Eur. Phys. J. Plus (2022) 137: 367
https://doi.org/10.1140/epjp/s13360-022-02591-0

Regular Article

Modeling the dynamics of tumor–immune cells interactions via fractional calculus

¹ International Intercollegiate Ph.D. Program, National Tsing Hua University, 30013, Hsinchu, Taiwan
² Department of Internal Medicine, E-Da Hospital, 82445, Kaohsiung, Taiwan
³ School of Medicine, College of Medicine, I-Shou University, 82445, Kaohsiung, Taiwan
⁴ Department of Family and Community Medicine, E-Da Hospital, 82445, Kaohsiung, Taiwan
⁵ Department of Engineering and System Science, National Tsing Hua University, 30013, Hsinchu, Taiwan
⁶ Department of Mathematical Sciences, University of Lakki Marwat, 28420, Lakki Marwat, KPK, Pakistan
⁷ Department of Mathematics, University of Swabi, 23561, Swabi, Pakistan
⁸ Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, 21589, Jeddah, Saudi Arabia

^b zahir@ulm.edu.pk

Received: 13 February 2022
Accepted: 9 March 2022
Published online: 19 March 2022

Abstract

The immune response in the tumor micro-environment is a complicated biological phenomenon that needs to be investigated further. It is eminent that tumor is murderous and almost effects all the sectors of humans life. It is reported that patients, families, businesses, and community as a whole lose economic resources due to the treatment of cancer. These losses include financial loss, disease, a worse quality of life, and early death. Therefore, it is of great importance to investigate the dynamical behavior of tumor–immune cells interactions to point out the critical factors of the system to the researchers and health officials for the prevention of infection. Here, we structure the interactions of tumor–immune in the framework of fractional derivative. Moreover, we focused on the qualitative analysis and dynamical behavior of tumor–immune cells interactions in our work. The uniqueness and existence of the solution of the proposed model of tumor system are investigated through Banach’s and Schaefer’s fixed point theorem. We determine adequate conditions for the Ulam–Hyers stability of our proposed fractional model. The solution pathways are scrutinized with the help of a novel numerical scheme to highlight the impact of the parameters on the dynamics of tumor–immune. To be more specific, we highlighted the dynamical and chaotic behavior of the proposed system with the variation of fractional order and other parameters of the system. The most important aspects of tumor–immune cell dynamics are identified and recommended to the policymakers. The result of this dynamic is a powerful predictor of clinical success, and immune cell penetration into, or exclusion from, a tumor is also crucial to immunotherapy effectiveness. Hence, it is valuable to look into the dynamical behavior of tumor–immune cell interactions in order to alert policymakers and health authorities to the system’s critical components for management and prevention.