Agent-Based Models and Optimal Control in Biology: A Discrete Approach - Mathematical Concepts and Methods in Modern Biology - page 168

5.9 TRANSLATING AGENT-BASED MODELS INTO POLYNOMIAL
DYNAMICAL SYSTEMS
This section is authored jointly by Franziska Hinkelmann, Matt Oremland, Hussein
Al-Asadi, Atsya Kumano, Laurel Ohm, Alice Toms, Reinhard Laubenbacher
Discrete models, including agent-based models, are important tools for model-
ing biological systems, but model complexity may hinder complete analysis. Rep-
resentation as a PDS provides a framework for efficient analysis using theory from
abstract algebra. In this section, we provide polynomials that describe common agent
interactions. In the previous section we described how to interpolate agent behav-
ior and to generate the appropriate polynomial. However, for a variable with many
different states, this method of interpolation results in long and complex polyno-
mials that are difficult to expand, simplify, or alter. Thus we provide some general
“shortcut rules” for constructing polynomials that describe key agent and patch inter-
actions present in many ABMs. Each of the following polynomials exists in the finite
field F p .

Since we are particularly interested in ABMs describing complex biological sys-
tems, we use the term concentration to describe the states of a patch variable (for
example, concentration of white blood cells on a patch). In this chapter we describe
several polynomials that describe both basic movement, and movement according to
the state of the neighboring patches.

5.9.1 Basic Movement Function
One can construct various polynomials to describe the movement of an agent on an
n -by- n grid where the x - and y -coordinates of patches are numbered 0 to n 1 from

left to right with torus topology, i.e., there are no boundaries to the grid, so that if
an agent on the left most patch moves to the left, he appears on the right side of the
grid, and similarly for top and bottom. By moving forward we mean moving to the
right, and potentially “wrapping” around to the left edge of the grid, and by moving
backward we mean moving to the left.
For an agent moving forward one patch per time step, we want an agent on patch
x to move to patch x

−

+ 1 unless the agent is on patch x = n − 1, in which case the

agent will move to patch x 0 on the next step due to the torus topology of the grid

(see Table 5.1 ). The polynomial describing this movement is given in Eq. ( 5.2 ).
One can construct a similar polynomial for an agent moving backward one step per
timestep(seeEq.( 5.3 )), and using this polynomial along with the forward movement
one can create a polynomial to describe movement of several steps along the x -axis
as specified by a series of elements in

=

(representing forward, backward,
or no movement), (Eq. 5.4 ). Furthermore, one can generalize these polynomials to
describe movement of a fixed step length m ,(Eq. 5.5 ) and (Eq. 5.6 ).

{ 1 , − 1 , 0 }