Biomathematics: Case Studies and Computer Laboratory Exercises

• Modeling the Dynamics of Single Species Population Growth

Using the computer software package Berkeley Madonna, we examine discrete and continuous models including unlimited growth, logistic growth, and delay models. Physical meaning of model parameters is emphasized.

• Modeling Physiological Mechanisms of Drug Elimination from the Bloodstream.

Why do you need to take two acetaminophen tablets every four to six hours when you have a headache? Why is there a warning on the label which cautions you not to take any more than four doses in a given 24-hour period? And why does your head start to ache again after four hours when the above warning suggests that you really ought to wait until six hours have elapsed before you take the next dose? We examine these questions by developing a mathematical model that describes the dynamics of concentration changes of physiologically active substances in the bloodstream cased by multiple dose intakes.

• Equilibrium States , Stability, and Long-term Behavior of Dynamic Models.

In the context of single-species populations, we discuss the dependence of a model (discrete and continuous models are considered) upon the initial conditions and the effect that small changes in the initial conditions may have on the long-term behavior of the solution. Convergence of trajectories toward the equilibrium states is studied, and the effect from harvesting on the equilibrium states of a model is considered.

• Populations of Interacting Species - Dynamical Models; Phase Plane Analysis; Periodicity.

We use Berkeley Madonna to examine the solutions of coupled systems of differential equations that model the dynamics of predator-prey systems. Phase plane solution profiles are studied to determine the long-term behavior of the solution, convergence to equilibrium, and periodicity. Fast Fourier Transform (FFT) algorithms are introduced and applied for identifying periodic components of the solutions.

• Models involving Delay; Oscillations in Biological Systems.

An oscillatory system must include a restorative process that, like inertia in physical systems, leads to overshoot of the equilibrium value. For most cellular, endocrine, and neuronal oscillations, the process that provides an overshoot is delay . In this laboratory exercise we consider in detail how introducing delay to some of the models we have previously considered could lead to oscillatory solutions. Such models include the logistic growth model and an S - I - S epidemic model.

• Epidemic Models and Applications.

We use Berkeley Madonna in combination with standard mathematical techniques to examine the threshold of an epidemic and the meaning of the relative removal rate , the contact rate, and the basic reproduction rate for an infections disease and the severity of the epidemic. The long-term solutions of the model are studied by means of time plots and phase plane trajectories. We also examine epidemic models without a removal group, as well as epidemic models with intermediate groups. The latter are appropriate for modeling the spread of an epidemic for infectious diseases with an incubation period.

• Selection in Genetics; Disappearance of a Toxic Gene.

We examine the effect caused by selective gene disadvantage has in a closed population. We develop a mathematical method that tracks the dynamical changes in gene frequencies and examine the long-term implications and disappearance of "defective" genes. Similar questions were investigated for genes that provide selective advantage.   Examples such as Tay-Sachs disease and sickle cell anemia are considered.

• Endocrine Oscillators; Modeling and Analysis of the Growth Hormone Network.

When a system involves more than one variable and the variables interact with one another, it is possible, that one of them inhibits the growth of the other. This is, for example, the case in predator-prey models of type Lotka-Volterra. Such interaction is referred to as negative feedback and is often a factor that creates oscillations. For example, is well known that the concentration of many hormones in the blood stream rises and falls multiple times per day and exhibits a periodic pulsatile patterns. Thus, the mathematical models describing the dynamics of these hormones should be able to capture and explain the oscillatory behavior they exhibit. We study the effects of negative feedback for the endocrine network of the Growth Hormone (GH) and focus on the ability of our models to generate a specific oscillatory pattern found in experimental data.   

• Hormone Pulsatility in Reproductive Endocrinology.

There is overwhelming evidence that the pattern of hormone delivery to target organs is crucial to the effectiveness of their action.   If the required hormone is supposed to be delivered in a pulsatile pattern and it is instead delivered at a constant level, the target organ will probably not respond.   If the required hormone is supposed to be delivered in 24 pulses per day and instead is delivered in 12 pulses, the target organ will probably not respond. In particular, if the female reproductive hormones do not follow the "correct" frequency and amplitude patterns of secretion, this may cause infertility. Hormone concentrations however are measured in the bloodstream where the secretion patterns are obscured due to the process of simultaneous elimination. We test a mathematical algorithm separating secretion and elimination components, allowing for determining the secretion profile form hormone concentration. The process is carried out through specialized computer software developed at the University of Virginia.  

• Risk Analysis of Blood Glucose Fluctuation Characteristics in Type 1 vs. Type 2 Diabetes Mellitus.

Diabetes is a complex of disorders, characterized by a common final element of hyperglycemia (a condition of very high level of blood glucose). Diabetes mellitus has two major types: Type 1 (T1DM) caused by autoimmune destruction of insulin producing pancreatic b -cells, and Type 2 (T2DM), caused by defective insulin action (insulin resistance) combined with progressive loss of insulin secretion. . The standard daily control of T1DM involves multiple insulin injections, or a continuous insulin infusion (insulin pump) that lowers BG. The control of T2DM may include any combination of diet, exercise, pills, or insulin injection. However, with the treatment, the chances for hypoglycemia (a potentially lethal condition of very low level of blood glucose) increase significantly. Thus patients with diabetes face a life-long clinical optimization problem : to maintain strict glycemic control without increasing their risk for hypoglycemia. We examine and test a mathematical model aimed at assessing the risk for hypo- and hyperglycemia. The model is derived from self-monitoring data obtained by blood glucose monitors. An algorithm based on this model is currently being implemented for commercial use.

• Using Heartbeat Characteristics and Patterns to Predict Sepsis in Neonates.

Approximately 40,000 very low birth weight (VLBW) infants (less than 1,500 gm) are born in the United States each year. Survival of this group has generally improved with advances in neonatal intensive care in the past decade, but specific diseases such as late-onset sepsis continue to be a major cause of morbidity and mortality. It is often the case, that when the clinical symptoms of sepsis are detected, immediate and massive antibiotic treatment may not be effective due to uncontrollably quick spread of the infection and death.   Thus, developing mathematical models capable of predicting possible sepsis episodes 12 to 24 hours prior to clinical symptoms is invaluable and could potentially save lives.   We study a model based on heart rate variability data and examine its effectiveness in predicting sepsis in prematurely born babies.