Chapter 7.1, Problem 70E

Imagine that you are diabetic and have to pay close attention to how your body metabolizes glucose. Letg(t) represent the excess glucose concentration in your blood, usually measured in milligrams of glucose per 100 milliliters of blood. (Excess means that we measure how much the glucose concentration deviates from your fasting level, i.e., the level your system approaches after many hours of fasting.) A negative value of g(t) indicates that the glucose concentration is below fasting level at time t. Shortly after you eat a heavy meal, the function g(t) will reach a peak, and then it will slowly return to 0. Certain hormones help regulate glucose, especially the hormone insulin. Let h(t) represent the excess hormone concentration in your blood. Researchers have developed mathematical models for the glucose regulatory system. The following is one such model, in slightly simplified form (these formulas apply between meals; obviously, the system is disturbed during and right after a meal):
$\left|\begin{array}{c}g\left(t+1\right)=ag\left(t\right)-bh\left(t\right)\\ h\left(t+1\right)=cg\left(t\right)+dh\left(t\right)\end{array}\right|$ ,
where time t is measured in minutes; a and d are constants slightly less than 1; and b and c are small positive constants. For your system, the equations might be $\left|\begin{array}{c}g\left(t+1\right)=0.978g\left(t\right)-0.006h\left(t\right)\\ h\left(t+1\right)=0.004g\left(t\right)+0.992h\left(t\right)\end{array}\right|$ . The term −0.006h(t) in the first equation is negative, because insulin helps your body absorb glucose. The term 0.004g(t) is positive, because glucose in your blood stimulates the cells of the pancreas to secrete insulin. (For a more thorough discussion of this model, read E. Ackerman et al., “Blood glucose regulation and diabetes,” Chapter 4 in Concepts and Models of Biomathematics, Marcel Dekker, 1969.)
Consider the coefficient matrix
$A=\left[\begin{array}{cc}0.978& -0.006\\ 0.004& 0.992\end{array}\right]$ of this dynamical system.
a. We are told that $\left[\begin{array}{l}-1\\ 2\end{array}\right]$ and $\left[\begin{array}{c}3\\ -1\end{array}\right]$ are eigenvectors of A. Find the associated eigenvalues.
b. After you have consumed a heavy meal, the concentrations in your blood are $g_0=100$ and $h_0=0$ .Find closed formulas for g(t) and h(t). Sketch the trajectory. Briefly describe the evolution of this system in practical terms.
c. For the case discussed in part (b), how long does it take for the glucose concentration to fall below fasting level? (This quantity is useful in diagnosing diabetes: A period of more than four hours may indicate mild diabetes.)