**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.