# SHORT COURSES ON THE MATHEMATICS OF BIOLOGICAL COMPLEXITY

## Supported by National Institutes of Health Award GM59924-01

### Optimal Control Theory in Application to Biology: July 9-12, 2003

The short course leaders were: Dr. Suzanne Lenhart, Dr. Renee Fister, and Dr. Hem Raj Joshi.

This course gives an introduction to optimal control techniques for systems of ordinary differential equations. These techniques are applied to a variety of biological and medical examples. Models with one differential equation and one control are treated first and then the extension to systems of differential equations and multiple controls is given. Examples include disease, cancer, bioreactor, bear and fishery models. Bang-bang and singular control cases are covered. To illustrate an extension to a hybrid system of discrete time and continuous space variables, an integro-difference population model paper is presented.

Below are links to four lectures given at the short course and several papers on the material presented at the course.

There are also MATLAB examples to illustrate the ideas. The "lab" examples are designed to be user-friendly.

• Short Course Lecture Schedule Text file
• Introductory Lecture, lecture by Lenhart PDF file
• Introductory Lecture part II, lecture by Lenhart PDF file
• Lecture on Bang-Bang and Singular Controls, lecture by Lenhart PDF file
• Lecture on Systems, lecture by Fister PDF file
• Two-strain Tuberculosis Model, paper by Jung, Lenhart, and Feng PDF file
• HIV Immunology Model, paper by Joshi PDF file
• Integro-difference Population Model, paper by Joshi, Lenhart, and Gaff PDF file
• References Text file

• The file, labsheets.pdf, gives the explanations for 10 examples of optimal control problems. The examples are designed to request the appropriate inputs interactively and display the resulting graphs. To work these examples, you need to download the 15 .m files below and use them with MATLAB (version 6 or newer). Note that five examples have 2 parts, for example lab1.m and labone.m. These examples were prepared by John Workman and Suzanne Lenhart.

• labsheets.pdf, lab1.m, labone.m, lab2.m, lab3.m, lab4.m, lab5.m, labfive.m, lab6.m, lab7.m, lab8.m, labeight.m, lab9.m, labnine.m, lab10.m, labten.m