A stochastic model with jumps for smoking incorporating media coverage

Media coverage is an important tool in the fight against smoking. So, in this paper we will incorporate media coverage in a deterministic SIRS model for smoking. Those who have studied this deterministic model have shown that by setting the constants of this model, we can control the tobacco epidemic. But this model is not very realistic: it does not take into account the action of media coverage and some other random factors. Thus, we incorporate the media coverage into this model and obtain a deterministic model with media coverage. Also, to take into account some randomness in the contact between individuals or sudden events that could disrupt the action of media coverage, we introduce in our deterministic model with media coverage white noise and jumps. We first prove the boundness of the solutions and the stability of the smoking-free equilibrium state of the deterministic model with media coverage. We prove that the solution of the stochastic differential equation with jumps of the stochastic model is unique, positive and global. Under certain conditions, we show that this solution oscillates respectively around each equilibrium state of our deterministic model. This allows us to consider conditions that lead to converge towards an extinction or persistence of smoking. The paper is ended by numerical simulations that corroborate our theoretical results.