Z. Naturforsch. **69a,** 225 – 231
(2014)

doi:10.5560/ZNA.2014-0020

Optimal Campaign Strategies in Fractional-Order Smoking Dynamics

Anwar
Zeb^{1,2},
Gul
Zaman^{2},
Il
Hyo Jung^{3}, and
Madad
Khan^{1}

^{1 }Department of Mathematics, COMSATS Institute of Information Technology Abbottabad, K.P.K, Pakistan

^{2 }Department of Mathematics, University of Malakand, Chakdara, Dir (L), K.P.K, Pakistan

^{3 }Department of Mathematics, Pusan National University, San 30, Geumjeong-Gu, Busan 609-735, South Korea

Received December 23, 2013 / revised March 10, 2014 / published online May 21, 2014

This paper deals with the optimal control problem in the giving up smoking model of fractional order. For the eradication of smoking in a community, we introduce three control variables in the form of education campaign, anti-smoking gum, and anti-nicotive drugs/medicine in the proposed fractional order model. We discuss the necessary conditions for the optimality of a general fractional optimal control problem whose fractional derivative is described in the Caputo sense. In order to do this, we minimize the number of potential and occasional smokers and maximize the number of ex-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.

*Key words:* Giving up Smoking Model; Fractional Order Derivatives; Optimal Control; Numerical Analysis.

*Mathematics Subject Classification 2000:* 92D25; 49J15; 93D20