American Institute of Mathematical Sciences

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July  2015, 20(5): 1573-1582. doi: 10.3934/dcdsb.2015.20.1573

Global behaviour of a delayed viral kinetic model with general incidence rate

Hong Yang 1, and  Junjie Wei 2,

1. 

Department of Mathematics, Heilongjiang Bayi Agricultural University, Daqing, Heilongjiang, 163319, China
2. 

Department of Mathematics, Harbin Institute of Technology(Weihai), Weihai, Shandong, 264209, China

Received  June 2014 Revised  October 2014 Published  May 2015

This paper aims to show the global behaviour of a viral kinetic model with two time delays and general incidence rate. For the basic reproduction number $R_{0}<1$, the disease-free equilibrium is shown to be globally asymptotically stable by constructing Lyapunov functional and using LaSalle invariance principle. For the basic reproduction number $R_{0}>1$, the interior equilibrium of model exists and is also globally asymptotically stable. Our work show more general conclusion than other known papers on delayed viral models.
Keywords: time delay, immune response, Viral model, global asymptotic stability..
Mathematics Subject Classification: Primary: 34K20; Secondary: 92B0.
Citation: Hong Yang, Junjie Wei. Global behaviour of a delayed viral kinetic model with general incidence rate. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1573-1582. doi: 10.3934/dcdsb.2015.20.1573
References:
[1]

E. Beretta and Y. Kuang, Geometric stability switches criteria in delay differential systems with delay dependent parameters, Siam. J. Math. Anal., 33 (2002), 1144-1165. doi: 10.1137/S0036141000376086.

[2]

K. Hattaf, N. Yousfi and A. Tridane, A delay virus dynamics model with general incidence rate, Differ. Equ. Dyn. Syst., 22 (2014), 181-190. doi: 10.1007/s12591-013-0167-5.

[3]

S. Hews, S. Eikenberry, J. D. Nagy and Y. Kuang, Rich dynamics of a hepatitis B viral infection model with logistic hepatocyte growth, Mathematical Biology, 60 (2010), 573-590. doi: 10.1007/s00285-009-0278-3.

[4]

A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883. doi: 10.1016/j.bulm.2004.02.001.

[5]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, 1993.

[6]

J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM Philadelphia, PA, 1976.

[7]

D. Li and W. Ma, Asymptotic properties of a HIV-1 infection model with time delay, Journal of Mathematical Analysis and Applications, 335 (2007), 683-691. doi: 10.1016/j.jmaa.2007.02.006.

[8]

M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay, Bulletin of Mathematical Biology, 72 (2010), 1492-1505. doi: 10.1007/s11538-010-9503-x.

[9]

Y. Nakata, Global dynamics of a viral infection model with a latent period and Beddington-DeAngelis response, Nonlinear Analysis, 74 (2011), 2929-2940. doi: 10.1016/j.na.2010.12.030.

[10]

M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79.

[11]

A. S. Perelson, D. E. Kirschner and R. D. Boer, Dynamics of HIV infection of CD$4^+$ T-cells, Math. Biosci., 114 (1993), 81-125. doi: 10.1016/0025-5564(93)90043-A.

[12]

Y. Qu and J. Wei, Bifurcation analysis in a predator-prey system with stage-structure and harvesting, Journal of Franklin Institute, 347 (2010), 1097-1113. doi: 10.1016/j.jfranklin.2010.03.017.

[13]

S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 10 (2003), 863-874.

[14]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995.

[15]

Y. Song, Y. Peng and J. Wei, Bifurcations for a predator-prey system with two delays, J. Math. Anal. Appl., 337 (2008), 466-479. doi: 10.1016/j.jmaa.2007.04.001.

[16]

J. P. Tian, S. Liao and J. Wang, Analyzing the infection dynamics and control strategies of cholera, Discrete and Continuous Systems, (2013), 747-757.

[17]

J. P. Tian and J. Wang, Global stability for cholera epidemic models, Mathematical Bio-sciences, 232 (2011), 31-41. doi: 10.1016/j.mbs.2011.04.001.

[18]

J. Wei and S. Ruan, Stability and bifurcation in a neural network model with two delays, Physica D: Nonlinear Phenomena, 130 (1999), 255-272. doi: 10.1016/S0167-2789(99)00009-3.

[19]

J. Wei and M. Y. Li, Global existence of periodic solutions in a tri-neuron network model with delays, Physica D, 198 (2004), 106-119. doi: 10.1016/j.physd.2004.08.023.

show all references

References:
[1]

E. Beretta and Y. Kuang, Geometric stability switches criteria in delay differential systems with delay dependent parameters, Siam. J. Math. Anal., 33 (2002), 1144-1165. doi: 10.1137/S0036141000376086.

[2]

K. Hattaf, N. Yousfi and A. Tridane, A delay virus dynamics model with general incidence rate, Differ. Equ. Dyn. Syst., 22 (2014), 181-190. doi: 10.1007/s12591-013-0167-5.

[3]

S. Hews, S. Eikenberry, J. D. Nagy and Y. Kuang, Rich dynamics of a hepatitis B viral infection model with logistic hepatocyte growth, Mathematical Biology, 60 (2010), 573-590. doi: 10.1007/s00285-009-0278-3.

[4]

A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883. doi: 10.1016/j.bulm.2004.02.001.

[5]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, 1993.

[6]

J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM Philadelphia, PA, 1976.

[7]

D. Li and W. Ma, Asymptotic properties of a HIV-1 infection model with time delay, Journal of Mathematical Analysis and Applications, 335 (2007), 683-691. doi: 10.1016/j.jmaa.2007.02.006.

[8]

M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay, Bulletin of Mathematical Biology, 72 (2010), 1492-1505. doi: 10.1007/s11538-010-9503-x.

[9]

Y. Nakata, Global dynamics of a viral infection model with a latent period and Beddington-DeAngelis response, Nonlinear Analysis, 74 (2011), 2929-2940. doi: 10.1016/j.na.2010.12.030.

[10]

M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79.

[11]

A. S. Perelson, D. E. Kirschner and R. D. Boer, Dynamics of HIV infection of CD$4^+$ T-cells, Math. Biosci., 114 (1993), 81-125. doi: 10.1016/0025-5564(93)90043-A.

[12]

Y. Qu and J. Wei, Bifurcation analysis in a predator-prey system with stage-structure and harvesting, Journal of Franklin Institute, 347 (2010), 1097-1113. doi: 10.1016/j.jfranklin.2010.03.017.

[13]

S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 10 (2003), 863-874.

[14]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995.

[15]

Y. Song, Y. Peng and J. Wei, Bifurcations for a predator-prey system with two delays, J. Math. Anal. Appl., 337 (2008), 466-479. doi: 10.1016/j.jmaa.2007.04.001.

[16]

J. P. Tian, S. Liao and J. Wang, Analyzing the infection dynamics and control strategies of cholera, Discrete and Continuous Systems, (2013), 747-757.

[17]

J. P. Tian and J. Wang, Global stability for cholera epidemic models, Mathematical Bio-sciences, 232 (2011), 31-41. doi: 10.1016/j.mbs.2011.04.001.

[18]

J. Wei and S. Ruan, Stability and bifurcation in a neural network model with two delays, Physica D: Nonlinear Phenomena, 130 (1999), 255-272. doi: 10.1016/S0167-2789(99)00009-3.

[19]

J. Wei and M. Y. Li, Global existence of periodic solutions in a tri-neuron network model with delays, Physica D, 198 (2004), 106-119. doi: 10.1016/j.physd.2004.08.023.

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