Agent-Based Models and Optimal Control in Biology: A Discrete Approach - Mathematical Concepts and Methods in Modern Biology - page 149

author's intentions explicit. One proposition to overcome this lack of formalization
is presented in [ 9 ]; this work proposes a rigorous mathematical representation for
agent-based models as polynomial dynamical systems. A further description of such
systems can be found in [ 10 ] and is described in Section 5.8 .
5.5 OPTIMIZATION AND OPTIMAL CONTROL
Agent-based models are typically created to simulate some real-world process in
order to aid investigation. Once a model has been created, a natural next step is to ask:
what are we to do with it? In this section we give a brief overview of optimal control
and optimization, and introduce these ideas as they apply to agent-based models.
Optimization is the process of finding the best solution with respect to a particular
goal. For example, suppose we have a model of the immune system battling a bacterial
infection, and we wish to study the effects of certain drugs on this battle. We may
wish to find out which drug does the least amount of tissue damage while curing
the infection—this is an optimization problem, because we are searching for the best
drug with respect to the stated goal. On the other hand, perhaps our goal is to cure
the infection in the shortest time possible, regardless of the tissue damage caused.
This would be another optimization problem. It is likely, then, that the solution to
the optimization problem depends on the optimization goal. In this scenario, the drug
which cures the infection most quickly may consequently do more tissue damage
than other drugs (though not necessarily); on the other hand, the drug that causes
the least tissue damage might not cure the infection in the shortest possible time.
In this example, optimization is a process of minimization: in one case we wish
to minimize tissue damage and in the other we wish to minimize the healing time.
However, it is also possible that the solution of an optimization problem is a process
of maximization: given a model of an immune system fighting a fatal illness, what
treatment will enable the patient to survive the longest? In fact, it may be that the goal
of the optimization problem is to minimize one value while maximizing another. For
example, our optimization goal may be to minimize tissue damage and maximize the
expected lifespan of the patient, given that we can only administer a particular drug
a fixed number of times. In general, however, optimization is the process of finding
the best solution, depending on the objective.
Typically, optimization is a complicated process, and becomes even more so when
realistic constraints are put in place. Through the use of agent-based models it may
be possible to obtain a solution to an optimization problem that is not feasible in
actuality: for example, the solution may exceed monetary limitations, may require
actions that are not permitted by health care regulations, or may require interaction
with the patient in an impractical way (e.g., if the solution calls for treatment every
hour for 100 consecutive hours, or for 100% of the population to receive vaccination).
Nevertheless, optimization remains a natural means of applying mathematics to solve
real-world problems. Such problems are often framed as questions of optimization:
what is the best outcome that can be produced based on the properties of the model?