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Chaotic dynamics in a non-linear tumor-immune model with Caputo–Fabrizio fractional operator

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Abstract

In this article, we investigate a tumor-immune and antigen-presenting cells population in the form of a mathematical model. To achieve greater accuracy in understanding the spread of tumor and immune cell populations, we apply the Caputo–Fabrizio (CF) fractional-order derivative. The fixed-point theorems are employed to analyze the uniqueness and existence of the model. The Laplace transform along with the Adomian decomposition approach is used to construct an algorithm for a semi-analytical solution under the CF fractional derivative. The chaotic dynamics of the cancer-immune model are confirmed by the Lyapunov exponents. The article examines how different vaccination protocols can affect tumor dormancy and recurrence. Furthermore, we offer a description for why adoptive immunotherapy techniques may potentially increase tumor growth rather than suppress it.

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Data availability

The authors declare that the data supporting the findings of this study are available within the manuscript.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Malakand, Dir(L), Chakdara, Khyber Pakhtunkhwa, Pakistan

    Amir Ali & Khalid Khan

  2. Department of Sciences and Technology, Ranyah University College, Taif University, P.O. Box 11099, Taif, 21944, Saudi Arabia

    Saad Althobaiti & Ali Althobaiti

  3. Institute of Energy Infrastructure (IEI), Department of Civil Engineering, College of Engineering, Universiti Tenaga Nasional (UNITEN), Putrajaya Campus, Jalan IKRAM-UNITEN, 43000, Kajang, Selangor, Malaysia

    Rashid Jan

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  1. Amir Ali
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It is also declared that all the authors have equal contribution in the manuscript. Furthermore, the authors have checked and approved the final version of the manuscript.

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Correspondence to Amir Ali.

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Ali, A., Althobaiti, S., Althobaiti, A. et al. Chaotic dynamics in a non-linear tumor-immune model with Caputo–Fabrizio fractional operator. Eur. Phys. J. Spec. Top. 232, 2513–2529 (2023). https://doi.org/10.1140/epjs/s11734-023-00929-y

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