## Modeling Complex Living Systems: A Kinetic Theory and Stochastic Game ApproachThesubjectofthisbookisthemodelingofcomplex systemsinthelife sciences constituted by a large number of interacting entities called active particles. Their physical state includes, in addition to geometrical and mechanical variables, a variable called the activity, which characterizes the speci?c living system to be modeled. Interactions among particles not only modify the microscopic state, but may generate proliferative and/or destructive phenomena. The aim of the book is to develop mathematical methods and tools, even a new mathematics, for the modeling of living systems. The background idea is that the modeling of living systems requires technically complex mathematical methods, which may be s- stantially di?erent from those used to deal with inert matter. The?rstpart ofthe bookdiscussesmethodological issues, namely the derivation of various general mathematical frameworks suitable to model particular systems of interest in the applied sciences. The second part presents the various models and applications. The mathematical approach used in the book is based on mathema- cal kinetic theoryfor active particles, whichleads tothederivation of evo- tion equations for a one-particle distribution function over the microscopic state. Two types of equations, to be regarded as a general mathematical framework for deriving the models, are derived corresponding to short and long range interactions. |

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### Contents

1 | |

12 Classical Models of Kinetic Theory | 4 |

13 Discrete Velocity Models | 13 |

14 Guidelines for Modeling Living Systems | 15 |

15 Purpose and Plan of the Book | 20 |

Mathematical Structures of the Kinetic Theory for Active Particles | 27 |

22 The Generalized Distribution Function | 29 |

23 Modeling Microscopic Interactions | 35 |

55 Critical Analysis and Further Developments | 104 |

Mathematical Modeling of Vehicular Traffic Flow Phenomena | 109 |

62 Scaling and Representation | 111 |

63 A Survey of Continuous Kinetic Traffic Models | 121 |

64 Discrete Velocity Models | 128 |

65 On the Model by Delitala and Tosin | 134 |

66 On the Modeling by Active Particle Methods | 144 |

Complex Biological Systems Mutations and Immune Competition | 147 |

24 Mathematical Frameworks | 44 |

25 Some Particular Frameworks | 47 |

26 Additional Concepts | 51 |

Additional Mathematical Structures for Modeling Complex Systems | 53 |

32 Models with MixedType Interactions | 54 |

33 Models with Exotic Proliferations | 55 |

34 Phenomenological Frameworks | 58 |

35 Open Systems | 61 |

Mathematical Frameworks for Discrete Activity Systems | 63 |

42 Motivations for a Discrete States Modeling | 65 |

43 On the Discrete Distribution Function | 66 |

44 Mathematical Framework | 70 |

45 Additional Generalizations | 74 |

46 Critical Analysis | 79 |

Modeling of Social Dynamics and Economic Systems | 81 |

52 A Model by Bertotti and Delitala | 83 |

53 Qualitative Analysis and Simulations | 89 |

54 Some Ideas on Further Modeling Perspectives | 96 |

72 Modeling the Immune Competition | 149 |

73 Mathematical Structures for Modeling | 151 |

74 An Example of Mathematical Models | 156 |

75 Modeling Developments and Perspectives | 163 |

Modeling Crowds and Swarms Congested and Panic Flows | 169 |

82 The Representation of Crowds and Swarms | 172 |

83 Modeling by Macroscopic Equations | 175 |

84 Modeling by Kinetic Theory Methods | 179 |

85 Looking Forward | 182 |

Additional Concepts on the Modeling of Living Systems | 189 |

92 Mathematical Problems | 191 |

93 Looking for New Mathematical Structures | 196 |

94 Additional Issues on Modeling | 199 |

95 Speculations on a Mathematical Theory for Living Systems | 203 |

209 | |

219 | |

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ability active particles activity variable additional analogous analyzed applied mathematicians assumption behavior Bellomo Boltzmann equation classical classical mechanics complex computational consider corresponding crowds and swarms defined Delitala denotes depends derivation described developed discrete velocity distribution function domain equilibrium evolution equation external actions field particle follows identified immune cells immune competition immune system individuals initial value problem instance interacting pairs interacting populations ith population large number living systems long range interactions macroscopic scale mathematical framework mathematical kinetic theory mathematical models mathematical problems mathematical structures methods microscopic interactions modeling approach modeling of microscopic modify Moreover number of particles obtained ordinary differential equations parameters phase space phenomena phenomenological probability density proliferative qualitative analysis research perspectives Section social classes space variable spatially homogeneous specific models strategy suitable technical term test particle theory for active tion traffic flow modeling value problem velocity variable

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