Mathematical analysis and numerical simulation for fractal-fractional cancer model

Noura Laksaci; Ahmed Boudaoui; Seham Mahyoub Al-Mekhlafi; Abdon Atangana; Noura Laksaci; Ahmed Boudaoui; Seham Mahyoub Al-Mekhlafi; Abdon Atangana

doi:10.3934/mbe.2023803

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 10: 18083-18103. doi: 10.3934/mbe.2023803

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Mathematical analysis and numerical simulation for fractal-fractional cancer model

1.
Laboratory of Mathematics Modeling and Applications. University of Adrar. National Road No. 06, Adrar 01000, Algeria
2.
Mathematics Department, Faculty of Education, Sana'a University, Yemen
3.
Department of Engineering Mathematics and Physics, Future University in Egypt, Egypt
4.
Faculty of Natural and Agricultural Sciences, University of the Free State, South Africa
5.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

Academic Editor: Carlo Cattani

Received: 06 July 2023 Revised: 27 August 2023 Accepted: 28 August 2023 Published: 20 September 2023

The mathematical oncology has received a lot of interest in recent years since it helps illuminate pathways and provides valuable quantitative predictions, which will shape more effective and focused future therapies. We discuss a new fractal-fractional-order model of the interaction among tumor cells, healthy host cells and immune cells. The subject of this work appears to show the relevance and ramifications of the fractal-fractional order cancer mathematical model. We use fractal-fractional derivatives in the Caputo senses to increase the accuracy of the cancer and give a mathematical analysis of the proposed model. First, we obtain a general requirement for the existence and uniqueness of exact solutions via Perov's fixed point theorem. The numerical approaches used in this paper are based on the Grünwald-Letnikov nonstandard finite difference method due to its usefulness to discretize the derivative of the fractal-fractional order. Then, two types of stabilities, Lyapunov's and Ulam-Hyers' stabilities, are established for the Incommensurate fractional-order and the Incommensurate fractal-fractional, respectively. The numerical results of this study are compatible with the theoretical analysis. Our approaches generalize some published ones because we employ the fractal-fractional derivative in the Caputo sense, which is more suitable for considering biological phenomena due to the significant memory impact of these processes. Aside from that, our findings are new in that we use Perov's fixed point result to demonstrate the existence and uniqueness of the solutions. The way of expressing the Ulam-Hyers' stabilities by utilizing the matrices that converge to zero is also novel in this area.
- cancer chaotic mode,
- fractal-fractional calculus,
- existence and uniqueness,
- Grünwald-Letnikov nonstandard finite difference method,
- stability
Citation: Noura Laksaci, Ahmed Boudaoui, Seham Mahyoub Al-Mekhlafi, Abdon Atangana. Mathematical analysis and numerical simulation for fractal-fractional cancer model[J]. Mathematical Biosciences and Engineering, 2023, 20(10): 18083-18103. doi: 10.3934/mbe.2023803

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Abstract

The mathematical oncology has received a lot of interest in recent years since it helps illuminate pathways and provides valuable quantitative predictions, which will shape more effective and focused future therapies. We discuss a new fractal-fractional-order model of the interaction among tumor cells, healthy host cells and immune cells. The subject of this work appears to show the relevance and ramifications of the fractal-fractional order cancer mathematical model. We use fractal-fractional derivatives in the Caputo senses to increase the accuracy of the cancer and give a mathematical analysis of the proposed model. First, we obtain a general requirement for the existence and uniqueness of exact solutions via Perov's fixed point theorem. The numerical approaches used in this paper are based on the Grünwald-Letnikov nonstandard finite difference method due to its usefulness to discretize the derivative of the fractal-fractional order. Then, two types of stabilities, Lyapunov's and Ulam-Hyers' stabilities, are established for the Incommensurate fractional-order and the Incommensurate fractal-fractional, respectively. The numerical results of this study are compatible with the theoretical analysis. Our approaches generalize some published ones because we employ the fractal-fractional derivative in the Caputo sense, which is more suitable for considering biological phenomena due to the significant memory impact of these processes. Aside from that, our findings are new in that we use Perov's fixed point result to demonstrate the existence and uniqueness of the solutions. The way of expressing the Ulam-Hyers' stabilities by utilizing the matrices that converge to zero is also novel in this area.

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