Some stochastic epidemiological models are less significant. They do not take into account some sudden events that could disrupt the behavior of the studied phenomenon. In this work, we introduce a white noise and jumps in a deterministic SIRS model for smoking to take into account of the effects of randomly fluctuation and such sudden factors respectively. First of all we prove that the solution of the stochastic differential equation with jumps of the new modelis positive. Then we study the asymptotic behavior around the smoking-free equilibrium state and the smoking-present equilibrium state of the original deterministic model. Under certain conditions, we show that the solution oscillate respectively around these equilibrium states. We prove that the intensity of these oscillations depends on the magnitude of noise and the jump diffusion coefficient of our stochastic differential equation with jumps. To support our theoretical results, we realise numerical simulations. The observations confirm our conclusions.

<abstract><p>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.</p></abstract>