PATTERNS OF ALCOHOLISM: A MATHEMATICAL MODEL

Abstract

In a first phenomenological part various forms of alcoholism and their classification are summarized. Our aim is to give an explanation of these patterns of alcohol consumption and addiction. Social, psychic, behavioral, and biochemico-physiological mechanisms currently discussed in the literature are briefly considered with respect to their potential for generating the dynamics of alcohol addiction. As central core those mechanisms turn out to be self-enhancing. By formulating these mechanisms in terms of mathematical models a tool is provided to study their consequences in isolation and in qualitative and quantitative detail. The mathematical modeling is put forward in several steps starting with a single (differential) equation for the process of self-enhancement. This process, acting alone, leads to exponential and unbounded increase of the average alcohol consumption. In a second step, inhibitory mechansims are added which result e.g. from negative effects of alcohol drinking. The combination and different relative weights of self-enhancement and inhibition can already explain several types of drinking behavior, in particular those of α, β, γ and δ drinkers according to the classification of Jellinek. Possible transitions from one type to another and also to abstinence become imaginable by discussion of bifurcations on a 'cusp catastrophe". Oscillatory phenomena in drinking behavior, e.g. the "episodic drinker", result if and only if, in a further step, the model is complemented by a second differential equation accounting for the dynamics of an internal state symbolically called "frustration level". More generally this variable may represent any factor which interacts in a circular fashion (feedback loop) with the alcohol consumption. In connection with the presented models, various ways and strategies for the transition from one type of drinking behavior to another (including low level drinking or abstinence) are discussed.