Analysis of a drinking epidemic model

Abstract

In this paper, we have developed a mathematical model of alcohol abuse which consists of four compartments corresponding to four population classes, namely, moderate and occasional drinkers, heavy drinkers, drinkers in treatment and temporarily recovered class. We have discussed about basic properties of the system. Sensitivity analysis of the system is also discussed. Next, Basic reproduction number (\(R_0\)) is calculated. The stability analysis of the model shows that the system is locally asymptotically stable at disease free equilibrium \(E_0\) when \(R_0 1\), endemic equilibrium \(E^*\) exists and the system becomes locally asymptotically stable at \(E^*\) and \(E_0\) becomes unstable. We have also discussed the global stability of the system at \(E_0.\) It is also found that a backward bifurcation may occur at \(R_0=1\). Next we have discussed the drinking epidemic model with treatment control. An objective functional is considered which is based on a combination of minimizing the number of heavy drinkers and the cost of treatment. Then an optimal control is obtained which minimizes the objective functional. Our numerical findings are illustrated through computer simulations using MATLAB, which show the reliability of our model from the practical point of view. Epidemiological implications of our analytical findings are addressed critically.