Optimal control of a fractional smoking system

Chongyang Liu; Wenjuan Sun; Xiaopeng Yi

doi:10.3934/jimo.2022071

Article Contents

2023, Volume 19, Issue 4: 2936-2954. Doi: 10.3934/jimo.2022071

This issue Previous Article Financial risk contagion and optimal control Next Article Optimal pricing strategy in a dual-channel supply chain: A two-period game analysis

Optimal control of a fractional smoking system

1.
School of Mathematics and Information Science, Shandong Technology and Business University, Yantai, Shandong, China
2.
School of Information and Electronic Engineering, Shandong Technology and Business University, Yantai, Shandong, China
3.
School of Computer Science and Technology, Shandong Technology and Business University, Yantai, Shandong, China

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

Received: September 2021

Revised: March 2022

Published: April 2023

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)

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Abstract

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.

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

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

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

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

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

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

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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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