See attachement.

Scenario

Consider a drug dosage problem for combating "anthrax". The doctor prescribes 500 mg of Cipro as a daily dosage. We assume that the kidney has diluted the drug by 45% every day. Model as a DDS and determine the long-term behavior of the system.

Problem Statement: Determine the long-term amount of Cipro in the body.

Assumptions: Nothing changes about the way the body processes Cipro for the duration of the model.

Model:

D(n) = Amount of Cipro in the body after day n.

D(n+1) = D(n)-.45*D(n)+500, D(0)=0

or

D(n+1) = .55 D(n) + 500, D(0)=0

1. Open a new worksheet

2. Enter your name and a brief problem description.

3. Enter the model

4. Create column headings for n, D(n) (assume we put n in cell A10, and D(n) in cell B10.

5. In the next row, enter the initial condition D(0)=0: put 0 in cell A11 and 0 in Cell B11.

6. In cell A12 put =a11+1

7. In cell B12 put =.55*b11+500

8. Copy those formulas into the remaining cells below until you get to a value of n that you desire.

This looks like:

Drug Problem

0

500

1000

1500

0

5

10

15

20

time

D(n)

Next, we might want to view a graph.

Highlight the columns and go to CHART WIZARD

Select SCATTERPLOT (x vs y)

We see a stable equilibrium at about 1111.

What is the true equilibrium value?

EXTENSION#1:

If you wanted to specify an equilibrium amount , E, in the blood, derive an equation that tells how many mg, K, would need to be in each daily dose?

HINT: Set D(n+1) = D(n) = E in the iteration equation, and replace the “500” with K. Solve for K in terms of E.

EXTENSION#2:

If you wanted to maintain an equilibrium of 1000 mg in the blood by giving two doses per day, how many mg would need to be in each dose?

HINT: Look at what happens if you start with 1000 mg in the blood with NO ADDITIONAL SHOTS. Does this look like exponential decay? What is the equation? If 0.55 of the original amount is left at the end of 24 hours, how much would be left at the end of just 12 hours? Now, set D(n+1/2) = D(n) = 1000 (so we get 1000 mg equilibrium with two doses per day), and solve for the dose (call it K).

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