AIMS Mathematics, 2020, 5(2): 1519-1531. doi: 10.3934/math.2020104.

Research article Special Issues

Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Investigating of an immune system-cancer mathematical model with Mittag-Leffler kernel

Department of Mathematics, Faculty of Science and Arts, Balıkesir University, Turkey

Special Issues: Recent Advances in Fractional Calculus with Real World Applications

Cancer that is difficult to treat, is a very common disease today and there are many types of cancer such as lung, colon, stomach. When cancer settles in the body, the immune system tries to resist it. In this study, the mathematical model of the interaction between immune system components and cancer is discussed and is modified by using Atangana-Baleanu derivative. After investigating the existence and uniqueness of the solution of the fractional immune system-cancer model, numerical simulations are given via predictor-corrector scheme.
  Figure/Table
  Supplementary
  Article Metrics

Keywords immune system-cancer; fractional calculus; Atangana-Baleanu derivative; dendritic cells and IL-2; fixed point theory

Citation: Necati Özdemir, Esmehan Uçar. Investigating of an immune system-cancer mathematical model with Mittag-Leffler kernel. AIMS Mathematics, 2020, 5(2): 1519-1531. doi: 10.3934/math.2020104

References

  • 1. L. Marsha, K. R. Conroy, J. L. Davis, et al. Atlas Pathophysiology, Lippincott Williams & Wilkins, 2010.
  • 2. V. Kumar, A. Abbas, J. Aster, Robbins and cotran pathologic basis of disease, Canada: Elsevier, 2014.
  • 3. M. Zanetti, Tapping CD4 T cells for cancer immunotherapy: The choice of personalized genomics, J. Immunol., 194 (2015), 2049-2056.    
  • 4. D. Cassell, J. Forman, Linked recognition of helper and cytotoxic antigenic determinants for he generation of cytotoxic T lymphocytes, Ann. N. Y. Acad. Sci., 532 (1998), 51-60.
  • 5. H. Choudhry, N. Helmi, W. H. Abdulaal, et al. Prospects of IL-2 in cancer immunotherapy, BioMed Res. Int., 2018 (2018), 9056173.
  • 6. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006.
  • 7. D. Baleanu, K. Diethelm, E. Scalas, et al. Fractional calculus models and numerical methods, World Scientific, 2012
  • 8. N. Özdemir, D. Karadeniz, B. B. Iskender, Fractional optimal control problem of a distributed system in cylindrical coordinates, Phys. Lett. A, 373 (2009), 221-226.    
  • 9. F. Evirgen, N. Özdemir, Multistage adomian decomposition method for solving NLP problems over a nonlinear fractional dynamical system, J. Comput. Nonlinear Dyn., 6 (2011), 21003.
  • 10. F. Evirgen, Analyze the optimal solutions of optimization problems by means of fractional gradient based system using VIM, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 6 (2016), 75-83.
  • 11. Z. Hammouch, T. Mekkaoui, Circuit design and simulation for the fractional-order chaotic behavior in a new dynamical system, Complex Intell. Syst., 4 (2018), 251-260.    
  • 12. E. Bonyah, A. Atangana, M. A. Khan, Modeling the spread of computer virus via Caputo fractional derivative and the beta derivative, Asia Pacific Journal on Computational Engineering, 4 (2017), 1-15.    
  • 13. N. Özdemir, M. Yavuz, Numerical solution of fractional Black-Scholes equation by using the multivariate pade approximation, Acta Phys. Pol. A., 132 (2017), 1050-1053.    
  • 14. E. Uçar, N. Özdemir, E. Altun, Fractional order model of immune cells influenced by cancer cells, Math. Model. Nat. Phenom., 14 (2019), 308.
  • 15. A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel: theory and applications to heat transfer model, Therm. Sci., 20 (2016), 763-769.    
  • 16. M. Yavuz, N. Özdemir, H. M. Baskonus, Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel, Eur. Phys. J. Plus, 133 (2018), 215.
  • 17. V. F. Morales-Delgadoa, J. F. Gomez-Aguilar, M. A. Taneco-Hernandez, et al. Mathematical modeling of the smoking dynamics using fractional differential equations with local and nonlocal kernel, J. Nonlinear Sci. Appl., 11 (2018), 994-1014.    
  • 18. N. A. Asif, Z. Hammouch, M. B. Riaz, et al. Analytical solution of a Maxwell fluid with slip effects in view of the Caputo-Fabrizio derivative, Eur. Phys. J. Plus, 133 (2018), 272.
  • 19. I. Koca, Analysis of rubella disease model with non-local and non-singular fractional derivatives, Int. J. Optim. Control Theor. Appl. IJOCTA, 8 (2018), 17-25.
  • 20. D. Avcı A. Yetim, Analytical solutions to the advection-diffusion equation with the AtanganaBaleanu derivative over a finite domain, J. BAUN Inst. Sci. Technol., 20 (2018), 382-395.
  • 21. S. Uçar, E. Uçar, N. Özdemir, et al. Mathematical analysis and numerical simulation for a smoking model with Atangana-Baleanu derivative, Chaos, Solitons & Fractals, 118 (2019), 300-306.
  • 22. D. Baleanu, A. Fernandez, On some new properties of fractional derivatives with Mittag-Leffler kernel, Commun. Nonlinear Sci. Numer. Simulat., 59 (2018), 444-462.    
  • 23. A. Fernandez, D. Baleanu, H. M. Srivastava, Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions, Commun. Nonlinear Sci. Numer. Simulat., 67 (2019), 517-527.    
  • 24. S. Uçar, Existence and uniqueness results for a smoking model with determination and education in the frame of non-singular derivatives, Discrete Continuous Dyn. Syst. Ser. S, in press.
  • 25. F. Evirgen, S. Uçar, N. Özdemir, et al. System response of an alcoholism model under the effect of immigration via non-singular kernel derivative, Discrete Continuous Dyn. Syst. Ser. S, in press.
  • 26. J. E. Solis-Perez, J. F. Gomez-Aguilar, A. Atangana, A factional mathematical model of breast cancer competition model, Chaos, Solitons and Fractals, 127 (2019), 38-54.    
  • 27. V. F. Morales-Delgado, J. F. Gomez-Aguilar, K. Saad, et al. Application of the Caputo-Fabrizio and Atangana-Baleanu fractional derivatives to mathematical model of cancer chemotherapy effect, Math. Methods Appl. Sci., 42 (2019), 1167-1193.    
  • 28. P. Vereesha, D. G. Prakasha, H. M. Baskonus, New numerical surfaces to the mathematical model of cancer chemotherapy effect in Caputo fractional derivatives, CHAOS, 29 (2019), 1-13.
  • 29. A. Minelli, F. Topputo, F. Bernelli, Controlled drug delivery in cancer immunotherapy: Stability, optimization and monte carlo analysis, SIAM J. Appl. Math., 71 (2011), 2229-2245.    
  • 30. L. G. De Pillis, A. Radunskaya, A mathematical tumour model with immune resistance and drug therapy: An optimal control approach, Journal of Theoretical Medicine, 3 (2001), 79-100.    
  • 31. F. Castiglione, B. Piccoli, Cancer immunotheraphy, mathematical modeling and optimal control, J. Theor. Biol., 247 (2007), 723-732.    
  • 32. D. Baleanu, A. Jajarmi, M. Hajipour, On the nonlinear dynamical systems within the generalized fractional derivatives with Mittag-Leffler kernel, Nonlinear Dyn., 94 (2018), 397-414.    

 

This article has been cited by

  • 1. Sümeyra Uçar, Necati Özdemir, İlknur Koca, Eren Altun, Novel analysis of the fractional glucose–insulin regulatory system with non-singular kernel derivative, The European Physical Journal Plus, 2020, 135, 6, 10.1140/epjp/s13360-020-00420-w

Reader Comments

your name: *   your email: *  

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved