysis)
연도 인용수 순위
1 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.   DOI   ScienceOn
2 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.
3 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.
4 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.   DOI   ScienceOn
5 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.   DOI   ScienceOn
6 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.   DOI   ScienceOn
7 A. S. Perelson, D. Kirschner and R. De Boer, Dynamics of HIV infection of $CD4^+$ T cells, Math. Biosci., 114(1) (1993), 81-125.   DOI   ScienceOn
8 A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol. 66 (2004), 879-883   DOI   ScienceOn
9 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.   DOI   ScienceOn
10 J. K. Hale and S. Verduyn Lunel, "Introduction to functional differential equations," Springer-Verlag, New York, 1993.
11 X. Song, A. U. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329 (2007), 281-297.   DOI   ScienceOn
12 A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886.   DOI   ScienceOn
13 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.   DOI   ScienceOn
14 R. Xu, Global stability of an HIV-1 infection model with saturation infection and intracellular delay, J. Math. Anal. Appl., 375 (2011), 75-81.   DOI   ScienceOn
15 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.   DOI   ScienceOn
16 R. Larson and B. H. Edwards, "Calculus of a single variable," Cengage Learning, Inc., USA, (2010).
17 M. A. Nowak and R. M. May, "Virus dynamics: Mathematical Principles of Immunology and Virology," Oxford Uni., Oxford, 2000.
18 M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79.   DOI   ScienceOn
19 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.   DOI   ScienceOn
20 A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44.   DOI   ScienceOn
21 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.   DOI
22 D. S. Callaway and A. S. Perelson, HIV-1 infection and low steady state viral loads, Bull. Math. Biol., 64 (2002), 29-64.   DOI   ScienceOn
23 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.   DOI   ScienceOn
24 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.   DOI   ScienceOn
25 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.   DOI   ScienceOn
26 A. M. Elaiw, Global properties of a class of virus infection models with multitarget cells, Nonlinear Dynam., 69 (2012), 423-435.   DOI
27 A. M. Elaiw and X. Xia, HIV dynamics: Analysis and robust multirate MPC-based treatment schedules, J. Math. Anal. Appl., 356 (2009), 285-301.
28 A. M. Elaiw, Global properties of a class of HIV models, Nonlinear Anal. Real World Appl., 11 (2010), 2253-2263.   DOI   ScienceOn
29 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.   DOI   ScienceOn
30 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.   DOI
31 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.   DOI   ScienceOn
32 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.   DOI   ScienceOn
33 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.   DOI   ScienceOn
34 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.   DOI   ScienceOn
35 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.   DOI   ScienceOn
36 P. Tanvi, G. Gujarati and G. Ambika, Virus antibody dynamics in primary and secondary dengue infections, J. Math. Biol., 69 (2014), 1773-1800.   DOI   ScienceOn
37 J. A. Deans and S. Cohen, Immunology of malaria, Ann. Rev. Microbiol. 37 (1983), 25-49.   DOI   ScienceOn
38 A. Murase, T. Sasaki and T. Kajiwara, Stability analysis of pathogen-immune interaction dynamics, J. Math. Biol., 51 (2005), 247-267.   DOI