Global Stability for a Binge Drinking Model with Two Stages

A more realistic two-stage model for binge drinking problem is introduced, where the youths with alcohol problems are divided into those who admit the problem and those who do not admit it. We also consider the direct transfer from the class of susceptible individuals towards the class of admitting drinkers. Mathematical analyses establish that the global dynamics of the model are determined by the basic reproduction number, . The alcohol-free equilibrium is globally asymptotically stable, and the alcohol problems are eliminated from the population if . A unique alcohol-present equilibrium is globally asymptotically stable if . Numerical simulations are also conducted in the analytic results. 1. Introduction Young people’s binge drinking problem is a major concern to public health. Recently, US surveys indicate that approximately 90% of college students have consumed alcohol at least once [1], and more than of college students have engaged in binge drinking [2, 3]. The binge drinking refers to youths in 17–30 age group who drink a large amount of alcohol and become so drunk; they are likely to exhibit antisocial behavior [4]. Although there have been many attempts to reduce the problem, alcohol abuse by college students has persisted and in some cases increased over the past several decades [5]. Prior studies have indicated that heavy alcohol drinkers are likely to engage in risky sexual behaviours and more likely to get sexually transmitted infection than social drinkers [6, 7]. There is a strong medical evidence that treatment of individuals with alcohol problems is a major issue [8–10]. Thus, it is very important to use a mathematical method to study the binge drinking problems in youths. A simple model for alcohol treatment is presented by Sanchez et al. [11]. Since then, there have been numerous studies investigating campus drinking and the associated consequences [12–15]. Manthey et al. [12] focus on a college campus, divide the student population into three classes: nondrinkers, social drinkers and problem drinkers, and show that campus alcohol abuse may be reduced by minimizing the ability of problem drinkers to directly recruit nondrinkers. Cintron-Arias et al. [13] focus on situations where relapse rates are high and conclude that the systematic removal of individuals from high-risk environments, or the development of programs that limit access or reduce the residence times in such environments (or both approaches combined), may reduce the level of alcohol abuse. Mubayi et al. [14] show that if the relative residence times of moderate drinkers