Stochastic Stability and Optimal Control Analysis for a Tobacco Smoking Model

In this paper, a smoking model, which takes snuffing class and Brownian motion into consideration and is thus an extension of previously studied deterministic smoking models. We analytically show that this extended model system has one and only one positively bounded solution for any nonnegative initial values for the state variables. Interestingly, we find that the model system can exhibit sharp threshold characteristics whatever values of the basic reproductive number. By analyzing persistence, extinction and stationary distribution, we also find that the stochastic system is ergodic only when the coefficients of the noise terms are small. To eliminate gradually the infection out of the community, we introduce a stochastic system of two control variables and perform analysis, with results that can provide guidelines for tobacco control department. Results obtained by theoretical analysis are verified by numerical simulations.