We mentioned earlier that systems of differential equations modeling specific behavior are often times very sensitive to changes in parameters. By now, you may have encountered solutions which were unrealistic or had problems which failed to converge when you supplied your own parameter values. Ideally, mathematical models are calibrated using data from field or clinical research. A model’s effectiveness is based on its ability to accurately portray behavior inside the realm of the original data. Providing parameters which are well beyond these bounds can cause the system to act unexpectedly. In optimal control problems, the optimality system can yield an optimal control which makes little sense physically; sometimes, the system fails to converge to provide any solution at all. In this lab, we examine an ill-conditioned problem with this type of behavior. In a study by Ackerman et al., a simplified, but highly accurate model
of the blood regulatory system was developed to improve the ability of the GTT (glucose tolerance test) to detect pre-diabetics and mild diabetics . The model considers the concentration of blood glucose g and net hormonal concentration h. It was shown that
g′(t) = c1g(t) + c2h(t), h′(t) = c3g(t) + c4h(t).