American Institute of Mathematical Sciences

Advanced
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

Yincui Yan 1, and  Wendi Wang 1,

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.
Keywords: Complete classification, Lyapuov functional, Reproductive number, Time delay, Invariance principle..
Mathematics Subject Classification: Primary: 92D30; Secondary: 34K2.
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), 234-241. 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), 27-39. 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), 29-45. 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, Springer-Verlag, New York, 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), 7247-7251. 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), 17-26. 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), 1492-1505. 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), 1117-1121. 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), 73-94. 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), 796-799. 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), 81-125. 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), 1582-1586. 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), 90-99. 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), 1593-1610. 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), 1743-1750. 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), 511-524.  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), 234-241. 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), 27-39. 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), 29-45. 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, Springer-Verlag, New York, 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), 7247-7251. 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), 17-26. 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), 1492-1505. 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), 1117-1121. 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), 73-94. 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), 796-799. 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), 81-125. 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), 1582-1586. 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), 90-99. 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), 1593-1610. 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), 1743-1750. 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), 511-524.  Google Scholar

[1]

Timothy C. Reluga, Jan Medlock, Alison Galvani. The discounted reproductive number for epidemiology. Mathematical Biosciences & Engineering, 2009, 6 (2) : 377-393. doi: 10.3934/mbe.2009.6.377
[2]

Zvi Artstein. Invariance principle in the singular perturbations limit. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3653-3666. doi: 10.3934/dcdsb.2018309
[3]

Ariel Cintrón-Arias, Carlos Castillo-Chávez, Luís M. A. Bettencourt, Alun L. Lloyd, H. T. Banks. The estimation of the effective reproductive number from disease outbreak data. Mathematical Biosciences & Engineering, 2009, 6 (2) : 261-282. doi: 10.3934/mbe.2009.6.261
[4]

Jifa Jiang, Fensidi Tang. The complete classification on a model of two species competition with an inhibitor. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 659-672. doi: 10.3934/dcds.2008.20.659
[5]

Igor Chueshov, Michael Scheutzow. Invariance and monotonicity for stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1533-1554. doi: 10.3934/dcdsb.2013.18.1533
[6]

Chunhong Li, Jiaowan Luo. Stochastic invariance for neutral functional differential equation with non-lipschitz coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3299-3318. doi: 10.3934/dcdsb.2018321
[7]

Anna-Lena Horlemann-Trautmann, Alessandro Neri. A complete classification of partial MDS (maximally recoverable) codes with one global parity. Advances in Mathematics of Communications, 2020, 14 (1) : 69-88. doi: 10.3934/amc.2020006
[8]

Chuangye Liu, Zhi-Qiang Wang. A complete classification of ground-states for a coupled nonlinear Schrödinger system. Communications on Pure & Applied Analysis, 2017, 16 (1) : 115-130. doi: 10.3934/cpaa.2017005
[9]

Cuicui Jiang, Wendi Wang. Complete classification of global dynamics of a virus model with immune responses. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1087-1103. doi: 10.3934/dcdsb.2014.19.1087
[10]

Seyedeh Marzieh Ghavidel, Wolfgang M. Ruess. Flow invariance for nonautonomous nonlinear partial differential delay equations. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2351-2369. doi: 10.3934/cpaa.2012.11.2351
[11]

Yaofeng Su. Almost surely invariance principle for non-stationary and random intermittent dynamical systems. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6585-6597. doi: 10.3934/dcds.2019286
[12]

Nikolaos Roidos, Yuanzhen Shao. Functional inequalities involving nonlocal operators on complete Riemannian manifolds and their applications to the fractional porous medium equation. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021026
[13]

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
[14]

Alfonso C. Casal, Jesús Ildefonso Díaz, José Manuel Vegas. Complete recuperation after the blow up time for semilinear problems. Conference Publications, 2015, 2015 (special) : 223-229. doi: 10.3934/proc.2015.0223
[15]

Xiaowei Tang, Xilin Fu. New comparison principle with Razumikhin condition for impulsive infinite delay differential systems. Conference Publications, 2009, 2009 (Special) : 739-743. doi: 10.3934/proc.2009.2009.739
[16]

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
[17]

Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687
[18]

J. P. Lessard, J. D. Mireles James. A functional analytic approach to validated numerics for eigenvalues of delay equations. Journal of Computational Dynamics, 2020, 7 (1) : 123-158. doi: 10.3934/jcd.2020005
[19]

Ismael Maroto, Carmen NÚÑez, Rafael Obaya. Dynamical properties of nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3939-3961. doi: 10.3934/dcds.2017167
[20]

Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (147)
  • HTML views (0)
  • Cited by (22)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences