#### Event Title

Mathematical Modeling and Optimal Control of Chemotherapy applied to HIV

#### Location

CSU 284A

#### Start Date

5-4-2011 9:00 AM

#### End Date

5-4-2011 10:30 AM

#### Student's Major

Mathematics and Statistics

#### Student's College

Science, Engineering and Technology

#### Mentor's Name

Namyong Lee

#### Mentor's Department

Mathematics and Statistics

#### Mentor's College

Science, Engineering and Technology

#### Description

Of great concern today is the treatment of patients infected with the human immunodeficiency virus (HIV). In this project, we built a series of mathematical models to understand the dynamics of HIV virus, immune system and chemotherapy interactions. Then we found the best chemotherapy strategy through optimal control theory.

Different chemotherapies are continuously being tested and these are under intense study to find the optimal strategy for administering the treatment. While chemotherapy can be effective at fighting HIV, at the same time it also can cause several negative side effects such as; nausea, diarrhea, anemia, neutropenia, and cytotoxicity. Further, HIV is capable of mutating and gaining drug resistance. Thus it is of vital importance to be able to find an optimal dosage strategy that both minimizes the negative side effects as well as the likelihood of mutation. We first developed a mathematical model for the interaction of chemotherapy with HIV and the immune system as well as the possibility of virus mutation.

Optimal control theory, an extension of the calculus of variations, is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and Richard Bellman.

The optimal control can be derived using Pontryagin's maximum principle (a necessary condition), or by solving the Hamilton-Jacobi-Bellman equation (a sufficient condition).

Utilizing optimal control theory, we have determined an optimal strategy with dosages of both reverse transcriptase (RT) inhibitors and protease inhibitors (PIs). The optimal strategy is to alternate dosages of RT and PI with a period of no treatment in between. This strategy reduces the amount of virus present while minimizing virus mutation and negative effects on the immune system. We will also present the computer simulation that supports our result.

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Mathematical Modeling and Optimal Control of Chemotherapy applied to HIV

CSU 284A

Of great concern today is the treatment of patients infected with the human immunodeficiency virus (HIV). In this project, we built a series of mathematical models to understand the dynamics of HIV virus, immune system and chemotherapy interactions. Then we found the best chemotherapy strategy through optimal control theory.

Different chemotherapies are continuously being tested and these are under intense study to find the optimal strategy for administering the treatment. While chemotherapy can be effective at fighting HIV, at the same time it also can cause several negative side effects such as; nausea, diarrhea, anemia, neutropenia, and cytotoxicity. Further, HIV is capable of mutating and gaining drug resistance. Thus it is of vital importance to be able to find an optimal dosage strategy that both minimizes the negative side effects as well as the likelihood of mutation. We first developed a mathematical model for the interaction of chemotherapy with HIV and the immune system as well as the possibility of virus mutation.

Optimal control theory, an extension of the calculus of variations, is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and Richard Bellman.

The optimal control can be derived using Pontryagin's maximum principle (a necessary condition), or by solving the Hamilton-Jacobi-Bellman equation (a sufficient condition).

Utilizing optimal control theory, we have determined an optimal strategy with dosages of both reverse transcriptase (RT) inhibitors and protease inhibitors (PIs). The optimal strategy is to alternate dosages of RT and PI with a period of no treatment in between. This strategy reduces the amount of virus present while minimizing virus mutation and negative effects on the immune system. We will also present the computer simulation that supports our result.

#### Recommended Citation

Branscombe, Daniel R.. "Mathematical Modeling and Optimal Control of Chemotherapy applied to HIV." *Undergraduate Research Symposium*, Mankato, MN, April 5, 2011.

http://cornerstone.lib.mnsu.edu/urs/2011/oral-session-12/3