July 30-August 2, 2008
Society for Mathematical Biology Conference

hosted by the Centre for Mathematical Medicine, Fields Institute
held at University of Toronto, Medical Sciences Bldg
.


SMB
2008

Back to mini-symposia index

3) Application of optimal control theory to oncology, ecology and epidemiology
Principal organiser: Holly Gaff and K. Renee Fister
Community and Environmental Health, College of Health Sciences, Old Dominion University and Dept. of Mathematics and Statistics, Murray State University

For many different kinds of problems, the natural question that arises is how to intervene to produce the “best” outcome, as measured by some predetermined goals. One approach to this is via optimal control theory. Optimal control theory is a branch of mathematics developed to find optimal ways to control a dynamical system. Optimal control theory governs strategies for maximizing a performance measure or minimizing a cost function as the state of a dynamic system evolves. This theoretical approach has been widely applied to many problems in engineering, physics, business and management, and economics. Optimal control theory opens new possibilities in ecology and epidemiology. In addition, from the practical viewpoint of developing optimal treatment strategies and therapies, optimal control theory is an important new tool for mathematical oncology. This will be the focus of this mini-symposium.

In the proposed mini-symposium, we bring together a group of distinguished researchers who have particular expertise in biological modeling and optimal control theory. The four talks in the proposed session provide a diversity of new developments in the area of optimal control theory of nonlinear biological systems.
The talk by Dr. Renee Fister will include optimal control of neuroblastoma therapy to design the most beneficial treatment methods for cancer patients. A mathematical model is used to investigate the effectiveness of the chemotherapy drug Topotecan against neuroblastoma. Neuroblastoma is mostly a pediatric cancer that consists of crest cells found in tumors of nerve tissues. Optimal control theory is applied to minimize the tumor volume. A state constraint is used to maintain the circulating neutrophils above a critical level. Existence and uniqueness of the optimality system are established.

The use of optimal control theory applied to species augmentation model will be presented by Erin Bodine. The ecological conservation method of “species augmentation” attempts to reduce species loss by augmenting declining or threatened populations with individuals from captive-bred or stable, wild populations. Optimal control theory is used to determine the augmentation strategy for a declining target population (the Florida panthers) where a growing reserve population is available.

Dr. Holly Gaff will address the impact of spatial heterogeneity on the spread and control of disease. Recent increases in reported outbreaks of vector-borne diseases throughout the world have led to increased interest in understanding and controlling epidemics involving transmission vectors. The talk presented here is motivated by an effort to control outbreak of human monocytic ehrlichiosis, recently-diagnosed as a tick-borne disease. Optimal control theory is used to define time- and location-dependent strategies for which acaricide might be most effectively placed.

The use of optimal control theory in resource allocation models will also be investigated by Dr. Hem Joshi. he increasing prevalence of HIV/AIDS in Africa over the last twenty-five years continues to erode the continent's health care and overall welfare. There have been various responses to the pandemic, led by Uganda which has had the greatest success in combating the disease. Part of Uganda's success has been attributed to a formalized Information, Education, and Communication (IEC) strategy, lowering estimated HIV/AIDS infection rates from 18.5\% in 1995 to 4.1\% in 2003. We formulate a model to investigate the effects of information and education campaigns
on the HIV epidemic in Uganda. These campaigns affect the people's behavior and can divide the susceptibles class into subclasses with different infectivity rates.

Our model is a system of ordinary differential equations and we use data about the epidemics and the number of organizations involved in the campaigns to estimate the model parameters. We compare our model with three types of susceptibles to a standard SIR model. This symposium will be a valuable resource for all scientists interested in applying mathematics to develop strategies for treating human diseases, controlling epidemics, developing health policies and addressing ecological problems.

Speakers:

Name: Renee Fister
Affiliation: Murray State University
Address: Dept. of Mathematics and Statistics, Murray State University
Title: Optimal Control Study of Neuroblastoma
Abstract: A mathematical model is used to investigate the e®ectiveness of the chemotherapy drug Topotecan against neuroblastoma. Optimal control theory is applied to minimize the tumor volume and the amount of drug utilized. The model incorporates a state constraint that requires the level of circulating neutrophils (white blood cells that form an integral part of the immune system) to remain above an acceptable value. The treatment schedule is designed to simultaneously satisfy this constraint and achieve the best results in treating the tumor. Existence and uniqueness of the optimality system, which is the state system coupled with the adjoint system, is established. Numerical simulations are given to demonstrate the behavior of the tumor and the immune system
components represented in the model.

Name: Hem Joshi
Affiliation: Dept. of Mathematics and Computer Science
Title: Effect of Education on HIV Epidemic

Abstract: The increasing prevalence of HIV/AIDS in Africa over the last twenty-five years continues to erode the continent's health care and overall welfare. There have been various responses to the pandemic, led by Uganda which has had the greatest success in combating the disease. Part of Uganda's success has been attributed to a formalized Information, Education, and Communication (IEC) strategy, lowering estimated HIV/AIDS infection rates from 18.5\% in 1995 to 4.1\% in 2003. We formulate a model to investigate the effects of information and education campaigns on the HIV epidemic in Uganda. These campaigns affect the people's behavior and can divide the susceptibles class into subclasses with different infectivity rates. Our model is a system of ordinary differential equations and we use data about the epidemics and the number of organizations involved in the campaigns to estimate the model parameters. We compare our model with three types of susceptibles to a standard SIR model.

Name: Holly Gaff
Affiliation: Community and Environmental Health, College of Health Sciences, Old Dominion University
Title:Optimal control of a metapopulation tick model
Abstract: Human monocytic ehrlichiosis is a tick-transmitted, rickettsial disease that has recently increased substantially in the USA from 142 reported cases in 2001 to 506 reported cases in 2005. In an effort to explore the dynamics of this disease and to gain insights into risk factors for outbreaks, we have developed a mathematical model to allow simulations of the disease dynamics assuming various environmental conditions and tick control strategies. This model is a set of four nonlinear differential equations linked by population and disease dynamics for both a single tick and a single host species. The model is then expanded spatially using discrete geographic patches linked by host migration patterns. Each patch is assigned appropriate habitat and environmental conditions. We present the results of this model that show the importance of spatial and temporal heterogeneity on the prevalence of the disease. We also apply the mathematical technique of optimal control with patch-specific strategies to identify intervention strategies that would reduce the risk of disease while preventing decimation of the tick populations. Numerical simulations of the model find that while blanket application of acaricides can reduce prevalence of the disease, further reductions in the diseased population can be obtained with less effort by using a more refined approach. This model provides the basis for more sophisticated models of HME dynamics.

Name: Erin Bodine
Affiliation: The Institute for Environmental Modeling and the Mathematics Department at The University of Tennessee
Title: Optimal control applied to a species augmentation model: the continuous time case
Abstract: Species augmentation is a method of reducing species loss via augmenting declining or threatened populations with individuals from captive-bred or stable, wild populations. We developed a differential equations model and optimal control formulation for a continuous time augmentation of a general declining population. We found a characterization for the optimal control and show numerical results for scenarios of different illustrative parameter sets. These work and results are a first step toward building a general theory of population augmentation which accounts for the complexities inherent in many conservation biology applications.

Back to