Dataset of figures

<pre>In this study, the author propose a fractional order smoking model in the Caputo sense by extending the model in [15]. The extended model incorporates hospitalized smokers and possible crucial parameter such as recovery and quit rate for snuffing and light smokers, and disease-induced mortality rates for light, chronic, and quitters. The addition of the hospitalized smokers compartment reflects the substantial health burden caused by smoking-related diseases, which according to the WHO. This dual extension offers a more realistic representation of smoking progression than classical models. As per the available literature, no study has yet introduced a fractional-order smoking model with a hospitalized class, highlighting the novelty of the present study. From the numerical perspective, the spectral method has not yet been widely applied to such smoking models. In this study, we address this gap by employing the spectral collocation</pre><pre> method based on Chebyshev polynomials to analyze the proposed model. The method provides high accuracy, computational efficiency and transforms the system of fractional differential</pre><pre>equations into a nonlinear algebraic system, and solves it efficiently using the Newton–Raphson method. Additionally, an optimal control problem is formulated by incorporating two intervention strategies: public awareness and nicotine replacement therapy and studied their cost effectiveness. This integrated framework provides a comprehensive mathematical basis for understanding smoking dynamics and evaluating effective and cost efficient control measures for smoking cessation.</pre>