January  2012, 17(1): 401-416. doi: 10.3934/dcdsb.2012.17.401

Global stability of a five-dimensional model with immune responses and delay

1. 

Key Laboratory of Eco-environments in Three Gorges Reservoir Region, School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China, China

Received  September 2010 Revised  December 2010 Published  October 2011

In this article, we study a virus model with immune responses and an intracellular delay which is relatively large compared with virus life-cycle and is an indispensable factor in understanding virus infections, such as HIV and HBV infections. By constructing suitable Liapunov functionals, the global dynamics of the model is completely determined by the reproductive numbers for viral infection $R_0$, for CTL immune response $R_1$, for antibody immune response $R_2$, for CTL immune competition $R_3$ and for antibody immune competition $R_4$. The global stability of the model precludes the existence of periodic solution and other complex dynamical behaviors.
Citation: 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
References:
[1]

A. A. Canabarro, I. M. Gléria and M. L. Lyra, Periodic solutions and chaos in a non-linear model for the delayed cellular immune response,, Physica A, 342 (2004), 234.  doi: 10.1016/j.physa.2004.04.083.  Google Scholar

[2]

R. V. Culshaw and S. G. Ruan, A delay-differential equation model of HIV infection of CD4+ T-cells,, Math. Biosci., 165 (2000), 27.  doi: 10.1016/S0025-5564(00)00006-7.  Google Scholar

[3]

N. Eshima, M. Tabata, T. Okada and S. Karukaya, Population dynamics of HTLV-I infection: A discrete-time mathematical epidemic model approach,, Math. Med. Biol., 20 (2003), 29.  doi: 10.1093/imammb/20.1.29.  Google Scholar

[4]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations,", Applied Mathematical Sciences, 99 (1993).   Google Scholar

[5]

V. Herz, S. Bonhoeffer, R. Anderson, R. May and M. Nowak, Viral dynamics in vivo: Limitations on estimations on intracellular delay and virus decay,, Proc. Nat. Acad. Sci., 93 (1996), 7247.  doi: 10.1073/pnas.93.14.7247.  Google Scholar

[6]

Y. Iwasa, M. Franziska and M. A. Nowak, Virus evolution with patients increases pathogenicity,, J. Theor. Biol., 232 (2005), 17.  doi: 10.1016/j.jtbi.2004.07.016.  Google Scholar

[7]

M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay,, Bull. Math. Biol., 72 (2010), 1492.  doi: 10.1007/s11538-010-9503-x.  Google Scholar

[8]

J. M. Murray, R. H. Purcell and S. F. Wieland, The half-life of hepatitis B virions,, Hepatology, 44 (2006), 1117.  doi: 10.1002/hep.21364.  Google Scholar

[9]

P. W. Nelson and A. S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection,, Math. Biosci., 179 (2002), 73.  doi: 10.1016/S0025-5564(02)00099-8.  Google Scholar

[10]

H. Pang, W. Wang and K. Wang, Global properties of virus dynamics model with immune response,, Journal of Southwest China Normal University (Natural Science), 30 (2005), 796.   Google Scholar

[11]

A. S. Perelson, D. E. Kirschner and R. De Boer, Dynamics of HIV-I infection of CD4 T cells,, Math. Biosci., 114 (1993), 81.  doi: 10.1016/0025-5564(93)90043-A.  Google Scholar

[12]

A. S. Perelson, A. U. Neumann, M. Markowitz, J. M. Leonard and D. D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time,, Science, 271 (1996), 1582.  doi: 10.1126/science.271.5255.1582.  Google Scholar

[13]

K. Wang, W. Wang and X. Liu, Viral infection model with periodic lytic immune response,, Chaos Solitons Fractals, 28 (2006), 90.  doi: 10.1016/j.chaos.2005.05.003.  Google Scholar

[14]

K. Wang, W. Wang and X. Liu, Global stability in a viral infection model with lytic and nonlytic immune responses,, Compute. Math. Appl., 51 (2006), 1593.  doi: 10.1016/j.camwa.2005.07.020.  Google Scholar

[15]

D. Wodarz, Hepatitis C virus dynamics and pathology: The role of CTL and antibody responses,, J. Gen. Virol., 84 (2003), 1743.  doi: 10.1099/vir.0.19118-0.  Google Scholar

[16]

H. Zhu and X. Zou, Dynamics of a HIN-1 infection model with cell-mediated immune response and intracellular delay,, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 511.   Google Scholar

show all references

References:
[1]

A. A. Canabarro, I. M. Gléria and M. L. Lyra, Periodic solutions and chaos in a non-linear model for the delayed cellular immune response,, Physica A, 342 (2004), 234.  doi: 10.1016/j.physa.2004.04.083.  Google Scholar

[2]

R. V. Culshaw and S. G. Ruan, A delay-differential equation model of HIV infection of CD4+ T-cells,, Math. Biosci., 165 (2000), 27.  doi: 10.1016/S0025-5564(00)00006-7.  Google Scholar

[3]

N. Eshima, M. Tabata, T. Okada and S. Karukaya, Population dynamics of HTLV-I infection: A discrete-time mathematical epidemic model approach,, Math. Med. Biol., 20 (2003), 29.  doi: 10.1093/imammb/20.1.29.  Google Scholar

[4]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations,", Applied Mathematical Sciences, 99 (1993).   Google Scholar

[5]

V. Herz, S. Bonhoeffer, R. Anderson, R. May and M. Nowak, Viral dynamics in vivo: Limitations on estimations on intracellular delay and virus decay,, Proc. Nat. Acad. Sci., 93 (1996), 7247.  doi: 10.1073/pnas.93.14.7247.  Google Scholar

[6]

Y. Iwasa, M. Franziska and M. A. Nowak, Virus evolution with patients increases pathogenicity,, J. Theor. Biol., 232 (2005), 17.  doi: 10.1016/j.jtbi.2004.07.016.  Google Scholar

[7]

M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay,, Bull. Math. Biol., 72 (2010), 1492.  doi: 10.1007/s11538-010-9503-x.  Google Scholar

[8]

J. M. Murray, R. H. Purcell and S. F. Wieland, The half-life of hepatitis B virions,, Hepatology, 44 (2006), 1117.  doi: 10.1002/hep.21364.  Google Scholar

[9]

P. W. Nelson and A. S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection,, Math. Biosci., 179 (2002), 73.  doi: 10.1016/S0025-5564(02)00099-8.  Google Scholar

[10]

H. Pang, W. Wang and K. Wang, Global properties of virus dynamics model with immune response,, Journal of Southwest China Normal University (Natural Science), 30 (2005), 796.   Google Scholar

[11]

A. S. Perelson, D. E. Kirschner and R. De Boer, Dynamics of HIV-I infection of CD4 T cells,, Math. Biosci., 114 (1993), 81.  doi: 10.1016/0025-5564(93)90043-A.  Google Scholar

[12]

A. S. Perelson, A. U. Neumann, M. Markowitz, J. M. Leonard and D. D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time,, Science, 271 (1996), 1582.  doi: 10.1126/science.271.5255.1582.  Google Scholar

[13]

K. Wang, W. Wang and X. Liu, Viral infection model with periodic lytic immune response,, Chaos Solitons Fractals, 28 (2006), 90.  doi: 10.1016/j.chaos.2005.05.003.  Google Scholar

[14]

K. Wang, W. Wang and X. Liu, Global stability in a viral infection model with lytic and nonlytic immune responses,, Compute. Math. Appl., 51 (2006), 1593.  doi: 10.1016/j.camwa.2005.07.020.  Google Scholar

[15]

D. Wodarz, Hepatitis C virus dynamics and pathology: The role of CTL and antibody responses,, J. Gen. Virol., 84 (2003), 1743.  doi: 10.1099/vir.0.19118-0.  Google Scholar

[16]

H. Zhu and X. Zou, Dynamics of a HIN-1 infection model with cell-mediated immune response and intracellular delay,, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 511.   Google Scholar

[1]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048

[2]

Simone Göttlich, Elisa Iacomini, Thomas Jung. Properties of the LWR model with time delay. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020032

[3]

Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331

[4]

Chongyang Liu, Meijia Han, Zhaohua Gong, Kok Lay Teo. Robust parameter estimation for constrained time-delay systems with inexact measurements. Journal of Industrial & Management Optimization, 2021, 17 (1) : 317-337. doi: 10.3934/jimo.2019113

[5]

Maoli Chen, Xiao Wang, Yicheng Liu. Collision-free flocking for a time-delay system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1223-1241. doi: 10.3934/dcdsb.2020251

[6]

Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003

[7]

Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341

[8]

Marc Homs-Dones. A generalization of the Babbage functional equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 899-919. doi: 10.3934/dcds.2020303

[9]

Giuseppe Capobianco, Tom Winandy, Simon R. Eugster. The principle of virtual work and Hamilton's principle on Galilean manifolds. Journal of Geometric Mechanics, 2021  doi: 10.3934/jgm.2021002

[10]

Xiangrui Meng, Jian Gao. Complete weight enumerator of torsion codes. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020124

[11]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, 2021, 14 (1) : 89-113. doi: 10.3934/krm.2020050

[12]

Nguyen Thi Kim Son, Nguyen Phuong Dong, Le Hoang Son, Alireza Khastan, Hoang Viet Long. Complete controllability for a class of fractional evolution equations with uncertainty. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020104

[13]

Dandan Wang, Xiwang Cao, Gaojun Luo. A class of linear codes and their complete weight enumerators. Advances in Mathematics of Communications, 2021, 15 (1) : 73-97. doi: 10.3934/amc.2020044

[14]

Riadh Chteoui, Abdulrahman F. Aljohani, Anouar Ben Mabrouk. Classification and simulation of chaotic behaviour of the solutions of a mixed nonlinear Schrödinger system. Electronic Research Archive, , () : -. doi: 10.3934/era.2021002

[15]

Yancong Xu, Lijun Wei, Xiaoyu Jiang, Zirui Zhu. Complex dynamics of a SIRS epidemic model with the influence of hospital bed number. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021016

[16]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[17]

Shuang Liu, Yuan Lou. A functional approach towards eigenvalue problems associated with incompressible flow. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3715-3736. doi: 10.3934/dcds.2020028

[18]

Shudi Yang, Xiangli Kong, Xueying Shi. Complete weight enumerators of a class of linear codes over finite fields. Advances in Mathematics of Communications, 2021, 15 (1) : 99-112. doi: 10.3934/amc.2020045

[19]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[20]

Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (109)
  • HTML views (0)
  • Cited by (21)

Other articles
by authors

[Back to Top]