Stability analysis of nonlinear time-delayed systems with application to biological models

H.a. Kruthika; Arun D. Mahindrakar; Ramkrishna Pasumarthy

International Journal of Applied Mathematics and Computer Science (2017)

  • Volume: 27, Issue: 1, page 91-103
  • ISSN: 1641-876X

Abstract

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In this paper, we analyse the local stability of a gene-regulatory network and immunotherapy for cancer modelled as nonlinear time-delay systems. A numerically generated kernel, using the sum-of-squares decomposition of multivariate polynomials, is used in the construction of an appropriate Lyapunov-Krasovskii functional for stability analysis of the networks around an equilibrium point. This analysis translates to verifying equivalent LMI conditions. A delay-independent asymptotic stability of a second-order model of a gene regulatory network, taking into consideration multiple commensurate delays, is established. In the case of cancer immunotherapy, a predator-prey type model is adopted to describe the dynamics with cancer cells and immune cells contributing to the predator-prey population, respectively. A delay-dependent asymptotic stability of the cancer-free equilibrium point is proved. Apart from the system and control point of view, in the case of gene-regulatory networks such stability analysis of dynamics aids mimicking gene networks synthetically using integrated circuits like neurochips learnt from biological neural networks, and in the case of cancer immunotherapy it helps determine the long-term outcome of therapy and thus aids oncologists in deciding upon the right approach.

How to cite

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H.a. Kruthika, Arun D. Mahindrakar, and Ramkrishna Pasumarthy. "Stability analysis of nonlinear time-delayed systems with application to biological models." International Journal of Applied Mathematics and Computer Science 27.1 (2017): 91-103. <http://eudml.org/doc/288096>.

@article{H2017,
abstract = {In this paper, we analyse the local stability of a gene-regulatory network and immunotherapy for cancer modelled as nonlinear time-delay systems. A numerically generated kernel, using the sum-of-squares decomposition of multivariate polynomials, is used in the construction of an appropriate Lyapunov-Krasovskii functional for stability analysis of the networks around an equilibrium point. This analysis translates to verifying equivalent LMI conditions. A delay-independent asymptotic stability of a second-order model of a gene regulatory network, taking into consideration multiple commensurate delays, is established. In the case of cancer immunotherapy, a predator-prey type model is adopted to describe the dynamics with cancer cells and immune cells contributing to the predator-prey population, respectively. A delay-dependent asymptotic stability of the cancer-free equilibrium point is proved. Apart from the system and control point of view, in the case of gene-regulatory networks such stability analysis of dynamics aids mimicking gene networks synthetically using integrated circuits like neurochips learnt from biological neural networks, and in the case of cancer immunotherapy it helps determine the long-term outcome of therapy and thus aids oncologists in deciding upon the right approach.},
author = {H.a. Kruthika, Arun D. Mahindrakar, Ramkrishna Pasumarthy},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {time-delay; cancer immunotherapy; gene-regulatory network; sum of squares},
language = {eng},
number = {1},
pages = {91-103},
title = {Stability analysis of nonlinear time-delayed systems with application to biological models},
url = {http://eudml.org/doc/288096},
volume = {27},
year = {2017},
}

TY - JOUR
AU - H.a. Kruthika
AU - Arun D. Mahindrakar
AU - Ramkrishna Pasumarthy
TI - Stability analysis of nonlinear time-delayed systems with application to biological models
JO - International Journal of Applied Mathematics and Computer Science
PY - 2017
VL - 27
IS - 1
SP - 91
EP - 103
AB - In this paper, we analyse the local stability of a gene-regulatory network and immunotherapy for cancer modelled as nonlinear time-delay systems. A numerically generated kernel, using the sum-of-squares decomposition of multivariate polynomials, is used in the construction of an appropriate Lyapunov-Krasovskii functional for stability analysis of the networks around an equilibrium point. This analysis translates to verifying equivalent LMI conditions. A delay-independent asymptotic stability of a second-order model of a gene regulatory network, taking into consideration multiple commensurate delays, is established. In the case of cancer immunotherapy, a predator-prey type model is adopted to describe the dynamics with cancer cells and immune cells contributing to the predator-prey population, respectively. A delay-dependent asymptotic stability of the cancer-free equilibrium point is proved. Apart from the system and control point of view, in the case of gene-regulatory networks such stability analysis of dynamics aids mimicking gene networks synthetically using integrated circuits like neurochips learnt from biological neural networks, and in the case of cancer immunotherapy it helps determine the long-term outcome of therapy and thus aids oncologists in deciding upon the right approach.
LA - eng
KW - time-delay; cancer immunotherapy; gene-regulatory network; sum of squares
UR - http://eudml.org/doc/288096
ER -

References

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  1. Aluru, S. (2005). Handbook of Computational Molecular Biology, CRC Press, Boca Raton, FL. Zbl1115.68387
  2. Andrew, S.M., Baker, C.T. and Bocharov, G.A. (2007). Rival approaches to mathematical modelling in immunology, Journal of Computational and Applied Mathematics 205(2): 669-686. Zbl1119.92037
  3. Babbs, C.F. (2011). Predicting success or failure of immunotherapy for cancer: Insights from a clinically applicable mathematical model, American Journal of Cancer Research 2(2): 204-213. 
  4. Banerjee, S. (2008). Immunotherapy with interleukin-2: A study based on mathematical modeling, International Journal of Applied Mathematics and Computer Science 18(3): 389-398, DOI: 10.2478/v10006-008-0035-6. Zbl1176.93005
  5. Bell, G.I. (1973). Predator-prey equations simulating an immune response, Mathematical Biosciences 16(3): 291-314. Zbl0253.92003
  6. Bernot, G., Comet, J.-P., Richard, A., Chaves, M., Gouzé, J.-L. and Dayan, F. (2013). Modeling and analysis of gene regulatory networks, in F. Cazals and P. Kornprobst (Eds.), Modeling in Computational Biology and Biomedicine: A Multidisciplinary Endeavor, Springer, Berlin/Heidelberg, pp. 47-80. 
  7. Bo, W., Yang, L. and Jianquan, L. (2012). New results on global exponential stability for impulsive cellular neural networks with any bounded time-varying delays, Mathematical and Computer Modelling 55(3): 837-843. Zbl1255.93103
  8. Bodnar, M. (2015). General model of a cascade of reactions with time delays: Global stability analysis, Journal of Differential Equations 259(2): 777-795. Zbl1330.34121
  9. Chen, L. and Aihara, K. (2002). Stability of genetic regulatory networks with time delay, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 49(5): 602-608. 
  10. De Jong, H. (2002). Modeling and simulation of genetic regulatory systems: A literature review, Journal of Computational Biology 9(1): 67-103. 
  11. d'Onofrio, A. (2005). A general framework for modeling tumor-immune-system competition and immunotherapy: Mathematical analysis and biomedical inferences, Physica D: Nonlinear Phenomena 208(3): 220-235. Zbl1087.34028
  12. d'Onofrio, A. (2008). Metamodeling tumor-immune-system interaction, tumor evasion and immunotherapy, Mathematical and Computer Modelling 47(5): 614-637. Zbl1148.92026
  13. d'Onofrio, A., Gatti, F., Cerrai, P. and Freschi, L. (2010). Delay-induced oscillatory dynamics of tumour-immune system interaction, Mathematical and Computer Modelling 51(5): 572-591. Zbl1190.34088
  14. Eduardo, L. and Ruiz-Herrera, A. (2013). Attractivity, multistability, and bifurcation in delayed Hopfield's model with non-monotonic feedback, Journal of Differential Equations 255(11): 4244-4266. Zbl1291.34121
  15. Goodwin, B.C. (1963). Temporal Organization in Cells: A Dynamic Theory of Cellular Control Processes, Academic Press, London/New York, NY. 
  16. Gu, K., Chen, J. and Kharitonov, V.L. (2003). Stability of TimeDelay Systems, Springer, New York, NY. 
  17. Kao, C.-Y. and Pasumarthy, R. (2012). Stability analysis of interconnected Hamiltonian systems under time delays, IET Control Theory and Applications 6(4): 570-577. 
  18. Kao, C.-Y. and Rantzer, A. (2007). Stability analysis of systems with uncertain time-varying delays, Automatica 43(6): 959-970. Zbl1282.93203
  19. Kauffman, S.A. (1969). Metabolic stability and epigenesis in randomly constructed genetic nets, Journal of Theoretical Biology 22(3): 437-467. 
  20. Kolmanovskii, V. and Myshkis, A. (1999). Introduction to the Theory and Applications of Functional Differential Equations, Springer, Dordrecht. Zbl0917.34001
  21. Liu, Y., Xu, P., Lu, J. and Liang, J. (2016a). Global stability of Clifford-valued recurrent neural networks with time delays, Nonlinear Dynamics 84(2): 767-777. Zbl1354.93132
  22. Liu, Y., Zhang, D., Lu, J. and Cao, J. (2016b). Global μ-stability criteria for quaternion-valued neural networks with unbounded time-varying delays, Information Sciences 360: 273-288. 
  23. Loiseau, J.J., Michiels, W., Niculescu, S.-I. and Sipahi, R. (2009). Topics in Time Delay Systems: Analysis, Algorithms and Control, Springer, Berlin/Heidelberg. 
  24. Mazenc, F. and Niculescu, S.-I. (2001). Lyapunov stability analysis for nonlinear delay systems, Systems & Control Letters 42(4): 245-251. Zbl0974.93059
  25. Melief, C.J. (2005). Cancer immunology: Cat and mouse games, Nature 437(7055): 41-42. 
  26. Papachristodoulou, A. (2004). Analysis of nonlinear time-delay systems using the sum of squares decomposition, American Control Conference, Boston, MA, USA, pp. 4153-4158. 
  27. Papachristodoulou, A. and Prajna, S. (2005). A tutorial on sum of squares techniques for systems analysis, American Control Conference, Portland, OR, USA, pp. 2686-2700. Zbl1138.93391
  28. Parrilo, P.A. (2000). Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization, PhD thesis, California Institute of Technology, Pasadena, CA. 
  29. Pasumarthy, R. and Kao, C.-Y. (2009). On stability of time delay Hamiltonian systems, American Control Conference, St. Louis, MO, USA, pp. 4909-4914. 
  30. Richard, J.-P. (2003). Time-delay systems: An overview of some recent advances and open problems, Automatica 39(10): 1667-1694. Zbl1145.93302
  31. Saleem, M. and Agrawal, T. (2012). Complex dynamics in a mathematical model of tumor growth with time delays in the cell proliferation, International Journal of Scientific and Research Publications 2(6): 1-7. 
  32. Sharma, A., Kohar, V., Shrimali, M. and Sinha, S. (2014). Realizing logic gates with time-delayed synthetic genetic networks, Nonlinear Dynamics 76(1): 431-439. Zbl1319.92031
  33. She, Z. and Li, H. (2013). Dynamics of a density-dependent stage-structured predator-prey system with beddingtondeangelis functional response, Journal of Mathematical Analysis and Applications 406(1): 188-202. Zbl1306.92050
  34. Thomas, R. (1991). Regulatory networks seen as asynchronous automata: A logical description, Journal of Theoretical Biology 153(1): 1-23. 
  35. Villasana, M. and Radunskaya, A. (2003). A delay differential equation model for tumor growth, Journal of Mathematical Biology 47(3): 270-294. Zbl1023.92014
  36. Wang, S., Wang, S. and Song, X. (2012). Hopf bifurcation analysis in a delayed oncolytic virus dynamics with continuous control, Nonlinear Dynamics 67(1): 629-640. Zbl1242.92036
  37. Yang, R., Wu, B. and Liu, Y. (2015). A Halanay-type inequality approach to the stability analysis of discrete-time neural networks with delays, Applied Mathematics and Computation 265: 696-707. 
  38. Zhivkov, P. and Waniewski, J. (2003). Modelling tumour-immunity interactions with different stimulation functions, International Journal of Applied Mathematics and Computer Science 13(3): 307-315. Zbl1035.92024
  39. Zhong, J., Lu, J., Liu, Y. and Cao, J. (2014). Synchronization in an array of output-coupled Boolean networks with time delay, IEEE Transactions on Neural Networks and Learning Systems 25(12): 2288-2294. 

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