# American Institute of Mathematical Sciences

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

 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.
Citation: Hong Yang, Junjie Wei. Global behaviour of a delayed viral kinetic model with general incidence rate. Discrete & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, 1993.  Google Scholar [6] J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM Philadelphia, PA, 1976.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, 1993.  Google Scholar [6] J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM Philadelphia, PA, 1976.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
 [1] Yu Ji, Lan Liu. Global stability of a delayed viral infection model with nonlinear immune response and general incidence rate. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 133-149. doi: 10.3934/dcdsb.2016.21.133 [2] Jinliang Wang, Lijuan Guan. Global stability for a HIV-1 infection model with cell-mediated immune response and intracellular delay. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 297-302. doi: 10.3934/dcdsb.2012.17.297 [3] Alexander Rezounenko. Stability of a viral infection model with state-dependent delay, CTL and antibody immune responses. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1547-1563. doi: 10.3934/dcdsb.2017074 [4] Yincui Yan, Wendi Wang. Global stability of a five-dimensional model with immune responses and delay. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 401-416. doi: 10.3934/dcdsb.2012.17.401 [5] Hong Yang, Junjie Wei. Dynamics of spatially heterogeneous viral model with time delay. Communications on Pure & Applied Analysis, 2020, 19 (1) : 85-102. doi: 10.3934/cpaa.2020005 [6] Aiping Wang, Michael Y. Li. Viral dynamics of HIV-1 with CTL immune response. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 2257-2272. doi: 10.3934/dcdsb.2020212 [7] Cuicui Jiang, Kaifa Wang, Lijuan Song. Global dynamics of a delay virus model with recruitment and saturation effects of immune responses. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1233-1246. doi: 10.3934/mbe.2017063 [8] Lianwen Wang, Zhijun Liu, Yong Li, Dashun Xu. Complete dynamical analysis for a nonlinear HTLV-I infection model with distributed delay, CTL response and immune impairment. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 917-933. doi: 10.3934/dcdsb.2019196 [9] Jinhu Xu, Yicang Zhou. Bifurcation analysis of HIV-1 infection model with cell-to-cell transmission and immune response delay. Mathematical Biosciences & Engineering, 2016, 13 (2) : 343-367. doi: 10.3934/mbe.2015006 [10] Huiyan Zhu, Xingfu Zou. Dynamics of a HIV-1 Infection model with cell-mediated immune response and intracellular delay. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 511-524. doi: 10.3934/dcdsb.2009.12.511 [11] Jinliang Wang, Jiying Lang, Xianning Liu. Global dynamics for viral infection model with Beddington-DeAngelis functional response and an eclipse stage of infected cells. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3215-3233. doi: 10.3934/dcdsb.2015.20.3215 [12] Lifeng Han, Changhan He, Yang Kuang. Dynamics of a model of tumor-immune interaction with time delay and noise. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2347-2363. doi: 10.3934/dcdss.2020140 [13] Haitao Song, Weihua Jiang, Shengqiang Liu. Virus dynamics model with intracellular delays and immune response. Mathematical Biosciences & Engineering, 2015, 12 (1) : 185-208. doi: 10.3934/mbe.2015.12.185 [14] Alexander Rezounenko. Viral infection model with diffusion and state-dependent delay: Stability of classical solutions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1091-1105. doi: 10.3934/dcdsb.2018143 [15] Yinshu Wu, Wenzhang Huang. Global stability of the predator-prey model with a sigmoid functional response. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1159-1167. doi: 10.3934/dcdsb.2019214 [16] Kaifa Wang, Yu Jin, Aijun Fan. The effect of immune responses in viral infections: A mathematical model view. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3379-3396. doi: 10.3934/dcdsb.2014.19.3379 [17] Anatoli F. Ivanov, Musa A. Mammadov. Global asymptotic stability in a class of nonlinear differential delay equations. Conference Publications, 2011, 2011 (Special) : 727-736. doi: 10.3934/proc.2011.2011.727 [18] Songbai Guo, Wanbiao Ma. Global dynamics of a microorganism flocculation model with time delay. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1883-1891. doi: 10.3934/cpaa.2017091 [19] Hongying Shu, Lin Wang. Global stability and backward bifurcation of a general viral infection model with virus-driven proliferation of target cells. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1749-1768. doi: 10.3934/dcdsb.2014.19.1749 [20] Yu Ji. Global stability of a multiple delayed viral infection model with general incidence rate and an application to HIV infection. Mathematical Biosciences & Engineering, 2015, 12 (3) : 525-536. doi: 10.3934/mbe.2015.12.525

2020 Impact Factor: 1.327