A Stochastic Model with Jumps for Smoking

Main Article Content

Mohamed Coulibaly
Modeste N'Zi


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 model
is 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.

Smoking model, jump perturbation, global positive solution, asymptotic behavior, numerical simulation.

Article Details

How to Cite
Coulibaly, M., & N’Zi, M. (2019). A Stochastic Model with Jumps for Smoking. Journal of Advances in Mathematics and Computer Science, 34(3), 1-16. https://doi.org/10.9734/jamcs/2019/v34i3-430207
Original Research Article


Castillo-Garsow C, Jordan-Salivia G, Rodriguez-Herrera A. Mathematical models for the dynamics of tobacco use, recovery and relapse. Technical Report Series No.BU-1505-M, BU- , Ithaca, NY: Mathematical and Theoretical Biology Institute, Cornell University; 1997.
Available: https://mtbi.asu.edu/1997-12

Sharomi O, Gumel AB. Curtailing smoking dynamics : A mathematical modeling approach. Applied Mathematics and Computation. 2008;195(2):475-499.Elsevier

Lahrouz A, Omari L, Kiouach D, Belmaati A. Deterministic and stochastic stability of a mathematical model of smoking. Statistics and Probability Letters.2011;81(8):1276-Elsevier

World Health Organization. WHO report on the global tobacco epidemic, 2015 - Raising taxes on tobacco; 2015.
Available:https://www.who.int/tobacco/global report/2015/en/

Zhanga X, Wang K. Stochastic SIR model with jumps. Applied Mathematics Letters.2013;26(8):867-874.Elsevier

Yingjia Guo. Stochastic regime switching SIS epidemic model with vaccination driven by Lvy noise. Advances in Difference Equations.2017;375 (2017). Springer

Berrhazi B, El Fatini M, Caraballo T, Pettersson R. A stochastic SIRI epidemic model with Lvy noise. Discrete & Continuous Dynamical Systems - B. 2018;23(6):2415-2431. American Institute of Mathematical Sciences.

Higham DJ. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Review.2009;43(3):525-546.

Zou X, Wang K. Numerical simulations and modeling for stochastic biological systems with jumps. Applied Mathematics Letters.2013;19(5):1557-1568.Elsevier

Bao J, Yuan C. Stochastic population dynamics driven by Levy noise. Journal of Mathematical Analysis and Applications.2012;391(2):363-375.

Alkhudhari Z, Al-Sheikh S, Al-Tuwairqi S. Global dynamics of a mathematical model on smoking. ISRN Applied Mathematics.2014;7.Article ID 847075.Indawi