May  2018, 23(3): 1091-1105. doi: 10.3934/dcdsb.2018143

Viral infection model with diffusion and state-dependent delay: Stability of classical solutions

1. 

V.N.Karazin Kharkiv National University, Kharkiv, 61022, Ukraine

2. 

Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, P.O. Box 18,182 08 Praha, CR

This paper is dedicated to the memory of Igor D. Chueshov

Received  February 2017 Revised  June 2017 Published  February 2018

A class of reaction-diffusion virus dynamics models with intracellular state-dependent delay and a general non-linear infection rate functional response is investigated. We are interested in classical solutions with Lipschitz in-time initial functions which are adequate to the discontinuous change of parameters due to, for example, drug administration. The Lyapunov functions technique is used to analyse stability of interior infection equilibria which describe the cases of a chronic disease.

Citation: 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
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]

G. CarloniA. CremaM. B. ValliA. Ponzetto and M. Clementi, HCV infection by cell-to-cell transmission: Choice or necessity?, Current Molecular Medicine, 12 (2012), 83-95.  doi: 10.2174/156652412798376152.  Google Scholar

[3]

I. D. Chueshov and A. V. Rezounenko, Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay, Communications on Pure and Applied Analysis, 14 (2015), 1685-1704.  doi: 10.3934/cpaa.2015.14.1685.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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.  Google Scholar

[8]

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

[9]

F. Hartung, T. Krisztin, H. -O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications, In: Canada, A., Drabek., P. and A. Fonda (Eds. ) Handbook of Differential Equations, Ordinary Differential Equations, Elsevier Science B. V., North Holland, 3 (2006), 435-545. doi: 10.1016/S1874-5725(06)80009-X.  Google Scholar

[10]

K. Hattaf and N. Yousfi, A generalized HBV model with diffusion and two delays, Computers and Mathematics with Applications, 69 (2015), 31-40.  doi: 10.1016/j.camwa.2014.11.010.  Google Scholar

[11]

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.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

J. Mallet-ParetR. D. Nussbaum and P. Paraskevopoulos, Periodic solutions for functional-differential equations with multiple state-dependent time lags, Topol. Methods Nonlinear Anal., 3 (1994), 101-162.  doi: 10.12775/TMNA.1994.006.  Google Scholar

[16]

R. H. Martin Jr. and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590.  Google Scholar

[17]

C. McCluskey and Yu. Yang, Global stability of a diffusive virus dynamics model with general incidence function and time delay, Nonlinear Anal. Real World Appl, 25 (2015), 64-78.  doi: 10.1016/j.nonrwa.2015.05.003.  Google Scholar

[18]

J. M. MurrayA. D. Kelleher and D. A. Cooper, Timing of the Components of the HIV Life Cycle in Productively Infected CD4+ T Cells in a Population of HIV-Infected Individuals, J. Virol., 85 (2011), 10798-10805.  doi: 10.1128/JVI.05095-11.  Google Scholar

[19]

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

[20]

J. M. Pawlotsky, New hepatitis C virus (HCV) drugs and the hope for a cure: Concepts in anti-HCV drug development, Semin Liver Dis., 34 (2014), 22-29.  doi: 10.1055/s-0034-1371007.  Google Scholar

[21]

A. Pazy, Semigroups of Linear Operators and Applications to partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. ⅷ+279 pp. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[22]

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.  Google Scholar

[23]

A. V. Rezounenko, Partial differential equations with discrete and distributed state-dependent delays, Journal of Mathematical Analysis and Applications, 326 (2007), 1031-1045.  doi: 10.1016/j.jmaa.2006.03.049.  Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

A. V. Rezounenko and P. Zagalak, Non-local PDEs with discrete state-dependent delays: well-posedness in a metric space, Discrete and Continuous Dynamical Systems -Series A, 33 (2013), 819-835.  doi: 10.3934/dcds.2013.33.819.  Google Scholar

[29]

A. V. Rezounenko, Stability of a viral infection model with state-dependent delay, CTL and antibody immune responses, Discrete and Continuous Dynamical Systems -Series B, 22 (2017), 1547-1563; Preprint arXiv: 1603.06281v1, [math.DS], 20 March 2016, arXiv.org/abs/1603.06281v1. doi: 10.3934/dcdsb.2017074.  Google Scholar

[30]

A. V. Rezounenko, Continuous solutions to a viral infection model with general incidence rate, discrete state-dependent delay, CTL and antibody immune responses, Electron. J. Qual. Theory Differ. Equ., 79 (2016), 1-15.  doi: 10.14232/ejqtde.2016.1.79.  Google Scholar

[31]

E. ShudoR. M. RibeiroA. H. Talal and A. S. Perelson, A hepatitis C viral kinetic model that allows for time-varying drug effectiveness, Antiviral Therapy, 13 (2008), 919-926.   Google Scholar

[32]

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

[33]

H. Smith, An Introduction to Delay Differential Equations with Sciences Applications to the Life, Texts in Applied Mathematics, vol. 57, Springer, New York, Dordrecht, Heidelberg, London, 2011.  Google Scholar

[34]

C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Transactions of AMS, 200 (1974), 395-418.  doi: 10.1090/S0002-9947-1974-0382808-3.  Google Scholar

[35]

H.-O. Walther, The solution manifold and C1-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

[36]

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

[37]

F.-B. WangY. Huang and X. Zou, Global dynamics of a PDE in-host viral model, Applicable Analysis: An International Journal, 93 (2014), 2312-2329.  doi: 10.1080/00036811.2014.955797.  Google Scholar

[38]

K. Wang and W. Wang, Propagation of HBV with spatial dependence, Math. Biosci., 201 (2007), 78-95.  doi: 10.1016/j.mbs.2007.05.004.  Google Scholar

[39]

J. WangJ. Yang and T. Kuniya, Dynamics of a PDE viral infection model incorporating cell-to-cell transmission, Journal of Mathematical Analysis and Applications, 444 (2016), 1542-1564.  doi: 10.1016/j.jmaa.2016.07.027.  Google Scholar

[40]

World Health Organization, Global Hepatitis Report-2017, April 2017, ISBN: 978-92-4-156545-5 http://apps.who.int/iris/bitstream/10665/255016/1/9789241565455-eng.pdf?ua=1 Google Scholar

[41]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

[42]

S. Xu, Global stability of the virus dynamics model with Crowley-Martin functional response, J. Qual. Theory Differ. Equ., 2012 (2012), 1-10.  doi: 10.14232/ejqtde.2012.1.9.  Google Scholar

[43]

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

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.  Google Scholar

[2]

G. CarloniA. CremaM. B. ValliA. Ponzetto and M. Clementi, HCV infection by cell-to-cell transmission: Choice or necessity?, Current Molecular Medicine, 12 (2012), 83-95.  doi: 10.2174/156652412798376152.  Google Scholar

[3]

I. D. Chueshov and A. V. Rezounenko, Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay, Communications on Pure and Applied Analysis, 14 (2015), 1685-1704.  doi: 10.3934/cpaa.2015.14.1685.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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.  Google Scholar

[8]

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

[9]

F. Hartung, T. Krisztin, H. -O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications, In: Canada, A., Drabek., P. and A. Fonda (Eds. ) Handbook of Differential Equations, Ordinary Differential Equations, Elsevier Science B. V., North Holland, 3 (2006), 435-545. doi: 10.1016/S1874-5725(06)80009-X.  Google Scholar

[10]

K. Hattaf and N. Yousfi, A generalized HBV model with diffusion and two delays, Computers and Mathematics with Applications, 69 (2015), 31-40.  doi: 10.1016/j.camwa.2014.11.010.  Google Scholar

[11]

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.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

J. Mallet-ParetR. D. Nussbaum and P. Paraskevopoulos, Periodic solutions for functional-differential equations with multiple state-dependent time lags, Topol. Methods Nonlinear Anal., 3 (1994), 101-162.  doi: 10.12775/TMNA.1994.006.  Google Scholar

[16]

R. H. Martin Jr. and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590.  Google Scholar

[17]

C. McCluskey and Yu. Yang, Global stability of a diffusive virus dynamics model with general incidence function and time delay, Nonlinear Anal. Real World Appl, 25 (2015), 64-78.  doi: 10.1016/j.nonrwa.2015.05.003.  Google Scholar

[18]

J. M. MurrayA. D. Kelleher and D. A. Cooper, Timing of the Components of the HIV Life Cycle in Productively Infected CD4+ T Cells in a Population of HIV-Infected Individuals, J. Virol., 85 (2011), 10798-10805.  doi: 10.1128/JVI.05095-11.  Google Scholar

[19]

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

[20]

J. M. Pawlotsky, New hepatitis C virus (HCV) drugs and the hope for a cure: Concepts in anti-HCV drug development, Semin Liver Dis., 34 (2014), 22-29.  doi: 10.1055/s-0034-1371007.  Google Scholar

[21]

A. Pazy, Semigroups of Linear Operators and Applications to partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. ⅷ+279 pp. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[22]

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.  Google Scholar

[23]

A. V. Rezounenko, Partial differential equations with discrete and distributed state-dependent delays, Journal of Mathematical Analysis and Applications, 326 (2007), 1031-1045.  doi: 10.1016/j.jmaa.2006.03.049.  Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

A. V. Rezounenko and P. Zagalak, Non-local PDEs with discrete state-dependent delays: well-posedness in a metric space, Discrete and Continuous Dynamical Systems -Series A, 33 (2013), 819-835.  doi: 10.3934/dcds.2013.33.819.  Google Scholar

[29]

A. V. Rezounenko, Stability of a viral infection model with state-dependent delay, CTL and antibody immune responses, Discrete and Continuous Dynamical Systems -Series B, 22 (2017), 1547-1563; Preprint arXiv: 1603.06281v1, [math.DS], 20 March 2016, arXiv.org/abs/1603.06281v1. doi: 10.3934/dcdsb.2017074.  Google Scholar

[30]

A. V. Rezounenko, Continuous solutions to a viral infection model with general incidence rate, discrete state-dependent delay, CTL and antibody immune responses, Electron. J. Qual. Theory Differ. Equ., 79 (2016), 1-15.  doi: 10.14232/ejqtde.2016.1.79.  Google Scholar

[31]

E. ShudoR. M. RibeiroA. H. Talal and A. S. Perelson, A hepatitis C viral kinetic model that allows for time-varying drug effectiveness, Antiviral Therapy, 13 (2008), 919-926.   Google Scholar

[32]

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

[33]

H. Smith, An Introduction to Delay Differential Equations with Sciences Applications to the Life, Texts in Applied Mathematics, vol. 57, Springer, New York, Dordrecht, Heidelberg, London, 2011.  Google Scholar

[34]

C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Transactions of AMS, 200 (1974), 395-418.  doi: 10.1090/S0002-9947-1974-0382808-3.  Google Scholar

[35]

H.-O. Walther, The solution manifold and C1-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

[36]

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

[37]

F.-B. WangY. Huang and X. Zou, Global dynamics of a PDE in-host viral model, Applicable Analysis: An International Journal, 93 (2014), 2312-2329.  doi: 10.1080/00036811.2014.955797.  Google Scholar

[38]

K. Wang and W. Wang, Propagation of HBV with spatial dependence, Math. Biosci., 201 (2007), 78-95.  doi: 10.1016/j.mbs.2007.05.004.  Google Scholar

[39]

J. WangJ. Yang and T. Kuniya, Dynamics of a PDE viral infection model incorporating cell-to-cell transmission, Journal of Mathematical Analysis and Applications, 444 (2016), 1542-1564.  doi: 10.1016/j.jmaa.2016.07.027.  Google Scholar

[40]

World Health Organization, Global Hepatitis Report-2017, April 2017, ISBN: 978-92-4-156545-5 http://apps.who.int/iris/bitstream/10665/255016/1/9789241565455-eng.pdf?ua=1 Google Scholar

[41]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

[42]

S. Xu, Global stability of the virus dynamics model with Crowley-Martin functional response, J. Qual. Theory Differ. Equ., 2012 (2012), 1-10.  doi: 10.14232/ejqtde.2012.1.9.  Google Scholar

[43]

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

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