A fractional-order model for drinking alcohol behaviour leading to road accidents and violence

In this paper, we propose a new fractional-order model of alcohol drinking involving the Caputo derivative and six groups of individuals.  We introduce road accidents and violence related to alcohol consumption as separate classes to highlight the role of alcoholism in the aggressive and risky behaviour of heavy drinkers.  We show the existence and uniqueness of the non-negative solutions, and we determine the basic reproduction number $R_{0}$.  The sensitivity analysis of the model parameters is performed to characterize the important parameters that have the most effects on the reproduction number.  Furthermore, the stability analysis of the model shows that the system is locally and globally asymptotically stable at drinking-free equilibrium $E^{0}$ when $R _{0}<1$, and the drinking present equilibrium $E^{\ast }$ exists.  The system is locally and globally asymptotically stable at $E^{\ast }$ when ${R _{0}>1}$.  Finally, numerical simulations are carried out to illustrate the theoretical results for different values of the order of the fractional derivative.