Theoretical Existence in Banach Spaces and Numerical Simulation of Fractional Smoking Model with Caputo-Fabrizio Derivatives

Smoking remains a significant public health burden whose population-level dynamics are inadequately captured by classical integer-order compartmental models, which lack the capacity to encode historical dependence in behavioral transitions. Using the Caputo-Fabrizio (CF) derivative, we propose and analyze a five-compartment fractional-order smoking model, which includes potential ([Formula: see text]), occasional ([Formula: see text]), heavy ([Formula: see text]), quitting ([Formula: see text]), and recovered ([Formula: see text]) individuals. Using the Banach contraction principle and Picard iteration, the existence and uniqueness of solutions in a Banach function space are established, and Ulam-Hyers stability is verified. To simulate fractional orders numerically, we develop a two-step Adams-Bashforth scheme applied to the CF integral. Finally, to recover biologically consistent parameter values, a Physics-Informed Neural Network (PINN) framework is constructed that approximates the five model states and estimates all ten biological parameters by minimising a composite loss enforcing the CF fractional equations, initial conditions, and data fidelity against CDC National Health Interview Survey cigarette smoking prevalence records for 2000–2023. Based on the estimated fractional order [Formula: see text], memory effects are measurable in smoking dynamics and in the recovered parameter values. The findings show that smoking initiation and preventive interventions dominate initial dynamics, whereas cessation and recovery parameters determine long-term asymptotic behavior. In the results, decreasing [Formula: see text] leads to decline of the potential smoker class, reduces infection peak, and increases recovery rate.