• Abstract
• Keywords
• References
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• ### Abstract

In a recent paper [Anwar Zeb, Gul Zaman, Shaher Momani, Square-root Dynamics of a Giving Up Smoking Model, Appl. Math. Model., 37 (2013) 5326-5334], the authors presented a new model of giving up smoking model. In this paper, we introduce three control variables in the form of education campaign, anti-smoking gum, and anti-nicotive drugs/medicine for the eradication of smoking in a community. Using the optimal control theory, the optimal levels of the three controls are characterized, and then the existence and uniqueness for the optimal control pair are established. In order to do this, we minimize the number of potential and occasional smokers and maximize the number of quit smokers. We use Pontryagin's maximum principle to characterize the optimal levels of the three controls. The resulting optimality system is solved numerically by Matlab.

• ### Keywords

Mathematical model; Square root dynamics; Non-standard; Finite difference scheme; Numerical analysis; Optimal control.

• ### References

[1] C. Castillo-Garsow, G. Jordan-Salivia and A. Rodriguez Herrera Mathematical Models for Dynamics of Tobacco Use, Recovery and Relapse" Technical Report Series BU-1505-M, Cornell University, (2000).

[2] O. Sharomi and A.B. Gumel Curtailing smoking dynamics: A mathematical modeling pproach" Applied Mathematics and Computation, 195 (2008) 475-499.

[3] O.K. Ham Stages and Processes of Smoking Cessation among Adolescents" West J. Nurs. Res., 29 (2007) 301-315.

[4] G. Zaman Qualitative behavior of giving up smoking models" Bulletin of the Malaysian Mathematical Sciences Society, 34 (2011) 403-415.

[5] A. Zeb, G. Zaman and S. MomaniSquare-root dynamics of a giving up smoking model" Applied Mathematical Modelling, 37 (2013) 53265334

[6] A.G. Radwan, K. Moaddy and S. Momani Stability and non-standard finite difference method of the generalized Chua's circuit" Computer and Mathematics with Applications, 62 (3) (2011) 961-970.

[7] D. Kirschner, S. Lenhart, S. Serbin, Optimal control of chemotherapy of HIV, J. Math. Biol., 35 (1997) 775-792.

[8] G. Zaman, Y.H. Kang, I.H. Jung,Optimal treatment of an SIR epidemic model with time delay, Bio., 98 (1) (2009) 43-50.

[9] M.I. Kamien, N.L. Schwartz, Dynamics Optimization, The Calculus of Variations and Optimal Control in Economics and Management, 1991.

[10] G. Zaman, H. Jung, Optimal vaccination and treatment in the SIR epidemic model, Proc. KSIAM, 3 (2007) 31-33.

[11] F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E., 53 (1996) 1890-1899.

[12] A.B. Gumel, P.N. Shivakumar, B.M. Sahai,A mathematical model for the dynamics of HIV-1 during the typical course of infection, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 47 (3) (2001) 17731783.

[13] J. Karrakchou, M. Rachik, S. Gourari, Optimal control and infectiology: Application to an HIV/AIDS model, Appl. Math. Comput., 177 (2006) 807-818.