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Methods and models in mathematical biology. Deterministic and stochastic approaches. (English) Zbl 1331.92002
Lecture Notes on Mathematical Modelling in the Life Sciences. Berlin: Springer (ISBN 978-3-642-27250-9/pbk; 978-3-642-27251-6/ebook). xiii, 711 p. (2015).
Chapter 1 is focused on methods: the basic techniques for linear ordinary differential equations, linear stochastic differential equations, Markov chains, and stochastic processes describing independent particles.
Chapter 2 gives some overview of the most important ecological models: tools to deal with these nonlinear models, e.g., bifurcation theory or the time to extinction. Therefore, Chapters 1 and 2 form the backbone of the book – they establish the linear as well as the nonlinear theory of compartmental modelling and present the classical results of mathematical biology in ecology. A new quality of models comes in with structure: the handling of space, size, and age requires additional ideas.
The authors develop these ideas primarily by considerations that occur in an ecological context (Chapter 3). The discussion of evolution is very much focused on the one hand on some basic, classical models as the Fisher-Wright-Haldane model, and on the other hand on adaptive dynamics. In this chapter, an individual is characterised by more properties than just its species. Typical attributes are location, age or size. In particular, if we focus on these properties, a continuous variable is adequate to describe this additional feature (e.g., age is continuous). Central questions in mathematical epidemiology are, to start with, an accurate description of the mechanisms that allow diseases to spread. The aim is, e.g., the prediction of the time course of outbreaks, or the model-based evaluation of surveillance data. Also, the a posteriori analysis of an epidemic is useful, as models can be checked and critical parameters can be identified. In Chapter 4, the challenges in mathematical epidemiology, especially diseases among humans, are described and these are the complexity and unpredictability of human behavior.
In many applications, especially when dealing with molecular processes, (bio-)chemical reactions are considered. In Chapter 5, the authors provide a short introduction explaining how such reactions can be formulated adequately by mathematical models. Neglecting spatial structure and stochasticity (by assuming sufficiently large numbers of involved players) leads mainly to ODE systems. But also the stochastic processes describing molecular dynamics are used, e.g., for simulations, in the so-called Gillespie algorithm. Often, the reactions involved in a system take place on different time scales – a fact that can be used to simplify the model equations. Chapter 6, dedicated to neuronal activity, starts with the description of a single ganglion (a neuronal cell, nerve cell) that is stimulated by a given input. Experiments show basically that a certain minimal activation is necessary to provoke a reaction. This reaction has the form of one spike in the case of a short input signal, or periodic spiking in case of a constant input. This behaviour is described by the Hodgkin-Huxley model, or by the simpler Fitzhugh-Nagumo model. Next, one considers the simplest network: the output signal is fed in again as input signal. The authors aim to understand the origin of this observation. In general, one is not able to state any theorem about large networks of neurons. However, a certain topology is provided (a two-dimensional lattice with interaction of nearest neighbours only) and also a further abstraction is derived, the Greenberg-Hastings automata, a special case of cellular automata. Using combinatorial methods, it is possible to prove theorems about the behaviour of these automata. In particular, it is possible to derive a condition under which a network stays activated and can never go completely to the resting state.

92-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to biology
92D25 Population dynamics (general)
92D40 Ecology
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60J28 Applications of continuous-time Markov processes on discrete state spaces
92D30 Epidemiology
92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
92C20 Neural biology
92D15 Problems related to evolution
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