A fractional-order mathematical model of smoking dynamics with liver damage and relapse pathways

Abstract

This study evaluates the dynamics of the smoking model using two distinct fractional operators—Caputo ($\c$) and modified Mittag-Leffler kernel ($\m$) derivatives. These fractional-order derivatives are widely used in modeling complex systems due to their ability to capture memory effects and hereditary properties. We conduct a mathematical analysis of the fractional model, including positivity of the solutions. Furthermore, the existence and uniqueness of solutions are demonstrated using fixed-point theory. In addition, we perform a detailed stability analysis of the model using the Ulam-Hyers stability criterion to ensure the robustness and reliability of the proposed fractional-order system. For numerical simulations, we employ a generalized predictor-corrector  method for the $\c$ derivative and $\m$ derivative. The models are solved computationally, and the results are graphically illustrated for various fractional-order values. Our findings highlight significant differences between the $\c$ and $\m$ derivatives, with the latter showing enhanced memory effects and smoother transitions across compartments.