Stability analysis of general viral infection models with humoral immunity
-
1958
Downloads
-
2224
Views
Authors
A. M. Elaiw
- Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia.
N. H. AlShamrani
- Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia.
Abstract
We present two nonlinear viral infection models with humoral immune response and investigate their global
stability. The first model describes the interaction of the virus, uninfected cells, infected cells and B cells.
This model is an improvement of some existing models by incorporating more general nonlinear functions
for: (i) the intrinsic growth rate of uninfected cells; (ii) the incidence rate of infection; (iii) the removal rate of
infected cells; (iv) the production, death and neutralize rates of viruses; (v) the activation and removal rate
of B cells. In the second model, we introduce an additional population representing the latently infected
cells. The latent-to-active conversion rate is also given by a more general nonlinear function. For each
model, we derive two threshold parameters and establish a set of conditions on the general functions which
are sufficient to determine the global dynamics of the models. By using suitable Lyapunov functions and
LaSalle's invariance principle, we prove the global asymptotic stability of all equilibria of the models.
Share and Cite
ISRP Style
A. M. Elaiw, N. H. AlShamrani, Stability analysis of general viral infection models with humoral immunity, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 2, 684--704
AMA Style
Elaiw A. M., AlShamrani N. H., Stability analysis of general viral infection models with humoral immunity. J. Nonlinear Sci. Appl. (2016); 9(2):684--704
Chicago/Turabian Style
Elaiw, A. M., AlShamrani, N. H.. "Stability analysis of general viral infection models with humoral immunity." Journal of Nonlinear Sciences and Applications, 9, no. 2 (2016): 684--704
Keywords
- Viral infection
- global stability
- humoral immune response
- Lyapunov function.
MSC
References
-
[1]
B. Buonomo, 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.
-
[2]
J. A. Deans, S. Cohen, Immunology of malaria , Ann. Rev. Microbiol., 37 (1983), 25-50.
-
[3]
D. Ebert, C. D. Zschokke-Rohringer, H. J. Carius, Dose effects and density-dependent regulation of two microparasites of Daphnia magna, Oecologia, 122 (2000), 200-209.
-
[4]
S. Eikenberry, S. Hews, J. D. Nagy, Y. Kuang, The dynamics of a delay model of HBV infection with logistic hepatocyte growth, Math. Biosci. Eng., 6 (2009), 283-299.
-
[5]
A. M. Elaiw, Global properties of a class of HIV models , Nonlinear Anal. Real World Appl., 11 (2010), 2253-2263.
-
[6]
A. M. Elaiw, Global properties of a class of virus infection models with multitarget cells, Nonlinear Dynam., 69 (2012), 423-435.
-
[7]
A. M. Elaiw , Global stability analysis of humoral immunity virus dynamics model including latently infected cells, J. Biol. Dyn., 9 (2015), 215-228.
-
[8]
A. M. Elaiw, Global threshold dynamics in humoral immunity viral infection models including an eclipse stage of infected cells, J. Korean Soc. Ind. Appl. Math., 19 (2015), 137-170.
-
[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. 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.
-
[11]
A. M. Elaiw, 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.
-
[12]
A. M. Elaiw, X. Xia, HIV dynamics: Analysis and robust multirate MPC-based treatment schedules, J. Math. Anal. Appl., 359 (2009), 285-301.
-
[13]
P. Georgescu, Y. H. Hsieh, Global stability for a virus dynamics model with nonlinear incidence of infection and removal, SIAM J. Appl. Math., 67 (2006), 337-353.
-
[14]
S. A. Gourley, Y. Kuang, J. D. Nagy , Dynamics of a delay differential equation model of hepatitis B virus infection, J. Biol. Dyn., 2 (2008), 140-153.
-
[15]
J. K. Hale, S. M. Verduyn Lunel, Introduction to functional differential equations, Springer-Verlag, New York (1993)
-
[16]
K. Hattaf, N. Yousfi, A. Tridane, Mathematical analysis of a virus dynamics model with general incidence rate and cure rate, Nonlinear Anal. Real World Appl., 13 (2012), 1866-1872.
-
[17]
G. Huang, Y. Takeuchi, W. Ma, Lyapunov functionals for delay differential equations model Of viral infections , SIAM J. Appl. Math., 70 (2010), 2693-2708.
-
[18]
A. Korobeinikov , Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883.
-
[19]
A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886.
-
[20]
D. C. Krakauer, M. Nowak , T-cell induced pathogenesis in HIV: Bystander effects and latent infection , Proc. R. Soc. Lond. B , 266 (1999), 1069-1075.
-
[21]
R. Larson, B. Edwards, Calculus of a single variable, Cengage Learning, USA (2010)
-
[22]
M. Y. Li, H. Shu, Impact of intracellular delays and target-cell dynamics on in vivo viral infections, SIAM J. Appl. Math., 70 (2010), 2434-2448.
-
[23]
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.
-
[24]
J. Li, K. Wang, Y. Yang , Dynamical behaviors of an HBV infection model with logistic hepatocyte growth, Math. Comput. Modelling, 54 (2011), 704-711.
-
[25]
S. Liu, L. Wang, Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy , Math. Biosci. Eng., 7 (2010), 675-685.
-
[26]
A. R. McLean, C. J. Bostock, Scrapie infections initiated at varying doses: an analysis of 117 titration experiments, Philos. Trans. R. Soc. Lond., Ser. B, Biol. Sci., 355 (2000), 1043-1050.
-
[27]
L. Min, Y. Su, Y. Kuang, Mathematical analysis of a basic model of virus infection with application to HBV infection, Rocky Mountain J. Math., 38 (2008), 1573-1585.
-
[28]
A. Murase, T. Sasaki, T. Kajiwara , Stability analysis of pathogen-immune interaction dynamics, J. Math. Biol., 51 (2005), 247-267.
-
[29]
M. A. Nowak, R. M. May, Virus dynamics: mathematical principles of immunology and virology, Oxford University, Oxford (2000)
-
[30]
M. A. Obaid, A. M. Elaiw, Stability of virus infection models with antibodies and chronically infected cells, Abstr. Appl. Anal., 2014 (2014), 12 pages.
-
[31]
H. Shu, L. Wang, J. Watmough , Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL imune responses, SIAM J. Appl. Math., 73 (2013), 1280-1302.
-
[32]
Y. Tian, X. Liu, Global dynamics of a virus dynamical model with general incidence rate and cure rate , Nonlinear Anal. Real World Appl., 16 (2014), 17-26.
-
[33]
K. Wang, A. Fan, A. Torres, Global properties of an improved hepatitis B virus model , Nonlinear Anal. Real World Appl., 11 (2010), 3131-3138.
-
[34]
T. Wang, Z. Hu, F. Liao, Stability and Hopf bifurcation for a virus infection model with delayed humoral immunity response, J. Math. Anal. Appl., 411 (2014), 63-74.
-
[35]
T. Wang, Z. Hu, F. Liao, W. Ma, Global stability analysis for delayed virus infection model with general incidence rate and humoral immunity, Math. Comput. Simulation, 89 (2013), 13-22.
-
[36]
L. Wang, M. Y. Li, Mathematical analysis of the global dynamics of a model for HIV infection of \(CD4^+\) T cells, Math. Biosci., 200 (2006), 44-57.
-
[37]
S. Wang, D. Zou , Global stability of in-host viral models with humoral immunity and intracellular delays, J. Appl. Math. Model, 36 (2012), 1313-1322.
-
[38]
S. Wang, D. Zou, Viral dynamics in a distributed time delayed HCV pathogenesis model, Int. J. Biomath., 2012 (2012), 15 pages.
-
[39]
R. Xu , Global stability of an HIV-1 infection model with saturation infection and intracellular delay, J. Math. Anal. Appl., 375 (2011), 75-81.
-
[40]
N. Yousfi, K. Hattaf, A. Tridane, Modeling the adaptive immune response in HBV infection, J. Math. Biol., 63 (2011), 933-957.
-
[41]
Z. Yuan, X. Zou, Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays, Math. Biosci. Eng., 10 (2013), 483-498.