A Unified Nonlinear Mathematical Model Based on Smoking-Vaping Dynamics with Spectral Graph Analysis and Simulation

Abstract

This research presents a nonlinear mathematical approach that integrates smoking and vaping dynamics in a one population. The model is defined as a system of nonlinear ordinary differential equations and rigorously analyzed to provied the existence, uniqueness and boundedness of solutions. The disease-free and endemic equilibrium states are obtained to investigate the major dynamical characteristics of the disease as well as the basic reproduction number is obtained using the next-generation matrix method. Stability analyses encompassing both local and global perspectives are conducted to detail the threshold conditions and the future behavior of the system. To examine the structure and the interaction of the model diagonally, a signal flow graph is plotted and analyzed using spectral graph theory. The measures include graph energy and Estrada index, which are used to establish the connectivity of the network, the feedback and flow of information among the variables. Moreover, sensitivity analysis is conducted to indicate the key parameters which have the greatest influence on the transmission of smoking and vaping, thereby causing the control strategies to be developed. The numerical experiments confirm the analytical findings besides showing the strength of the suggested framework to represent the complex correlation between the behaviors of smoking and vaping accurately. The simulations were further supported by a MATLAB Simulink model that demonstrated the system equations as an animated block diagram which besides allowing the visual exploration of state transitions and parameter sensitivity in real-time also provides a user-friendly tool for scenario testing and educational demonstration.