bloodstream by 1 mg/liter.
mt+1 = 1mt + 1 with m0 = 0. 3
In class we studied the following model for the level of medication in the bloodstream of a patient who receives a fixed daily dose of medication, that raises the concentration of the medication in the
The equation m0 = 0 means that on day 0 (before the treatment starts) there is no medication in the blood. The '+1' represents the dosage and the 1 is the fraction of medication left in the
3
bloodstream by the patient's body tissue each day (which means the other 2 is absorbed out). 3
The fraction of the medication that is absorbed out of the blood into the tissue can be different for different people. Below we explore what would happen if the patient absorbed more of the medication.
1. Assume that the patient absorbs 80% of the medication that is present in the bloodstream (so 80% of what is in the blood stream on day t is absorbed out of the bloodstream into the tissue by the next day, t + 1). The daily dose is still 1 mg/liter, and the patient begins with no medication in the blood on day 0. Give the new discrete dynamical system that models this situation (including the initial condition).
Note: For the remaining questions in this section assume the discrete dynamical system is given by mt+1 = .25mt + 1 with m0 = 0 .
2. Write an excel document that calculates the medication levels in the bloodstream for the first two weeks of taking this medication (i.e., for 14 days). What do you expect to happen with the amount of medication in the bloodstream if the patient continues to take this dose of medication for a longer period of time?
3. How is this different from what happened for the first system (mt+1 = 1 mt + 1)? What 3
conclusions can you draw about the relationship between the level of absorption by the body tissues and the final levels of the medication concentration? Does this make sense?
4. Find the equilibrium point(s) of this system. How is this related to the answer of your previous question?
5. Draw a cobwebbing diagram that determines whether this equilibrium is stable or unstable. Does this answer agree with what you would have expected from your Excel table?
1For questions 6-8, we change the daily dose. For a daily dose of d mg/liter the system is given by:
mt+1 = .25mt + d , with m0 = 0 .
6. What would the equilibrium concentration be in this case? Give your answer as a function of
d.
7. Suppose that a doctor wants to give the patient a daily dose such that eventually the concen-
tration of medication will be 2 mg/liter. What should that dose be?