Math 4 - Proposal
Applications of Calculus to Medicine and Biology

Applications of Calculus to Medicine and Biology. In this course we will establish the relevance of calculus to medicine. We will develop mathematical tools extending the techniques of introductory calculus, including some matrix algebra and solution techniques for first order differential equations. Then we will use these methods to construct simple and elegant models of phenomena such as the mutation of HIV, spread of infectious disease, biological disposition of drugs and inorganic toxins, enzyme kinetics and population growth.

Prerequisite: Mathematics 3. Note: This is a second-term calculus course, but it does not cover the same material as Math 8, and does not serve as a prerequisite for Math 13.

Textbooks: An Introduction to the Mathematics of Biology by Yeargers, Shonkwiler and Herod; Lake Victoria, by Abonyo, Cornell, Fixman, von Rittman, and Wallace.

Sample Syllabus

Week 1:

• Introduction to course and discussion of mathematical models in biology.
• Estimation of model parameters via transformation and least squares fits.
• Multiple regression.

Week 2:

• Differential Equations - separable, first order linear, direction fields and numerical solutions.
• Systems of linear differential equations, phase portraits, matrix eigenvalues and long-term behavior of systems.

Week 3:

• Statistical representation of data through histograms, mean, median, standard deviation.
• Elementary probability with an introduction to the law of total probability and Baye's rule.

Week 4: First half of ``Lake Victoria: A Mathematical Ecohistory" Includes discussion of history and colonization of Kenya, Uganda and Tanzania. Development of the Malthusian model of population growth.

Week 5: Second half of "Lake Victoria: A Mathematical Ecohistory" Continued discussion of Africa together with models for ecosystems with carrying capacity. Predator prey models.

Week 6: Discussion of organ systems and how drugs and inorganic toxins move through the body. Clinical effects of lead poisioning and discussion of the prevalence of lead in inner cities.

Week 7: Development of the compartment model for lead in the body. Pharacokinetics and serum concentration cycles.

Week 8: Background information necessary to understand HIV and AIDS. Cell biology, dna, rna etc. Development of Perelson's model of AIDS dynamics within the human body.

Week 9:

• Solution of Perelson's model, determination of its parameters and discussion.
• Beginning information on genetics and probability through Mendel's pea experiment.

Week 10: Discussion of the Hardy-Weinberg principle (if mating is random and all genotypes are equally fit, the fractions of alleles remains the same from generation to generation).