We are going to develop two continuous models, one simple, and one more sophisticated, to
describe the one step of the metabolism of alcohol in the body, by focusing on the amount in the
bloodstream as one drinks, and the removal of this alcohol from the bloodstream by the liver. We
are not going to worry about what happens in the liver, how the waste products are disposed,
the effect on the brain cells, and so on. As this process is continuous, we will use continuous
dynamic models. In the first model we assume that the quantity of alcohol in the bloodstream
Q(t), measured in grams, as a function of time t, measured in hours, is changing by consumption
of c grams/hour and is being removed by the liver at a rate proportional to Q itself. Denote the
constant of proportionality by r, and write the model equation that governs the changing Q. If
you like you may use L(t) to denote the amount of alcohol absorbed by the liver, and write the
model equation for this as well, but we are not really interested in this part of the process, as I
indicated above. In humans the value of r is around 2.5 hr
−1
or 2.5/hr (this actually depends
on weight, body composition (alcohol is absorbed into all tissues that contain water, not just
the blood), and consequently, also gender). Since we are taking r > 0, be sure that your model
equation(s) have the correct sign(s); remember that Q(t) represents the amount of alcohol in the
bloodstream at time t. Let us assume that Q(0) = 0, and for starters that c = 14 g/hr (this
amounts to one “drink” such as a 12 oz. can of beer or a 5 oz. serving of wine or one oz. of
100 proof hard liquor per hour). Solve the model equation by obtaining an explicit formula for
Q(t) in terms of t. Is there an equilibrium value for Q? What happens to Q(t) as time goes on?
In this model, the liver continues to remove alcohol from the bloodstream no matter how much
is consumed (how can you tell this from the model?). Unfortunately in real life, this does not
happen; the liver can only take up so much so fast; in technical terms there is a level of “satiation”
analogous to the satiation of a predator when prey are abundant; so we must modify the model
to take this into account.
