Controlling smoking: A smoking epidemic model with different smoking degrees in deterministic and stochastic environments.

Engaging in smoking not only leads to substantial health risks but also imposes considerable financial burdens. To deepen our understanding of the mechanisms behind smoking transmission and to address the tobacco epidemic, we examined a five-dimensional smoking epidemic model that accounts for different degrees of smoking under both deterministic and stochastic conditions. In the deterministic case, we determine the basic reproduction number, analyze the stability of equilibria with and without smoking, and investigate the existence of saddle-node bifurcation. Our analysis reveals that the basic reproduction number cannot completely determine the existence of smoking, and the model possesses bistability, indicating its dynamic is susceptible to interference from environmental noises. In the stochastic case, we establish sufficient conditions for the ergodic stationary distribution and the elimination of smokers by constructing appropriate Lyapunov functions. Numerical simulations suggest that the effects of inevitable random fluctuations in the natural environment on controlling the smoking epidemic may be beneficial, harmful, or negligible, which are closely related to the noise intensities, initial smoking population sizes, and the effective exposure rate of smoking transmission (β). Given the uncontrollable nature of environmental random effects, effective smoking control strategies can be achieved by: (1) accurate monitoring of initial smoking population sizes, and (2) implementing effective measures to reduce β. Therefore, it is both effective and feasible to implement a complete set of strong MPOWER measures to control smoking prevalence.