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Qualitative and Ulam–Hyres stability analysis of fractional order cancer-immune model

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  • Xu, Changjin
  • Farman, Muhammad

Abstract

The primary objective of this study is to employ the novel fractional operator to investigate a mathematical model of cancer, specifically focusing on the dynamics of cancer cells when treated with a combination of IL-12 cytokines and an anti-PD-L1 inhibitor. We utilize nonlinear differential equations to represent this complex system. We aim to conduct a comprehensive mathematical analysis of the fractional-order tumor model using the Atangana–Baleanu–Caputo derivative. We intend to qualitatively and quantitatively assess the dynamics of the tumor-immune system to understand its real-world implications. We seek to establish the existence and uniqueness of solutions for our model using fixed-point theorems. We will analyze the stability of the system, verifying its equilibrium points through the Ulam–Hyres stability concept and demonstrating global stability using a Volterra-type Lyapunov function. We will explore the impact of the fractional operator on the dynamical system, employing the Atangana–Tufik scheme with a generalized Mittag-Leffler kernel and results are compared with Caputo fractional derivative. Finally, we will present numerical simulations using the proposed scheme to provide insights into the practical behavior of our model within the scientific community.

Suggested Citation

  • Xu, Changjin & Farman, Muhammad, 2023. "Qualitative and Ulam–Hyres stability analysis of fractional order cancer-immune model," Chaos, Solitons & Fractals, Elsevier, vol. 177(C).
    • Handle: RePEc:eee:chsofr:v:177:y:2023:i:c:s0960077923011797
    DOI: 10.1016/j.chaos.2023.114277
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    References listed on IDEAS

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    1. Asamoah, Joshua Kiddy K. & Okyere, Eric & Yankson, Ernest & Opoku, Alex Akwasi & Adom-Konadu, Agnes & Acheampong, Edward & Arthur, Yarhands Dissou, 2022. "Non-fractional and fractional mathematical analysis and simulations for Q fever," Chaos, Solitons & Fractals, Elsevier, vol. 156(C).
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    3. Li, Peiluan & Gao, Rong & Xu, Changjin & Li, Ying & Akgül, Ali & Baleanu, Dumitru, 2023. "Dynamics exploration for a fractional-order delayed zooplankton–phytoplankton system," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    4. Baleanu, Dumitru & Hasanabadi, Manijeh & Mahmoudzadeh Vaziri, Asadollah & Jajarmi, Amin, 2023. "A new intervention strategy for an HIV/AIDS transmission by a general fractional modeling and an optimal control approach," Chaos, Solitons & Fractals, Elsevier, vol. 167(C).
    5. Farman, Muhammad & Sarwar, Rabia & Akgul, Ali, 2023. "Modeling and analysis of sustainable approach for dynamics of infections in plant virus with fractal fractional operator," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
    6. Li, Zhongfei & Liu, Zhuang & Khan, Muhammad Altaf, 2020. "Fractional investigation of bank data with fractal-fractional Caputo derivative," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
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