Optimal control of a fractional smoking system

In this paper, considering the contact transmission between heavy smokers and potential smokers, we propose a novel fractional smoking model. Moreover, we introduce two intervention measures, i.e., media campaign and adjuvant drug treatment, to the fractional smoking model in order to control the spread of smoking. For this controlled fractional smoking model, some important properties are discussed. Then, we propose an optimal control problem involving the controlled fractional system. The existence of the optimal control for this problem is also established. By applying the calculus of variation, we derive the necessary optimality conditions for the fractional optimal control problem. Furthermore, we develop a numerical solution approach based on Adams-type predictor-corrector method to solve the derived optimality system. Finally, numerical results show that the obtained optimal control strategies are effective in reducing the susceptible individuals and smokers.