Article Contents
Article Contents

# Optimal control of a fractional smoking system

• *Corresponding author: Email address: chongyangliu@aliyun.com

This work was supported by the Natural Science Foundation of China (No. 11771008), and the Natural Science Foundation of Shandong Province, China (Nos. ZR2019MA031 and ZR2019QA009)

• 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.

Mathematics Subject Classification: Primary: 49J15; Secondary: 93C15.

 Citation:

• Figure 1.  Flow chart of controlled smoking system (1)

Figure 2.  The optimal control strategies when $\alpha = 0.6$

Figure 3.  The optimal state trajectories when $\alpha = 0.6$

Figure 4.  The optimal control strategies when $\alpha = 0.7, 0.8, 0.9$ and $1.0$

Figure 5.  The optimal state trajectories and total population density when $\alpha = 0.7, 0.8, 0.9$ and $1.0$

Table 1.  Parameter values used in system (3)

 Parameter $b$ $\mu$ $\beta_1$ $\beta_2$ $r$ Value $0.035$ (Assumed) $0.04$ ([24]) $0.6$ ([24]) $0.1$ (Assumed) $0.7$ ([24]) Parameter $e$ $m$ $d$ $\eta$ Value $0.5$ ([24]) $0.7$ ([24]) $0.05$ ([24]) $0.04$ (Assumed)
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