Stability of pathogen dynamics models with viral and cellular infections and immune impairment
Volume 11, Issue 4, pp 456--468
http://dx.doi.org/10.22436/jnsa.011.04.02
Publication Date: March 10, 2018
Submission Date: October 13, 2017
Revision Date: October 30, 2017
Accteptance Date: December 26, 2017
-
2717
Downloads
-
5679
Views
Authors
A. M. Elaiw
- Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia.
A. A. Raezah
- Department of Mathematics, Faculty of Science, King Khalid University, P. O. Box 25145, Abha 61466, Saudi Arabia.
B. S. Alofi
- Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia.
Abstract
We study the global stability analysis of pathogen infection models with
immune impairment. Both pathogen-to-susceptible and infected-to-susceptible
transmissions have been considered. We drive the basic reproduction
parameter \(\mathcal{R}_{0}\), which determines the global dynamics of
models. Using the method of Lyapunov function, we established the global
stability of the steady states of the models. Numerical simulations are used
to confirm the theoretical results.
Share and Cite
ISRP Style
A. M. Elaiw, A. A. Raezah, B. S. Alofi, Stability of pathogen dynamics models with viral and cellular infections and immune impairment, Journal of Nonlinear Sciences and Applications, 11 (2018), no. 4, 456--468
AMA Style
Elaiw A. M., Raezah A. A., Alofi B. S., Stability of pathogen dynamics models with viral and cellular infections and immune impairment. J. Nonlinear Sci. Appl. (2018); 11(4):456--468
Chicago/Turabian Style
Elaiw, A. M., Raezah, A. A., Alofi, B. S.. "Stability of pathogen dynamics models with viral and cellular infections and immune impairment." Journal of Nonlinear Sciences and Applications, 11, no. 4 (2018): 456--468
Keywords
- Global stability
- pathogen infection
- immune impairment transfer
- Lyapunov function
- cell-to-cell transmission
MSC
References
-
[1]
R. A. Arnaout, M. A. Nowak, D. Wodarz , HIV1 dynamics revisited: biphasic decay by cytotoxic T lymphocyte killing?, Proc. Roy. Soc. Lond. B, 267 (2000), 1347–1354.
-
[2]
E. Avila-Vales, N. Chan-Chí, G. García-Almeida , Analysis of a viral infection model with immune impairment, intracellular delay and general non-linear incidence rate , Chaos Solitons Fractals, 69 (2014), 1–9.
-
[3]
D. S. Callaway, A. S. Perelson , HIV-1 infection and low steady state viral loads, Bull. Math. Biol., 64 (2002), 29–64.
-
[4]
S.-S. Chen, C.-Y. Cheng, Y. Takeuchi , Stability analysis in delayed within-host viral dynamics with both viral and cellular infections, J. Math. Anal. Appl., 442 (2016), 642–672.
-
[5]
R. V. Culshaw, S. Ruan, G. Webb, A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay, J. Math. Biol., 46 (2003), 425–444.
-
[6]
A. M. Elaiw , Global properties of a class of HIV models, Nonlinear Anal. Real World Appl., 11 (2010), 2253–2263.
-
[7]
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), 13 pages.
-
[8]
A. M. Elaiw, Global properties of a class of virus infection models with multitarget cells, Nonlinear Dynam., 69 (2012), 423–435.
-
[9]
A. M. Elaiw, R. M. Abukwaik, 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., 2014 (2014), 25 pages.
-
[10]
A. M. Elaiw, N. A. Almuallem, Global dynamics of delay-distributed HIV infection models with differential drug efficacy in cocirculating target cells , Math. Methods Appl. Sci., 39 (2016), 4–31.
-
[11]
A. M. Elaiw, N. H. AlShamrani , Global properties of nonlinear humoral immunity viral infection models, Int. J. Biomath., 2015 (2015), 53 pages.
-
[12]
A. M. Elaiw, N. H. AlShamrani , Global stability of humoral immunity virus dynamics models with nonlinear infection rate and removal, Nonlinear Anal. Real World Appl., 26 (2015), 161–190.
-
[13]
A. M. Elaiw, 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.
-
[14]
A. M. Elaiw, I. Hassanien, S. A. Azoz , Global stability of HIV infection models with intracellular delays , J. Korean Math. Soc., 49 (2012), 779–794.
-
[15]
A. M. Elaiw, A. A. Raezah, K. Hattaf, Stability of HIV-1 infection with saturated virus-target and infected-target incidences and CTL immune response, Int. J. Biomath., 2017 (2017), 29 pages.
-
[16]
H. Gómez-Acevedo, M. Y. Li, Backward bifurcation in a model for HTLV-I infection of CD4+ T cells, Bull. Math. Biol., 67 (2005), 101–114.
-
[17]
Z. Hu, J. Zhang, H. Wang, W. Ma, F. Liao, Dynamics analysis of a delayed viral infection model with logistic growth and immune impairment, Appl. Math. Model., 38 (2014), 524–534.
-
[18]
D. Huang, X. Zhang, Y. Guo, H.Wang, Analysis of an HIV infection model with treatments and delayed immune response , Appl. Math. Model., 40 (2016), 3081–3089.
-
[19]
P. Krishnapriya, M. Pitchaimani , Analysis of time delay in viral infection model with immune impairment, J. Appl. Math. Comput., 55 (2017), 421–453.
-
[20]
P. Krishnapriya, M. Pitchaimani, Modeling and bifurcation analysis of a viral infection with time delay and immune impairment, Jpn. J. Ind. Appl. Math., 34 (2017), 99–139.
-
[21]
X. Lai, X. Zou, Modelling HIV-1 virus dynamics with both virus-to-cell infection and cell-to-cell transmission , SIAM J. Appl. Math., 74 (2014), 898–917.
-
[22]
X. Lai, X. Zou, Modeling cell-to-cell spread of HIV-1 with logistic target cell growth, J. Math. Anal. Appl., 426 (2015), 563–584.
-
[23]
B. Li, Y. Chen, X. Lu, S. Liu , A delayed HIV-1 model with virus waning term , Math. Biosci. Eng., 13 (2016), 135–157.
-
[24]
X. Li, S. Fu , Global stability of a virus dynamics model with intracellular delay and CTL immune response, Math. Methods Appl. Sci., 38 (2015), 420–430.
-
[25]
M. Y. Li, 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.
-
[26]
M. Y. Li, L. Wang , Backward bifurcation in a mathematical model for HIV infection in vivo with anti-retroviral treatment, Nonlinear Anal. Real World Appl., 17 (2014), 147–160.
-
[27]
C. Lv, L. Huang, Z. Yuan, Global stability for an HIV-1 infection model with Beddington–DeAngelis incidence rate and CTL immune response , Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 121–127.
-
[28]
C. Monica, M. Pitchaimani, Analysis of stability and Hopf bifurcation for HIV-1 dynamics with PI and three intracellular delays, Nonlinear Anal. Real World Appl., 27 (2016), 55–69.
-
[29]
A. U. Neumann, N. P. Lam, H. Dahari, D. R. Gretch, T. E. Wiley, T. J. Layden, A. S. Perelson, Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon-alpha therapy , Science, 282 (1998), 103–107.
-
[30]
M. A. Nowak, C. R. M. Bangham , Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74–79.
-
[31]
M. A. Nowak, R. May , Virus dynamics: Mathematical Principles of Immunology and Virology, Oxford Uni., UK (2000)
-
[32]
J. Pang, J.-A. Cui , Analysis of a hepatitis B viral infection model with immune response delay , Int. J. Biomath., 2017 (2017), 18 pages.
-
[33]
J. Pang, J.-A. Cui. J. Hui , The importance of immune responses in a model of hepatitis B virus , Nonlinear Dynam., 67 (2012), 723–734.
-
[34]
H. Pourbashash, S. S. Pilyugin, P. De Leenheer, C. McCluskey, Global analysis of within host virus models with cell-to-cell viral transmission, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3341–3357.
-
[35]
R. R. Regoes, D. Wodarz, M. A. Nowak, Virus dynamics: the effect to target cell limitation and immune responses on virus evolution , J. Theor. Biol., 191 (1998), 451–462.
-
[36]
P. K. Roy, A. N. Chatterjee, D. Greenhalgh, 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.
-
[37]
H. Shu, L.Wang, J.Watmough , Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL immune responses, SIAM J. Appl. Math., 73 (2013), 1280–1302.
-
[38]
X. Wang, A. M. Elaiw, X. Song , Global properties of a delayed HIV infection model with CTL immune response, Appl. Math. Comput., 218 (2012), 9405–9414.
-
[39]
K. Wang, A. Fan, A. Torres , Global properties of an improved hepatitis B virus model, Nonlinear Anal. Real World Appl., 11 (2010), 3131–3138.
-
[40]
J. Wang, M. Guo, X. Liu, Z. Zhao, Threshold dynamics of HIV-1 virus model with cell-to-cell transmission, cell-mediated immune responses and distributed delay, Appl. Math. Comput., 291 (2016), 149–161.
-
[41]
J. Wang, J. Lang, X. Zou , Analysis of an age structured HIV infection model with virus-to-cell infection and cell-to-cell transmission , Nonlinear Anal. Real World Appl., 34 (2017), 75–96.
-
[42]
L. Wang, M. Y. Li, D. Kirschner, Mathematical analysis of the global dynamics of a model for HTLV-I infection and ATL progression , Math. Biosci., 179 (2002), 207–217.
-
[43]
S. Wang, X. Song, Z. Ge, Dynamics analysis of a delayed viral infection model with immune impairment , Appl. Math. Model., 35 (2011), 4877–4885.
-
[44]
K. Wang, W. Wang, X. Liu, Global Stability in a viral infection model with lytic and nonlytic immune response, Comput. Math. Appl., 51 (2006), 1593–1610.
-
[45]
Y. Yang, L. Zou, S. Ruan , Global dynamics of a delayed within-host viral infection model with both virus-to-cell and cell-to-cell transmissions, Math. Biosci., 270 (2015), 183–191.
-
[46]
F. Zhang, J. Li, C. Zheng, L. Wang , Dynamics of an HBV/HCV infection model with intracellular delay and cell proliferation, Commun. Nonlinear Sci. Numer. Simul., 42 (2017), 464–476.
-
[47]
S. Zhang, X. Xu , Dynamic analysis and optimal control for a model of hepatitis C with treatment, Commun. Nonlinear Sci. Numer. Simul., 46 (2017), 14–25.
-
[48]
Y. Zhao, Z. Xu , Global dynamics for a delyed hepatitis C virus,infection model , Electron. J. Differential Equations, 2014 (2014), 18 pages.