Biology Reference
In-Depth Information
explicitly mathematical, so that we lack mathematical tools for model analysis. For
instance, many of these phenomena are connected to optimization and optimal control
problems, as pointed out in this chapter, but no systematic methods are available for
agent-basedmodels to solve these.We have attempted here to do two things. Firstly, we
described so-called heuristic local search methods, such as genetic algorithms, which
can be applied directly to agent-based models. And we described a way in which
one can translate an agent-based model into a mathematical object, in this case a
polynomial dynamical system over a finite field. Many computational and theoretical
tools are available for such systems. For instance, to compute the steady states of
a polynomial system
F
=
(
f
1
,...,
f
n
)
, one can solve the system of polynomial
equations
x
n
.
There are several computer algebra systems available to solve such problems. To
compute the steady states of an agent-based model, on the other hand, one is limited
to extensive simulation, without any guarantee of having found all possible steady
states.
The chapter provides a snapshot of ongoing research in the field. The approach
via polynomial dynamical systems, for instance, is very promising, but still lacks
appropriate algorithms that scale to larger models. In addition to searching for such
algorithms, further research in model reduction is taking place, as outlined earlier in
the chapter. At the same time, other mathematical frameworks, such as ordinary or
partial differential equations and Markov models are being explored for this purpose.
Much work remains to be done but, in the end, a combination of better algorithms,
improvements in hardware, and dimension reduction methods is likely to provide
for us a tool kit that will allow the solution of realistic large-scale optimization and
optimal control problems in ecology, biomedicine, and other fields related to the life
sciences.
f
1
(
x
1
,...,
x
n
)
=
x
1
,
f
2
(
x
1
,...,
x
n
)
=
x
2
,...,
f
n
(
x
1
,...,
x
n
)
=
5.11
SUPPLEMENTARY MATERIALS
All supplementary files and/or computer code associated with this article can be found
from the volume's website
http://booksite.elsevier.com/9780124157804
References
[1] LedzewiczU, NaghnaeianM, Schattler H. Optimal response to chemotherapy for
a mathematical model of tumor-immune dynamics. J Math Biol 2012;64:557-
577.
[2] Sontag E.D. Mathematical control theory. New York, NY: Springer Verlag 2nd
ed. 1998.
[3] Iglesias PA, Ingalls BP, editors.Control theory and systems biology. Cambridge,
MA: The MIT Press 2009.
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