Problem 27 - Calculus for Biology and Medicine

Question

If a patient takes ibuprofen every \(T\) hours, rather than every 6 hours then the concentration of ibuprofen in their blood one hour after each pill is taken (that is, after \(1,1+T, 1+2 T\), hours, and so on) is given by a recurrence equation: $$C_{n+1}=(0.7575)^{T} C_{n}+40$$ (a) Find the equilibrium point of this recurence equation, and show that it is locally stable for any value of \(T>0\). (b) Assume that \(T=1\) and \(C_{1}=40 .\) Make a cobweb plot to illustrate the behavior of the sequence \(C_{1}, C_{2}, C_{3}, \ldots .\)

Step-by-Step Solution

Answer

The equilibrium point is \( C_e = \frac{40}{1 - (0.7575)^T} \).

Step 1: Define Equilibrium Point

The equilibrium point of a recurrence relation is where the concentration remains constant: \( C_{n+1} = C_n = C_e \). Using the recurrence equation, we have \[ C_e = (0.7575)^T C_e + 40 \]. To solve for the equilibrium, rearrange to find:\[ C_e - (0.7575)^T C_e = 40 \]\[ C_e (1 - (0.7575)^T) = 40 \]\[ C_e = \frac{40}{1 - (0.7575)^T} \].

Key Concepts

Equilibrium Point

An equilibrium point in a recurrence relation is a steady state where the variables do not change between iterations. In simpler terms, the equilibrium occurs when the concentration of ibuprofen in the blood remains constant after each dose. For this exercise, the given recurrence relation is:\[ C_{n+1} = (0.7575)^T C_n + 40 \] To find the equilibrium point \( C_e \), we set \( C_{n+1} \) equal to \( C_n \), because the concentration does not change at equilibrium. Thus, we have:\[ C_e = (0.7575)^T C_e + 40 \] Rearranging this equation, we isolate \( C_e \) on one side:\[ C_e (1 - (0.7575)^T) = 40 \] Then, we solve for \( C_e \):\[ C_e = \frac{40}{1 - (0.7575)^T} \] This formula gives us the equilibrium concentration of ibuprofen in the patient's blood. It shows how the time interval \( T \) between doses affects the steady-state concentration if the dosing continues indefinitely. This concept is crucial for understanding how medicine remains effective in the body over time.

Stability

Stability in the context of recurrence relations refers to whether small changes in the initial conditions or parameters cause the system to return to equilibrium. A local stability implies that the system, when slightly perturbed, will return back to its equilibrium point over time.For this recurrence relation:\[ C_{n+1} = (0.7575)^T C_n + 40 \] The stability of the equilibrium point can be analyzed by looking at the multiplier \( (0.7575)^T \). A system is stable if this multiplier \( |a| \) is less than 1:- When \( T > 0 \), the base number 0.7575 raised to any positive power \( T \) will be less than one.- This results in \( |(0.7575)^T| < 1 \) ensuring the equation is stable.This means for any \( T > 0 \), the equilibrium point is attractive. If there is a small deviation from this point, the system will naturally steer back towards equilibrium over time. It's akin to a steady ship righting itself after a wave introduces some wobble.

Cobweb Plot

A cobweb plot is a visual representation used to illustrate the behavior of sequences defined by recurrence relations. It helps in understanding how the system evolves over each step, especially near equilibrium points.To create a cobweb plot for:\[ C_{n+1} = (0.7575)^T C_n + 40 \] we follow these steps:1. **Start** with an initial concentration value, like \( C_1 = 40 \).2. **Plot the Line**: Draw the line \( y = x \). This line helps visualize equilibrium, as points on this line mean \( C_{n+1} = C_n \).3. **Function Plot**: Sketch \( y = (0.7575)^T x + 40 \).4. **Cobweb Pattern**: Using the initial condition, plot \( C_1 = 40 \) on the x-axis. - Move vertically to intersect the curve \( y = (0.7575)^T x + 40 \). - Move horizontally to the line \( y = x \). - Repeat this vertical and horizontal movement to see how \( C_{2}, C_{3}, \ldots \) behave.The path traced out resembles a cobweb, hence the name. When \( T = 1 \), the sequence set by the cobweb plot will begin to clearly show whether it’s moving towards or away from the equilibrium. If your cobweb eventually settles into a straight line along \( y = x \), it indicates stability, confirming a tendency towards the equilibrium point.