A fractional calculus approach to smoking dynamics with bifurcation analysis

Smoking cigarettes is one of the world’s leading causes of death, with harmful chemicals affecting both smokers and those exposed to secondhand and thirdhand smoke. This study employs a Caputo–Fabrizio fractional derivative within a mathematical model to investigate the dynamics of smoking behavior. The analysis confirms the existence and uniqueness of solutions and examines critical properties such as positivity, boundedness, and the invariant region to ensure the model’s robustness. The reproduction number is calculated using the next-generation matrix method to estimate the potential for smoking transmission, parameterized by \(({\rho }_{3},\vartheta )\) $({\rho }_3,\vartheta )$ . Comprehensive sensitivity and stability analyses are performed, including an investigation of the local and global stabilities of both smoking-free and smoking-present equilibrium points. Bifurcation analysis is conducted with the bifurcation parameter \({\epsilon }_{1}^{\star }=1.523\) ${ϵ}_1^{\star }=1.523$ , highlighting the transitions between unstable and stable states. We also apply the system Lagrange interpolation scheme, which yields results validating our theoretical findings. Numerical simulations, executed using Python, further corroborate these results and provide deeper insights into the complex dynamics of smoking behavior. Additionally, the study discusses the implications of these findings for public health strategies and intervention planning.