Mathematical and computational analysis of a fractional-order drug abuse model with nonlinear incidence and logistic growth

This paper presents a novel mathematical model for analyzing the dynamics of drug addiction using a fractional-order system based on the Liouville–Caputo derivative. The proposed model incorporates a nonlinear saturated incidence rate, logistic growth in the addiction compartments, and seven interconnected subpopulations representing different stages of drug use and recovery, including relapse and awareness. We conduct a rigorous mathematical analysis to establish the existence, uniqueness, positivity, and boundedness of solutions, ensuring the epidemiological and physical validity of the model. The basic reproduction number R0 is derived, and the local and global stability of the equilibrium points is analyzed. A major contribution of this work is the application of a new domain decomposition spectral method based on second-kind Dickson polynomials, combined with the quasilinearization technique, to efficiently solve the complex nonlinear system. The convergence of the numerical method is theoretically validated. Numerical simulations are provided to illustrate the model’s dynamics and to explore the impact of various parameters and intervention strategies. Compared to existing models, this study offers an improved framework for understanding memory-dependent behavior in addiction dynamics and introduces a computationally efficient approach to solve fractional-order systems with high accuracy.