Computers, Materials & Continua CMC, vol.65, no.3, pp.1905-1924, 2020 CMC. doi:10.32604/cmc.2020.011623 www.techscience.com/journal/cmc Analysis and Dynamics of Illicit Drug Use Described by Fractional Derivative with Mittag-Leffler Kernel Berat Karaagac 1, 2 , Kolade Matthew Owolabi 1, 3, * and Kottakkaran Sooppy Nisar 4 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 1 α < ≤ . 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: Atangana-Baleanu fractional operator, illicit drug use, existence and uniqueness of solutions, stability analysis. 1 Introduction Illicit drug use is both a serious threat to human health and a common problem for the nations and the world due to material and moral losses, the decrease in national welfare, loss and misery of individual lives, serious infectious diseases, deaths, and the increase in crimes associated with addiction. Certain drugs are natural such as cannabis and cocaine or could be derived from natural substances such as heroin. Some may be synthetic such as amphetamines and various others. These drugs are classified into four main categories based on their effects: stimulants, depressants, opioids and hallucinogens. Stimulant substances improve attention, energy, blood pressure, pulse and respiratory rates. Depressant substances reduce pain and make the individuals feel sleepy. Furthermore, 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: koladematthewowolabi@tdtu.edu.vn. Received: 20 May 2020; Accepted: 03 August 2020.