Agent-Based Models and Optimal Control in Biology: A Discrete Approach - Mathematical Concepts and Methods in Modern Biology - page 152

information. Given that agent-based models are often stochastic in nature (meaning
that while the rules are fixed, the model dynamics depend on many random variables),
results obtained from one simulation are typically not reliable. We eliminate such
variation by simulating a given control schedule many times and using averages for
evaluation in the cost function.
Of all the possible 2 365 solutions, there must be one that is better (or at least
no worse) than every other solution. Strictly speaking, such a solution is referred
to as an optimal solution. However, in many cases we cannot simulate all possible
solutions because of the monumental amount of computing time involved. Thus, for
the purposes of this chapter, we refer to the best solution we are able to find as the
optimal solution , with the caveat that there may indeed be a better solution elsewhere
in the solution space.
We now define the following terms, each of which have been illustrated in this
example.
Definition 5.2 (Terms).
￿ Model parameters: quantities that are part of the model specification. They have
fixed values.
￿ Model dynamics: the relationships between the state variables in a model. In
general, the model dynamics will be affected by the model parameters and the
rules that govern the interaction of the variables.
￿ Control variable: a variable whose value can be specified by the user. Altering
the value of a control variable will (in general) have some effect on the resulting
model dynamics.

￿ State variable: a variable whose value is observed but cannot be directly specified

by the user (i.e., not a control variable). State variable values affect the model
dynamics; they are affected by the value of other state variables, model parameters,
and control variable values.

￿ Solution: a sequence of inputs to the control variables. A full solution assigns a

value to each control variable at each time step; a partial solution is a sequence
wherein values are either only assigned to some of the control variables, or are
assigned to the control variables at only certain time steps.

￿ Solution space: the set of all possible solutions. If p 1 , p 2 ,..., p n are the control

variables and each parameter p i (for 1 n ) can take on a i possible values,

and there are a total of t time steps, then the solution space will consist of

i

n

a i a 1 · a 2 · ... · a n = (

t

= a 1 a 2 ··· a n )

i = 1

solutions (thus each solution is a vector of length t , representing the sequence of
inputs to the control variables).
￿ Population: a subset of the solution space. The population may be the entire
solution space or a proper subset.