Modeling for dynamical aspects of smoking abuse with e-cigarette users: Threshold dynamics and optimal control strategies.

Smoking is one of the leading causes of mortality and morbidity around the globe, and e-cigarette use introduces new challenges to public health concerns. Understanding the connection between traditional smoking, e-cigarette adoption, and cessation dynamics is critical for developing effective tobacco-related harm reduction initiatives. In this paper, we present a compartmental mathematical model that captures the transitions between nonsmokers, smokers, e-cigarette users, and quitters while explicitly integrating relapse and cessation behaviors. In the paradigm, e-cigarettes serve as a harm-reduction transition for current smokers, from which users can either quit entirely or return to smoking. The model is analyzed using threshold dynamics to determine the conditions under which smoking behavior persists or declines in a population, and stability analysis is employed to characterize equilibrium states and identify crucial parameters that influence long-term outcomes. Building on this paradigm, we formulate an optimal-control problem by linking control functions to intervention-sensitive processes such as the transition from smoking to e-cigarette usage, the cessation rate among e-cigarette users, and the relapse rate from vaping to smoking. Following harm-reduction principles, the framework prioritizes reducing explosive smoking while discouraging consistent vaping, provided that doing so does not increase smoking prevalence. This paradigm enables researchers to investigate how policy-relevant levers may alter the trajectory of tobacco use over time; however, the report does not compare specific treatments. The model can be extended with numerical simulations that compare the efficacy of interventions like awareness campaigns, taxation, and cessation programs. Such simulations might also include cost-effectiveness studies to determine how limited public health resources could be best allocated across various initiatives. The proposed paradigm provides a theoretical foundation for embedding public health interventions into the dynamics of smoking and e-cigarette use, allowing policymakers and academics to explore how different strategies affect long-term prevalence and reduction outcomes. The graphs illustrate the difference between uncontrolled baseline dynamics and the implementation of an effective management method, demonstrating how interventions can accelerate smoking prevalence reductions. The findings provide a theoretical basis for evaluating how interventions can influence the trajectories of tobacco/vaping use, without claiming to have identified a single "most effective" policy. Future developments could include comparative simulations and cost-effectiveness assessments to help inform targeted decisions related to public health.