An analytical framework for smoking epidemic modeling using fuzzy logic and dual time-delay dynamics

• A novel fuzzy-based smoking epidemic model (PSRQE model) is developed, incorporating two distinct time delays to capture initiation and relapse behaviors. • Stability, sensitivity, and equilibrium analyses are performed to understand smoking dynamics under uncertainty. • Numerical simulations using Forward Euler, RK-4, and Non-Standard Finite Difference (NSFD) methods validate the model, with NSFD preserving positivity and boundedness. • Results highlight the critical influence of initiation delays and fuzzy uncertainty on smoking prevalence and intervention outcomes. • The proposed model offers a robust framework for designing targeted public health strategies to reduce smoking rates and their associated health and economic burdens. The process of smoking is divided into several stages and has a clear tendency towards uncertainty and variability, which are not reflected in the traditional models with presumed parameters. To overcome this difficulty, a fuzzy mathematical model is derived to represent smoking dynamics more accurately under uncertainty. The PSRQE model presented and comprises Potential, Social, Regular, Transitional Non-smokers, and Ex-smokers, integrates vital considerations like the chance of developing smoking and the chance of quitting smoking. The model is analyzed by a stability analysis, numerical simulations, and sensitivity analysis of the basic reproduction number R o . Three algorithms based on the Forward Euler scheme, the Fourth-Order Runge-Kutta (RK-4) treatment method, and the Non-Standard Finite Difference (NSFD) technique are used to obtain numerical solutions. The NSFD scheme is positive and bounded by convergence analysis, and simulation results have shown that it also preserves the structural properties of the model even when the step sizes are larger. Moreover, the influence of time deviations τ 1 and τ 2 on the smoking habits is also examined. It is demonstrated that this framework provides a valuable foundation for comprehending the leading patterns that govern smoking behavior that are required to reduce smoking rates and the related social, health, and economic impacts.