A mathematical approach to cancer growth: The role of smoking through fractional order models with Mittag-Leffler kernels

This study investigates the impact of smoking on cancer development through the application of nonsingular and nonlocal kernels of fractional differential equations, specifically focusing on the Atangana–Beleanu fractional-order derivative in the Caputo sense (ABC). To account for the periodic effects of smoking, a periodic function has been incorporated into the model. The existence and uniqueness of solutions have been analyzed using fixed point theory. Furthermore, the stability of the iterative method has been assessed through fixed-point G stability in conjunction with the Laplace transform. The conditions for the existence and local stability of the equilibrium points have been established. The numerical results are examined through a two-step back-and-forth Adams–Bashforth numerical scheme. The novelty of this study lies in the incorporation of a non-singular and non-local kernel in the tumor growth model, specifically considering the effects of smoking. The simulations performed provide valuable insights into the influence of fractional-order derivatives, the strength of immune cells, the sensitivity of initial concentrations, and the frequency of smoking on the dynamics of tumor growth. The chaotic behavior observed in the dynamics, characterized by a high frequency of smoking and the absence of immune flux under low initial conditions of host cells, has been analyzed for various fractional-order derivatives.