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)
• Accepted : 2014.11.27
• Published : 2014.12.25

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.

References

1. 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
2. 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
3. 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. https://doi.org/10.1016/j.mbs.2005.12.026
4. M. A. Nowak and R. M. May, "Virus dynamics: Mathematical Principles of Immunology and Virology," Oxford Uni., Oxford, 2000.
5. 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. 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. 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. https://doi.org/10.1016/S0025-5564(99)00055-3
9. P. W. Nelson and A. S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection, Math. Biosci., 179 (2002), 73-94. https://doi.org/10.1016/S0025-5564(02)00099-8
10. N. Bairagi, D. Adak, Global analysis of HIV-1 dynamics with Hill type infection rate and intracellular delay, Appl. Math. Model. (In press).
11. R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of $CD4^+$ T-cells, Math. Biosci., 165 (2000), 27-39. https://doi.org/10.1016/S0025-5564(00)00006-7
12. 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. https://doi.org/10.1016/S0025-5564(98)10027-5
13. 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. https://doi.org/10.1002/mma.2596
14. 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.
15. 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.
16. 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
17. 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.
18. 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.
19. 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.
20. 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. https://doi.org/10.1016/j.jtbi.2003.09.002
21. 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
22. 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
23. M. A. Nowak, 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
24. 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
25. 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
26. J. Li, K. Wang, 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
27. 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
28. 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
29. 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
30. 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
31. P. Tanvi, G. Gujarati, and G. Ambika , Virus antibody dynamics in primary and secondary dengue infections, J. Math. Biol., (In press).
32. 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
33. 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
34. 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. https://doi.org/10.1093/intimm/12.4.467
35. 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.
36. 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.
37. 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.
38. 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. https://doi.org/10.1016/j.matcom.2013.03.004
39. 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
40. A. Korobeinikov, Global properties of infectious disease models with nonlinear incdence, Bull. Math. Biol., 69 (2007), 1871-1886. https://doi.org/10.1007/s11538-007-9196-y
41. 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
42. 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. https://doi.org/10.1002/mma.2576
43. 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. https://doi.org/10.1016/j.amc.2013.07.005
44. 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. https://doi.org/10.1016/j.nonrwa.2011.12.015
45. 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. https://doi.org/10.1016/j.joems.2013.12.010
46. K. Hattaf, N. Yousfi, A. Tridane, A delay virus dynamics model with general incidence rate, Differ. Equ. Dyn. 22(2), (2014) 181-190. https://doi.org/10.1007/s12591-013-0167-5
47. J. K. Hale and S. Verduyn Lunel, Introduction to functional differential equations, Springer-Verlag, New York, 1993.

Cited by

1. GLOBAL THRESHOLD DYNAMICS IN HUMORAL IMMUNITY VIRAL INFECTION MODELS INCLUDING AN ECLIPSE STAGE OF INFECTED CELLS vol.19, pp.2, 2015, https://doi.org/10.12941/jksiam.2015.19.137
2. Global Properties of General Viral Infection Models with Humoral Immune Response vol.25, pp.3, 2017, https://doi.org/10.1007/s12591-015-0247-9
3. Global stability of a delayed humoral immunity virus dynamics model with nonlinear incidence and infected cells removal rates vol.5, pp.2, 2017, https://doi.org/10.1007/s40435-015-0200-3
4. Qualitative Analysis of a Generalized Virus Dynamics Model with Both Modes of Transmission and Distributed Delays vol.2018, pp.1687-9651, 2018, https://doi.org/10.1155/2018/9818372