Modeling Complex Living Systems

Using tools from mathematical kinetic theory, this work deals with the modeling of large complex systems in the applied sciences, particularly those comprised of several interacting individuals whose dynamics follow rules determined by some organized, or even "intelligent" ability. Traditionally, methods of mathematical kinetic theory have been applied to model the evolution of large systems of interacting classical or quantum particles. This book, on the other hand, examines the modeling of living systems as opposed to inert systems. The author develops new mathematical methods and tools'hopefully a "new" mathematics'toward the modeling of living systems. Such tools need to be far more complex than those dealing with systems of inert matter. The first part of the book deals with deriving general evolution equations that can be customized to particular systems of interest in the applied sciences. The second part of the book deals with various models and applications. The presentation unfolds with the following common style in each chapter: * Phenomenological interpretation of the physical system in the context of mathematical modeling * Derivation of the mathematical model using methods from mathematical kinetic theory for active particles * Simulations, parameter sensitivity analysis, and critical inspection of the derived model towards validation * Overview of presented ideas to improve existing models, with special emphasis on applications Specific topics covered include: * Modeling of the competition between cells of an aggressive invasive agent and cells of the immune system * Modeling of vehicular traffic flow * Modeling of swarms and crowd dynamics in complex geometric environments * Methodological aspects related to multiscale modeling of large systems viewed as interconnected subsystems Modeling Complex Living Systems will be a valuable resource for applied mathematicians, engineers, physicists, biologists, economists, and graduate students involved in modeling complex social systems and living matter in general.

Preface	p. xi
From Scaling and Determinism to Kinetic Theory Representation	p. 1
Scaling and Determinism	p. 1
Classical Models of Kinetic Theory	p. 4
Discrete Velocity Models	p. 13
Guidelines for Modeling Living Systems	p. 15
Purpose and Plan of the Book	p. 20
Mathematical Structures of the Kinetic Theory for Active Particles	p. 27
Introduction	p. 27
The Generalized Distribution Function	p. 29
Modeling Microscopic Interactions	p. 35
Mathematical Frameworks	p. 44
Some Particular Frameworks	p. 47
Additional Concepts	p. 51
Additional Mathematical Structures for Modeling Complex Systems	p. 53
Introduction	p. 53
Models with Mixed-Type Interactions	p. 54
Models with Exotic Proliferations	p. 55
Phenomenological Frameworks	p. 58
Open Systems	p. 61
Mathematical Frameworks for Discrete Activity Systems	p. 63
Introduction	p. 63
Motivations for a Discrete States Modeling	p. 65
On the Discrete Distribution Function	p. 66
Mathematical Framework	p. 70
Additional Generalizations	p. 74
Critical Analysis	p. 79
Modeling of Social Dynamics and Economic Systems	p. 81
Introduction	p. 81
A Model by Bertotti and Delitala	p. 83
Qualitative Analysis and Simulations	p. 89
Some Ideas on Further Modeling Perspectives	p. 96
Critical Analysis and Further Developments	p. 104
Mathematical Modeling of Vehicular Traffic Flow Phenomena	p. 109
Introduction	p. 109
Scaling and Representation	p. 111
A Survey of Continuous Kinetic Traffic Models	p. 121
Discrete Velocity Models	p. 128
On the Model by Delitala and Tosin	p. 134
On the Modeling by Active Particle Methods	p. 144
Complex Biological Systems: Mutations and Immune Competition	p. 147
Introduction	p. 147
Modeling the Immune Competition	p. 149
Mathematical Structures for Modeling	p. 151
An Example of Mathematical Models	p. 156
Modeling Developments and Perspectives	p. 163
Modeling Crowds and Swarms: Congested and Panic Flows	p. 169
Introduction	p. 169
The Representation of Crowds and Swarms	p. 172
Modeling by Macroscopic Equations	p. 175
Modeling by Kinetic Theory Methods	p. 179
Looking Forward	p. 182
Additional Concepts on the Modeling of Living Systems	p. 189
Introduction	p. 189
Mathematical Problems	p. 191
Looking for New Mathematical Structures	p. 196
Additional Issues on Modeling	p. 199
Speculations on a Mathematical Theory for Living Systems	p. 203
Collective Bibliography	p. 209
Index	p. 219
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Modeling Complex Living Systems

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