Browse > Article
http://dx.doi.org/10.12941/jksiam.2014.18.317

GLOBAL ANALYSIS FOR A DELAY-DISTRIBUTED VIRAL INFECTION MODEL WITH ANTIBODIES AND GENERAL NONLINEAR INCIDENCE RATE  

Elaiw, A.M. (DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE, KING ABDULAZIZ UNIVERSITY)
Alshamrani, N.H. (DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE, KING ABDULAZIZ UNIVERSITY)
Publication Information
Journal of the Korean Society for Industrial and Applied Mathematics / v.18, no.4, 2014 , pp. 317-335 More about this Journal
Abstract
In this work, we investigate the global stability analysis of a viral infection model with antibody immune response. The incidence rate is given by a general function of the populations of the uninfected target cells, infected cells and free viruses. The model has been incorporated with two types of intracellular distributed time delays to describe the time required for viral contacting an uninfected cell and releasing new infectious viruses. We have established a set of conditions on the general incidence rate function and determined two threshold parameters $R_0$ (the basic infection reproduction number) and $R_1$ (the antibody immune response activation number) which are sufficient to determine the global dynamics of the model. The global asymptotic stability of the equilibria of the model has been proven by using Lyapunov theory and applying LaSalle's invariance principle.
Keywords
Virus dynamics; Intracellular delay; global stability; antibody immune response; Lyapunov functional;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
연도 인용수 순위
1 A. M. Elaiw, A. Alhejelan and M. A. Alghamdi, A delayed viral infection model with antibody immune response, Life Science Journal 10(4) (2013) 695-700.
2 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.
3 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
4 A. Korobeinikov, Global properties of infectious disease models with nonlinear incdence, Bull. Math. Biol., 69 (2007), 1871-1886.   DOI
5 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
6 X. Wang and S. Liu, A class of delayed viral models with saturation infection rate and immune response, Math. Meth. Appl. Sci., 36 (2013), 125-142.   DOI
7 K. Hattaf, N. Yousfi, A. Tridane, Stability analysis of a virus dynamics model with general incidence rate and two delays, Applied Mathematics and Computation, 221 (2013) 514-521.   DOI
8 K. Hattaf, N. Yousfi, A. Tridane, Mathematical analysis of a virus dynamics model with general incidence rate and cure rate, Nonlinear Anal. RWA 13 (2012) 1866-1872.   DOI
9 K. Hattaf, N. Yousfi, Global stability of a virus dynamics model with cure rate and absorption, Journal of the Egyptian Mathematical Society, 22 (2014) 386-389.   DOI
10 K. Hattaf, N. Yousfi, A. Tridane, A delay virus dynamics model with general incidence rate, Differ. Equ. Dyn. 22(2), (2014) 181-190.   DOI
11 J. K. Hale and S. Verduyn Lunel, Introduction to functional differential equations, Springer-Verlag, New York, 1993.
12 J. Li, K. Wang, Y. Yang, Dynamical behaviors of an HBV infection model with logistic hepatocyte growth, Math. Comput. Modelling, 54 (2011), 704-711.   DOI
13 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
14 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
15 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
16 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
17 P. Tanvi, G. Gujarati, and G. Ambika , Virus antibody dynamics in primary and secondary dengue infections, J. Math. Biol., (In press).
18 J. A. Deans and S. Cohen, Immunology of malaria, Ann. Rev. Microbiol. 37 (1983), 25-49.   DOI
19 A. Murase, T. Sasaki and T. Kajiwara, Stability analysis of pathogen-immune interaction dynamics, J. Math. Biol., 51 (2005), 247-267.   DOI
20 W. Dominik, R. M. May and M. A. Nowak, The role of antigen-independent persistence of memory cytotoxic T lymphocytes, Int. Immunol. 12 (4) (2000), 467-477.   DOI
21 T.Wang, Z. Hu, F. Liao and Wanbiao, Global stability analysis for delayed virus infection model with general incidence rate and humoral immunity, Math. Comput. Simulation, 89 (2013), 13-22.   DOI
22 H. F. Huo, Y. L. Tang and L. X. Feng, A virus dynamics model with saturation infection and humoral immunity, Int. J. Math. Anal., 6 (2012), 1977-1983.
23 A. M. Elaiw, R. M. Abukwaik and E. O. Alzahrani, Global properties of a cell mediated immunity in HIV infection model with two classes of target cells and distributed delays, Int. J. Biomath., 7(5) (2014) 1450055, 25 pages.
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.   과학기술학회마을   DOI
25 A. M. Elaiw, Global dynamics of an HIV infection model with two classes of target cells and distributed delays, Discrete Dyn. Nat. Soc., 2012 (2012), Article ID 253703.
26 A. M. Elaiw and A. S. Alsheri, Global Dynamics of HIV Infection of CD4+ T Cells and Macrophages, Discrete Dyn. Nat. Soc., 2013, Article ID 264759.
27 A. M. Elaiw and M. A. Alghamdi, Global properties of virus dynamics models with multitarget cells and discrete-time delays, Discrete Dyn. Nat. Soc., 2011, Article ID 201274.
28 N. M. Dixit, and A. S. Perelson, Complex patterns of viral load decay under antiretroviral therapy: Influence of pharmacokinetics and intracellular delay, J. Theoret. Biol., 226 (2004), 95-109.   DOI
29 A. M. Elaiw, Global properties of a class of virus infection models with multitarget cells, Nonlinear Dynam., 69 (2012) 423-435.   DOI
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 A. M. Elaiw, Global properties of a class of HIV models, Nonlinear Anal. Real World Appl., 11 (2010), 2253-2263.   DOI
32 M. A. Nowak, C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79.   DOI   ScienceOn
33 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
34 M. A. Nowak and R. M. May, "Virus dynamics: Mathematical Principles of Immunology and Virology," Oxford Uni., Oxford, 2000.
35 Y. Zhao, D. T. Dimitrov, H. Liu and Y. Kuang, Mathematical insights in evaluating state dependent e ectiveness of HIV prevention interventions, Bull. Math. Biol., 75 (2013), 649-675.   DOI
36 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
37 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
38 R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of $CD4^+$ T-cells, Math. Biosci., 165 (2000), 27-39.   DOI   ScienceOn
39 P. W. Nelson and A. S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection, Math. Biosci., 179 (2002), 73-94.   DOI   ScienceOn
40 P. W. Nelson, J. Murray and A. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay, Math. Biosci., 163 (2000), 201-215.   DOI   ScienceOn
41 N. Bairagi, D. Adak, Global analysis of HIV-1 dynamics with Hill type infection rate and intracellular delay, Appl. Math. Model. (In press).
42 J. Mittler, B. Sulzer, A. Neumann and A. Perelson, Influence of delayed virus production on viral dynamics in HIV-1 infected patients, Math. Biosci., 152 (1998), 143-163.   DOI   ScienceOn
43 A.M. Elaiw and S.A. Azoz, Global properties of a class of HIV infection models with Beddington-DeAngelis functional response, Math. Method Appl. Sci., 36 (2013), 383-394.   DOI
44 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.
45 L. Wang, 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
46 M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79.   DOI   ScienceOn
47 A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44.   DOI   ScienceOn