June 2017, 22(4): 1547-1563. doi: 10.3934/dcdsb.2017074

Stability of a viral infection model with state-dependent delay, CTL and antibody immune responses

V.N.Karazin Kharkiv National University, Kharkiv, 61022, Ukraine, Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, P.O. Box 18, 18208 Praha, CR

Received  March 2016 Revised  July 2016 Published  February 2017

A virus dynamics model with intracellular state-dependent delay and nonlinear infection rate of Beddington-DeAngelis functional response is studied. The technique of Lyapunov functionals is used to analyze stability of the main interior infection equilibrium which describes the case of both CTL and antibody immune responses activated. We consider first a particular biologically motivated class of discrete state-dependent delays. The general case is investigated next. The stability of the infection-free and the immune-exhausted equilibria is also discussed.

Citation: 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
References:
[1]

J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, Journal of Animal Ecology, 44 (1975), 331-340. doi: 10.2307/3866.

[2]

D. L. DeAngelisR. A. Goldstein and R. V. O'Neill, A model for tropic interaction, Ecology, 56 (1975), 881-892. doi: 10.2307/1936298.

[3]

O. Diekmann, S. van Gils, S. Verduyn Lunel and H. -O. Walther, Delay Equations: Functional, Complex, and Nonlinear Analysis Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.

[4]

R. D. Driver, A two-body problem of classical electrodynamics: The one-dimensional case, Ann. Physics, 21 (1963), 122-142. doi: 10.1016/0003-4916(63)90227-6.

[5]

S. A. GourleyY. Kuang and J. D. Nagy, Dynamics of a delay differential equation model of hepatitis B virus infection, Journal of Biological Dynamics, 2 (2008), 140-153. doi: 10.1080/17513750701769873.

[6]

J. K. Hale, Theory of Functional Differential Equations Springer, Berlin-Heidelberg-New York, 1977.

[7]

D. Wodarz, Killer Cell Dynamics. Mathematical and Computational Approaches to Immunology Interdisciplinary Applied Mathematics, 32. Springer-Verlag, New York, 2007. xiv+220 pp. doi: 10.1016/S1874-5725(06)80009-X.

[8]

K. HattafM. Khabouze and N. Yousfi, Dynamics of a generalized viral infection model with adaptive immune response, International Journal of Dynamics and Control, 3 (2015), 253-261. doi: 10.1007/s40435-014-0130-5.

[9]

K. Hattaf and N. Yousfi, A class of delayed viral infection models with general incidence rate and adaptive immune response, International Journal of Dynamics and Control, 4 (2016), 254-265. doi: 10.1007/s40435-015-0158-1.

[10]

G. HuangW. Ma and Y. Takeuchi, Global analysis for delay virus dynamics model with Beddington-DeAngelis functional response, Applied Mathematics Letters, 24 (2011), 1199-1203. doi: 10.1016/j.aml.2011.02.007.

[11]

A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886. doi: 10.1007/s11538-007-9196-y.

[12]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics Mathematics in Science and Engineering, 191. Academic Press, Inc. , Boston, MA, 1993.

[13]

A. M. Lyapunov, The General Problem of the Stability of Motion Kharkov Mathematical Society, Kharkov, 1892, 251p.

[14]

M. Nowak and C. Bangham, Population dynamics of immune response to persistent viruses, Science, 272 (1996), 74-79. doi: 10.1126/science.272.5258.74.

[15]

A. PerelsonA. NeumannM. MarkowitzJ. Leonard and 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.

[16]

A. V. Rezounenko, Differential equations with discrete state-dependent delay: Uniqueness and well-posedness in the space of continuous functions, Nonlinear Analysis: Theory, Methods and Applications, 70 (2009), 3978-3986. doi: 10.1016/j.na.2008.08.006.

[17]

A. V. Rezounenko, Non-linear partial differential equations with discrete state-dependent delays in a metric space, Nonlinear Analysis: Theory, Methods and Applications, 73 (2010), 1707-1714. doi: 10.1016/j.na.2010.05.005.

[18]

A. V. Rezounenko, A condition on delay for differential equations with discrete state-dependent delay, Journal of Mathematical Analysis and Applications, 385 (2012), 506-516. doi: 10.1016/j.jmaa.2011.06.070.

[19]

A. V. Rezounenko, Local properties of solutions to non-autonomous parabolic PDEs with state-dependent delays, Journal of Abstract Differential Equations and Applications, 2 (2012), 56-71.

[20]

A. V. Rezounenko and P. Zagalak, Non-local PDEs with discrete state-dependent delays: Well-posedness in a metric space, Discrete Contin. Dyn. Syst., 33 (2013), 819-835. doi: 10.3934/dcds.2013.33.819.

[21]

A. V. Rezounenko, Stability of a viral infection model with state-dependent delay, CTL and antibody immune responses, March 20, 2016, preprint, arXiv: 1603.06281.

[22]

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.

[23]

H.-O. Walther, The solution manifold and $C$ -1-smoothness for differential equations with state-dependent delay, Journal of Differential Equations, 195 (2003), 46-65. doi: 10.1016/j.jde.2003.07.001.

[24]

X. Wang and S. Liu, A class of delayed viral models with saturation infection rate and immune response, Math. Methods Appl. Sci., 36 (2013), 125-142. doi: 10.1002/mma.2576.

[25]

J. WangJ. PangT. Kuniya and Y. Enatsu, Global threshold dynamics in a five-dimensional virus model with cell-mediated, humoral immune responses and distributed delays, Applied Mathematics and Computation, 241 (2014), 298-316. doi: 10.1016/j.amc.2014.05.015.

[26]

D. Wodarz, Hepatitis C virus dynamics and pathology: The role of CTL and antibody responses, Journal of General Virology, 84 (2003), 1743-1750. doi: 10.1099/vir.0.19118-0.

[27]

D. Wodarz, Killer Cell Dynamics. Mathematical and Computational Approaches to Immunology Interdisciplinary Applied Mathematics, 32. Springer-Verlag, New York, 2007. xiv+220 pp. doi: 10.1007/978-0-387-68733-9.

[28]

Y. Yan and W. Wang, Global stability of a five-dimesional model with immune responses and delay, Discrete and Continuous Dynamical Systems -Series B, 17 (2012), 401-416. doi: 10.3934/dcdsb.2012.17.401.

[29]

N. YousfiK. Hattaf and A. Tridane, Modeling the adaptive immune response in HBV infection, Journal of Mathematical Biology, 63 (2011), 933-957. doi: 10.1007/s00285-010-0397-x.

[30]

Y. Zhao and Z. Xu, Global dynamics for a delayed hepatitis C virus infection model, Electronic Journal of Differential Equations, 2014 (2014), 1-18.

[31]

H. Zhu and X. Zou, Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay, Discrete and Continuous Dynamical Systems Series B, 12 (2009), 511-524. doi: 10.3934/dcdsb.2009.12.511.

show all references

References:
[1]

J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, Journal of Animal Ecology, 44 (1975), 331-340. doi: 10.2307/3866.

[2]

D. L. DeAngelisR. A. Goldstein and R. V. O'Neill, A model for tropic interaction, Ecology, 56 (1975), 881-892. doi: 10.2307/1936298.

[3]

O. Diekmann, S. van Gils, S. Verduyn Lunel and H. -O. Walther, Delay Equations: Functional, Complex, and Nonlinear Analysis Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.

[4]

R. D. Driver, A two-body problem of classical electrodynamics: The one-dimensional case, Ann. Physics, 21 (1963), 122-142. doi: 10.1016/0003-4916(63)90227-6.

[5]

S. A. GourleyY. Kuang and J. D. Nagy, Dynamics of a delay differential equation model of hepatitis B virus infection, Journal of Biological Dynamics, 2 (2008), 140-153. doi: 10.1080/17513750701769873.

[6]

J. K. Hale, Theory of Functional Differential Equations Springer, Berlin-Heidelberg-New York, 1977.

[7]

D. Wodarz, Killer Cell Dynamics. Mathematical and Computational Approaches to Immunology Interdisciplinary Applied Mathematics, 32. Springer-Verlag, New York, 2007. xiv+220 pp. doi: 10.1016/S1874-5725(06)80009-X.

[8]

K. HattafM. Khabouze and N. Yousfi, Dynamics of a generalized viral infection model with adaptive immune response, International Journal of Dynamics and Control, 3 (2015), 253-261. doi: 10.1007/s40435-014-0130-5.

[9]

K. Hattaf and N. Yousfi, A class of delayed viral infection models with general incidence rate and adaptive immune response, International Journal of Dynamics and Control, 4 (2016), 254-265. doi: 10.1007/s40435-015-0158-1.

[10]

G. HuangW. Ma and Y. Takeuchi, Global analysis for delay virus dynamics model with Beddington-DeAngelis functional response, Applied Mathematics Letters, 24 (2011), 1199-1203. doi: 10.1016/j.aml.2011.02.007.

[11]

A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886. doi: 10.1007/s11538-007-9196-y.

[12]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics Mathematics in Science and Engineering, 191. Academic Press, Inc. , Boston, MA, 1993.

[13]

A. M. Lyapunov, The General Problem of the Stability of Motion Kharkov Mathematical Society, Kharkov, 1892, 251p.

[14]

M. Nowak and C. Bangham, Population dynamics of immune response to persistent viruses, Science, 272 (1996), 74-79. doi: 10.1126/science.272.5258.74.

[15]

A. PerelsonA. NeumannM. MarkowitzJ. Leonard and 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.

[16]

A. V. Rezounenko, Differential equations with discrete state-dependent delay: Uniqueness and well-posedness in the space of continuous functions, Nonlinear Analysis: Theory, Methods and Applications, 70 (2009), 3978-3986. doi: 10.1016/j.na.2008.08.006.

[17]

A. V. Rezounenko, Non-linear partial differential equations with discrete state-dependent delays in a metric space, Nonlinear Analysis: Theory, Methods and Applications, 73 (2010), 1707-1714. doi: 10.1016/j.na.2010.05.005.

[18]

A. V. Rezounenko, A condition on delay for differential equations with discrete state-dependent delay, Journal of Mathematical Analysis and Applications, 385 (2012), 506-516. doi: 10.1016/j.jmaa.2011.06.070.

[19]

A. V. Rezounenko, Local properties of solutions to non-autonomous parabolic PDEs with state-dependent delays, Journal of Abstract Differential Equations and Applications, 2 (2012), 56-71.

[20]

A. V. Rezounenko and P. Zagalak, Non-local PDEs with discrete state-dependent delays: Well-posedness in a metric space, Discrete Contin. Dyn. Syst., 33 (2013), 819-835. doi: 10.3934/dcds.2013.33.819.

[21]

A. V. Rezounenko, Stability of a viral infection model with state-dependent delay, CTL and antibody immune responses, March 20, 2016, preprint, arXiv: 1603.06281.

[22]

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.

[23]

H.-O. Walther, The solution manifold and $C$ -1-smoothness for differential equations with state-dependent delay, Journal of Differential Equations, 195 (2003), 46-65. doi: 10.1016/j.jde.2003.07.001.

[24]

X. Wang and S. Liu, A class of delayed viral models with saturation infection rate and immune response, Math. Methods Appl. Sci., 36 (2013), 125-142. doi: 10.1002/mma.2576.

[25]

J. WangJ. PangT. Kuniya and Y. Enatsu, Global threshold dynamics in a five-dimensional virus model with cell-mediated, humoral immune responses and distributed delays, Applied Mathematics and Computation, 241 (2014), 298-316. doi: 10.1016/j.amc.2014.05.015.

[26]

D. Wodarz, Hepatitis C virus dynamics and pathology: The role of CTL and antibody responses, Journal of General Virology, 84 (2003), 1743-1750. doi: 10.1099/vir.0.19118-0.

[27]

D. Wodarz, Killer Cell Dynamics. Mathematical and Computational Approaches to Immunology Interdisciplinary Applied Mathematics, 32. Springer-Verlag, New York, 2007. xiv+220 pp. doi: 10.1007/978-0-387-68733-9.

[28]

Y. Yan and W. Wang, Global stability of a five-dimesional model with immune responses and delay, Discrete and Continuous Dynamical Systems -Series B, 17 (2012), 401-416. doi: 10.3934/dcdsb.2012.17.401.

[29]

N. YousfiK. Hattaf and A. Tridane, Modeling the adaptive immune response in HBV infection, Journal of Mathematical Biology, 63 (2011), 933-957. doi: 10.1007/s00285-010-0397-x.

[30]

Y. Zhao and Z. Xu, Global dynamics for a delayed hepatitis C virus infection model, Electronic Journal of Differential Equations, 2014 (2014), 1-18.

[31]

H. Zhu and X. Zou, Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay, Discrete and Continuous Dynamical Systems Series B, 12 (2009), 511-524. doi: 10.3934/dcdsb.2009.12.511.

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