Modelling Alcoholism as a Contagious Disease: A Mathematical Model with Awareness Programs and Time Delay

A dynamic alcohol consumption model with awareness programs and time delay is formulated and analyzed. The aim of this model is to capture the effects of awareness programs and time delay in controlling the alcohol problems. We introduce awareness programs by media in the model as a separate class with growth rate of the cumulative density of them being proportional to the number of mortalities induced by heavy drinking. Susceptible population will isolate themselves and avoid contact with the heavy drinkers or become aware of risk of heavy drinking and decline to drink due to such programs. In particular, we incorporate time delay because the nonconsumer population will take a period of time to become an alcohol consumer. We find that the model has two equilibria: one without alcohol problems and one where alcohol problems are endemic in population. The model analysis shows that though awareness programs cannot eradicate alcohol problems, they are effective measures in controlling the alcohol problems. Further, we conclude that the time delay in alcohol consumption habit which develops in susceptible population may result in Hopf bifurcation by increasing the value of time delay. Some numerical simulation results are also given to support our theoretical predictions