Mathematical Modeling and Analysis of Khat-Chewing Dynamics

Abstract

Khat is a green leaf and greenish plant where its branches and leaves are chewed to discharge liquid having active chemicals that change the user’s mood. The purpose of this article is to develop and analyze a mathematical model that can be used to understand the dynamics of chewing Khat. The proposed model monitors the dynamics of five compartments, namely, a group of people who do not chew Khat, designated as \n \n N\n \n \n t\n \n \n \n ; a group of people who are surrounded by Khat chewers but do not chew at present and may chew Khat in the future, denoted this as \n \n Σ\n \n \n t\n \n \n \n ; a group of people who chew Khat, which is represented in \n \n C\n \n \n t\n \n \n \n ; a group of people contains individuals who consumed Khat quite temporarily for social, spiritual, and recreational purposes, and we describe this group in \n \n T\n \n \n t\n \n \n \n ; and a group of people those who constantly chew Khat, and they are denoted by \n \n H\n \n \n t\n \n \n \n . We determined the Khat chewing generation number \n \n \n \n R\n \n \n c\n 0\n \n \n \n using the next-generation matrix method, and we have examined the biological meaningfulness, mathematical wellposedness, and stability of both Khat chewing-free and Khat chewing-present equilibrium points of the model analytically. Numerical simulations were presented by solving our dynamical system using Matlabode45 to check the analytical results by considering parameter estimations. The results of this study show that, for \n \n \n \n R\n \n \n c\n 0\n \n \n =\n .00039\n \n , the Khat chewing-free equilibrium point is stable, and it is unstable for \n \n \n \n R\n \n \n c\n 0\n \n \n =\n 1.194\n \n , and the Khat chewing-present equilibrium point is stable if \n \n \n \n R\n \n \n c\n 0\n \n \n =\n 1.194\n \n , and it is unstable if \n \n \n \n R\n \n \n c\n 0\n \n \n =\n .00039\n \n . The stability of both equilibrium points implies that, for a high rate of conversion from non-Khat chewer to exposed groups \n \n \n \n ρ\n \n \n \n , the inflow of an insignificant number of Khat chewers to the community produces a significant number of Khat chewers , and if the return back from Khat chewing to the exposed group because of socio-economic, environmental, and religious influences\n \n \n \n \n \n α\n \n \n 2\n \n \n \n \n \n grows exponentially, the inflow of an insignificant number of Khat chewers to the community produces an insignificant number of Khat chewers. It is found that increasing the rate of conversion from non-Khat chewer to exposed groups \n \n \n \n ρ\n \n \n \n makes the disease eradication more challenging. We, therefore, strongly urge religious leaders, social committee leaders, elders, and health experts to teach their followers to reduce their Khat-chewing habits.