Numerical simulation of the smoking model using spectral collocation method

Smoking is one of the leading causes of health problems and remains one of the world’s most pressing public health challenges. This paper proposes a modified smoking model based on the Caputo fractional derivative, which turns out to be a system of five nonlinear differential equations. The study employs a dual approach, combining both theoretical and numerical perspectives to analyze the smoking model. In this paper, the existence and uniqueness of the solution are established using fixed-point theory and the Picard–Lindelöf method and the stability for both smoke-free and smoke-present equilibria are analyzed using both Jacobian matrix and Lyapunov functions. Moreover, the model is examined using the spectral collocation method, employing Chebyshev polynomials as basis functions. The convergence and stability of the numerical solutions are captured via the maximum residual error for both integer and fractional orders over different sets of collocation points. The study also examines the effects of different parameters for various fractional order values. Furthermore, the combined effects of the transmission rates, as well as their interactions with recovery, depart, and death rates due to smoking/ingestion, are explored. These results are represented in the form of tables and detailed graphs.