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# Analysis and Dynamics of Illicit Drug Use Described by Fractional Derivative with Mittag-Leffler Kernel

Berat Karaagac1, 2, Kolade Matthew Owolabi1, 3, *, Kottakkaran Sooppy Nisar4

1 Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam.
2 Department of Mathematics Education, Faculty of Education, Adıyaman University, Adıyaman, Turkey.
3 Department of Mathematical Sciences, Federal University of Technology, Akure, Nigeria.
4 Department of Mathematics, College of Arts and Sciences, Wadi Aldawaser, Prince Sattam Bin Abdulaziz University, Al Kharj, Saudi Arabia.

* Corresponding Author: Kolade Matthew Owolabi. Email: .

(This article belongs to this Special Issue: Mathematical aspects of the Coronavirus Disease 2019 (COVID-19): Analysis and Control)

Computers, Materials & Continua 2020, 65(3), 1905-1924. https://doi.org/10.32604/cmc.2020.011623

## Abstract

Illicit drug use is a significant problem that causes great material and moral losses and threatens the future of the society. For this reason, illicit drug use and related crimes are the most significant criminal cases examined by scientists. This paper aims at modeling the illegal drug use using the Atangana-Baleanu fractional derivative with Mittag-Leffler kernel. Also, in this work, the existence and uniqueness of solutions of the fractional-order Illicit drug use model are discussed via Picard-Lindelöf theorem which provides successive approximations using a convergent sequence. Then the stability analysis for both disease-free and endemic equilibrium states is conducted. A numerical scheme based on the known Adams-Bashforth method is designed in fractional form to approximate the novel Atangana-Baleanu fractional operator of order 0 < α ≤ 1. Finally, numerical simulation results based on different values of fractional order, which also serve as control parameter, are presented to justify the theoretical findings.

## Keywords

B. Karaagac, K. Matthew Owolabi and K. Sooppy Nisar, "Analysis and dynamics of illicit drug use described by fractional derivative with mittag-leffler kernel," Computers, Materials & Continua, vol. 65, no.3, pp. 1905–1924, 2020.

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