Dynamical behavior of tumor-immune system with fractal-fractional operator

Muhammad Farman; Aqeel Ahmad; Ali Akgül; Muhammad Umer Saleem; Kottakkaran Sooppy Nisar; Velusamy Vijayakumar; Muhammad Farman; Aqeel Ahmad; Ali Akgül; Muhammad Umer Saleem; Kottakkaran Sooppy Nisar; Velusamy Vijayakumar

doi:10.3934/math.2022489

AIMS Mathematics

2022, Volume 7, Issue 5: 8751-8773. doi: 10.3934/math.2022489

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Dynamical behavior of tumor-immune system with fractal-fractional operator

1.
Department of Mathematics and Statistics, University of Lahore, Lahore-54590, Pakistan
2.
Department of Mathematics, Ghazi University, D. G. Khan, Pakistan
3.
Art and Science Faculty, Department of Mathematics, Siirt University, 56100 Siirt, Turkey
4.
Department of Mathematics, University of Education, Lahore-54590, Pakistan
5.
Department of Mathematics, College of Arts and Sciences, Wadi Aldawaser, Prince Sattam bin Abdulaziz University, Saudi Arabia
6.
Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, 632014, Tamilnadu, India

Received: 17 October 2021 Revised: 10 January 2022 Accepted: 08 February 2022 Published: 03 March 2022
MSC : 37C75, 93B05, 65L07

In this paper, the dynamical behavior of the fractional-order cancer model has been analyzed with the fractal-fractional operator, which discretized the conformable cancer model. The fractional-order model consists of the system of nonlinear fractional differential equations. Also, we discuss the fractional-order model to check the relationship between the immune system and cancer cells by mixing IL-12 cytokine and anti-PD-L1 inhibitor. The tumor-immune model has been studied qualitatively as well as quantitatively via Atangana-Baleanu fractal-fractional operator. The nonlinear analysis is used to check the Ulam-Hyres stability of the proposed model. Moreover, the dynamical behavior for the fractional-order model has been checked by using a fractal-fractional operator with a generalized Mittag-Leffler Kernel and verifying the effect of fractional parameters. Finally, the obtained solutions are interpreted biologically, and simulations are carried out to illustrate cancer disease and support theoretical results, which will be helpful for further analysis and to control the effect of cancer in the community.
- tumor-immune model,
- IL-12 cytokin,
- anti-PD-L1 inhibitor,
- fractional operator,
- Mittag-Leffler kernel,
- existence-uniqueness,
- Ulam-Hyres stability
Citation: Muhammad Farman, Aqeel Ahmad, Ali Akgül, Muhammad Umer Saleem, Kottakkaran Sooppy Nisar, Velusamy Vijayakumar. Dynamical behavior of tumor-immune system with fractal-fractional operator[J]. AIMS Mathematics, 2022, 7(5): 8751-8773. doi: 10.3934/math.2022489

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Abstract

In this paper, the dynamical behavior of the fractional-order cancer model has been analyzed with the fractal-fractional operator, which discretized the conformable cancer model. The fractional-order model consists of the system of nonlinear fractional differential equations. Also, we discuss the fractional-order model to check the relationship between the immune system and cancer cells by mixing IL-12 cytokine and anti-PD-L1 inhibitor. The tumor-immune model has been studied qualitatively as well as quantitatively via Atangana-Baleanu fractal-fractional operator. The nonlinear analysis is used to check the Ulam-Hyres stability of the proposed model. Moreover, the dynamical behavior for the fractional-order model has been checked by using a fractal-fractional operator with a generalized Mittag-Leffler Kernel and verifying the effect of fractional parameters. Finally, the obtained solutions are interpreted biologically, and simulations are carried out to illustrate cancer disease and support theoretical results, which will be helpful for further analysis and to control the effect of cancer in the community.

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