This book deals with the modelling of complex systems in life sciences constituted by a large number of interacting entities called {\dc active particles}. Their physical state includes, in addition to geometrical and mechanical variables, also an additional variable called {\dc activity}, which characterizes the specific living system object of the modelling approach. Interactions not only modify the microscopic state, but may generate proliferative and/or destructive phenomena. The {\dc aim} of this book consists in developing mathematical methods and tools, hopefully a new mathematics, towards the modelling of living systems. The {\dc background idea} is that modelling living systems needs technically complex mathematical methods, which may be substantially different from those used to deal with the inert matter. The first part of the book deals with methodological issues, namely with the derivation of various general mathematical frameworks suitable to be specialized to model particular systems of interest in applied sciences; the second part with various models and applications. The mathematical approach, proposed in the first part of the book, is based on the mathematical kinetic theory for active particles which leads to the derivation of evolution equations for the one particle distribution function over the microscopic state. Two types of equations, that ought to be regarded as a general mathematical framework for the derivation of models, are derived corresponding to short and long range interactions. It is a new kinetic theory, that can be applied to derive various models of practical interest in life sciences, and that includes, as special cases, the classical models of the kinetic theory, namely the Boltzmann and Vlasov equations. The difference, with respect to the classical theory, is that interactions follow stochastic rules technically related to the strategy developed by individuals belonging living systems. Various models and applications, derived within the above mathematical framework, are presented in the second part of the book following a common style for all chapters: phenomenological interpretation of the physical system in view of the mathematical modelling; derivation of the mathematical model by methods of the mathematical kinetic theory for active particles; simulations, parameter sensitivity analysis, and critical analysis of the model; perspective ideas to improve the existing models with special attention to applications. \sm Specifically, the following classes of models are dealt with: social competition related to individuals characterized by a microscopic state referred to their social collocation; modelling of vehicular traffic flow; immune competition for multicellular systems; and modelling of crowds and swarms with special attention to the analysis of the transition from normal to panic situations. The complexity of the mathematical structures needed to deal with the above systems increases from the first to the last class of models. For instance, models of social dynamics refer to spatially homogeneous systems, while traffic flow models need a space structure; immune competition models refer to a variable number of active particles due to the presence of proliferative and/or destructive events. Finally. models of crowds and swarms take into account a complex interaction space dynamics with rules which may consistently be modified by changes in the environment. The common characteristic for all above models is the attempt to describe the collective behavior starting from the microscopic dynamics. The above selection is generated by the personal experience of the author. Hopefully, the reader interested in modelling living systems can generalize the approach to different field of applications, that may even need technical developments of the mathematical structures proposed in this book. The contents of each chapter are not limited to a review of the existing literature, while new ideas and perspectives are proposed. The presentation mainly refers to modelling and simulations, while analytic aspects, say good position of mathematical problems and qualitative analysis of their solutions, are critically analyzed and are brought to the attention of applied mathematicians as interesting research perspectives. The analysis of models generates a variety of analytic and computational problems which are sufficiently complex to generate a remarkable attraction towards the intellectual energies of applied mathematician interested in the challenging perspective of modelling living systems.

Modeling complex living systems - Kinetic theory and stochastic game approach / Bellomo, Nicola. - (2008).

Modeling complex living systems - Kinetic theory and stochastic game approach

BELLOMO, Nicola
2008

Abstract

This book deals with the modelling of complex systems in life sciences constituted by a large number of interacting entities called {\dc active particles}. Their physical state includes, in addition to geometrical and mechanical variables, also an additional variable called {\dc activity}, which characterizes the specific living system object of the modelling approach. Interactions not only modify the microscopic state, but may generate proliferative and/or destructive phenomena. The {\dc aim} of this book consists in developing mathematical methods and tools, hopefully a new mathematics, towards the modelling of living systems. The {\dc background idea} is that modelling living systems needs technically complex mathematical methods, which may be substantially different from those used to deal with the inert matter. The first part of the book deals with methodological issues, namely with the derivation of various general mathematical frameworks suitable to be specialized to model particular systems of interest in applied sciences; the second part with various models and applications. The mathematical approach, proposed in the first part of the book, is based on the mathematical kinetic theory for active particles which leads to the derivation of evolution equations for the one particle distribution function over the microscopic state. Two types of equations, that ought to be regarded as a general mathematical framework for the derivation of models, are derived corresponding to short and long range interactions. It is a new kinetic theory, that can be applied to derive various models of practical interest in life sciences, and that includes, as special cases, the classical models of the kinetic theory, namely the Boltzmann and Vlasov equations. The difference, with respect to the classical theory, is that interactions follow stochastic rules technically related to the strategy developed by individuals belonging living systems. Various models and applications, derived within the above mathematical framework, are presented in the second part of the book following a common style for all chapters: phenomenological interpretation of the physical system in view of the mathematical modelling; derivation of the mathematical model by methods of the mathematical kinetic theory for active particles; simulations, parameter sensitivity analysis, and critical analysis of the model; perspective ideas to improve the existing models with special attention to applications. \sm Specifically, the following classes of models are dealt with: social competition related to individuals characterized by a microscopic state referred to their social collocation; modelling of vehicular traffic flow; immune competition for multicellular systems; and modelling of crowds and swarms with special attention to the analysis of the transition from normal to panic situations. The complexity of the mathematical structures needed to deal with the above systems increases from the first to the last class of models. For instance, models of social dynamics refer to spatially homogeneous systems, while traffic flow models need a space structure; immune competition models refer to a variable number of active particles due to the presence of proliferative and/or destructive events. Finally. models of crowds and swarms take into account a complex interaction space dynamics with rules which may consistently be modified by changes in the environment. The common characteristic for all above models is the attempt to describe the collective behavior starting from the microscopic dynamics. The above selection is generated by the personal experience of the author. Hopefully, the reader interested in modelling living systems can generalize the approach to different field of applications, that may even need technical developments of the mathematical structures proposed in this book. The contents of each chapter are not limited to a review of the existing literature, while new ideas and perspectives are proposed. The presentation mainly refers to modelling and simulations, while analytic aspects, say good position of mathematical problems and qualitative analysis of their solutions, are critically analyzed and are brought to the attention of applied mathematicians as interesting research perspectives. The analysis of models generates a variety of analytic and computational problems which are sufficiently complex to generate a remarkable attraction towards the intellectual energies of applied mathematician interested in the challenging perspective of modelling living systems.
2008
9780817646004
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/1654521
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