Dynamical analysis and optimal control of a novel fractional-order compartmental model for phone addiction

This paper formulates and compares a new fractional-order mathematical model of the transmission dynamics and behavioral development of mobile phone addiction across distinct user subpopulations. The model is developed in the context of fractional derivatives. The population is categorized into susceptible population, three classes with addiction-intensity levels, a treatment population, and a recovered population. The existence, uniqueness, boundedness, and nonnegativity of the model solutions are rigorously proven, thereby establishing the biological feasibility of the trajectories. Equilibrium points are located, and the local property of asymptotic stability is described in terms of the basic reproduction number of addiction reinforcement. Sensitivity analysis identifies how the main behavioral and intervention-related parameters affect the reproduction threshold, thereby informing the most effective control mechanisms. Based on these analytical findings, an optimal control problem is developed to reduce the total prevalence of addiction in the population. Moreover, a generalized Euler scheme is developed to derive numerical solutions of the fractional system, and a set of computational experiments is performed to demonstrate the qualitative consistency of the results obtained by analysis with the dynamics of addiction simulated by the scheme. The results suggest that, compared to the classical formulation, the fractional-order formulation offers greater descriptive flexibility and more realistic temporal responses. This observation highlights the importance of memory-based structures in the modeling of techno-dependent behavioral processes.