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. 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.

When behaviour turns contagious: the use of deterministic epidemiological models in modeling social contagion phenomena

“…Treatment may be included in these models via a treatment class and the recovered class. Addictive behaviors modelled include alcoholism [39][40][41][42][43][44], smoking [45] and the use of ecstasy, crystal meth and heroin [46][47][48][49].…”

A Discrete Mathematical Modeling of the Influence of Alcohol Treatment Centers on the Drinking Dynamics Using Optimal Control

“…ey discussed the existence and stability of drinking-free and endemic equilibrium. Other mathematical models have also been widely used to study this phenomenon (for example, [5,[9][10][11][12]).…”

Mathematical Modeling and Analysis of an Alcohol Drinking Model with the Influence of Alcohol Treatment Centers

“…Mathematical models can be used to analyze the spread of infectious diseases or the social behavior of individuals [9][10][11][12][13][14][15][16][17][18][19][20]. As regard to drinking, several different mathematical models have been formulated and studied to help in reducing the number of drinkers [17,[21][22][23][24]. Agrawal et al [22] developed a nonlinear SHTR mathematical model of alcohol abuse with a nonlinear incidence rate.…”

“…ey demonstrated that backward bifurcation can occur when R 0 � 1. Sharma et al [24] developed a two-stage (four compartments) model for youths with serious drinking problems and their treatment having admitted and having not admitted. ey also analyzed the stability of all the equilibria.…”