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Smoking is one of the most common habits in the world. Due to the harmful substances contained in it cigarette disturb people's health and causes their death. Every year, many people lose their health from smoking or by direct exposure to cigarettes. In this study, we conducted a new application and stability analysis related to a fractional model of cigarette use. This model consists of three compartments: potential smokers (P), smokers (S) and definitely quitters of smoking (Qp). The fractional derivative is used in the sense of Caputo. In this article, we develop a fractional mathematical model to illustrate the consequences of smoking habit in Turkey by the Caputo fractional-order derivative. In addition to stability analysis, numerical solutions are obtained by using Euler method for different values of fractional order. [ABSTRACT FROM AUTHOR]