Department of Mathematics, Ghazi University, DG Khan, Pakistan.
Department of Mathematics and Statistics, University of Lahore, Lahore 54590, Pakistan.
Comput Math Methods Med. 2022 May 20;2022:9683187. doi: 10.1155/2022/9683187. eCollection 2022.
In recent years, there are many new definitions that were proposed related to fractional derivatives, and with the help of these definitions, mathematical models were established to overcome the various real-life problems. The true purpose of the current work is to develop and analyze Atangana-Baleanu (AB) with Mittag-Leffler kernel and Atangana-Toufik method (ATM) of fractional derivative model for the Smoking epidemic. Qualitative analysis has been made to `verify the steady state. Stability analysis has been made using self-mapping and Banach space as well as fractional system is analyzed locally and globally by using first derivative of Lyapunov. Also derive a unique solution for fractional-order model which is a new approach for such type of biological models. A few numerical simulations are done by using the given method of fractional order to explain and support the theoretical results.
近年来，已经提出了许多与分数阶导数相关的新定义，并且借助这些定义，建立了数学模型来克服各种现实生活中的问题。目前工作的真正目的是开发和分析具有 Mittag-Leffler 核的 Atangana-Baleanu (AB)和 Atangana-Toufik 分数阶导数方法 (ATM)模型，以研究吸烟流行现象。已经进行了定性分析以验证定态。稳定性分析是使用自映射和巴拿赫空间进行的，并且通过使用李雅普诺夫的一阶导数对分数阶系统进行局部和全局分析。还推导出分数阶模型的唯一解，这是此类生物模型的一种新方法。通过使用给定的分数阶方法进行了一些数值模拟，以解释和支持理论结果。
