# American Institute of Mathematical Sciences

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.  Google Scholar [2] D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for tropic interaction, Ecology, 56 (1975), 881-892.  doi: 10.2307/1936298.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] S. A. Gourley, Y. 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.  Google Scholar [6] J. K. Hale, Theory of Functional Differential Equations Springer, Berlin-Heidelberg-New York, 1977.  Google Scholar [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.  Google Scholar [8] K. Hattaf, M. 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.  Google Scholar [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.  Google Scholar [10] G. Huang, W. 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.  Google Scholar [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.  Google Scholar [12] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics Mathematics in Science and Engineering, 191. Academic Press, Inc. , Boston, MA, 1993.  Google Scholar [13] A. M. Lyapunov, The General Problem of the Stability of Motion Kharkov Mathematical Society, Kharkov, 1892, 251p. Google Scholar [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.  Google Scholar [15] A. Perelson, A. Neumann, M. Markowitz, J. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] J. Wang, J. Pang, T. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] N. Yousfi, K. 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.  Google Scholar [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.   Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [2] D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for tropic interaction, Ecology, 56 (1975), 881-892.  doi: 10.2307/1936298.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] S. A. Gourley, Y. 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.  Google Scholar [6] J. K. Hale, Theory of Functional Differential Equations Springer, Berlin-Heidelberg-New York, 1977.  Google Scholar [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.  Google Scholar [8] K. Hattaf, M. 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.  Google Scholar [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.  Google Scholar [10] G. Huang, W. 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.  Google Scholar [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.  Google Scholar [12] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics Mathematics in Science and Engineering, 191. Academic Press, Inc. , Boston, MA, 1993.  Google Scholar [13] A. M. Lyapunov, The General Problem of the Stability of Motion Kharkov Mathematical Society, Kharkov, 1892, 251p. Google Scholar [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.  Google Scholar [15] A. Perelson, A. Neumann, M. Markowitz, J. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] J. Wang, J. Pang, T. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] N. Yousfi, K. 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.  Google Scholar [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.   Google Scholar [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.  Google Scholar
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