Lecture Notes on Mathematical Modelling in the Life Sciences – serie

Mathematical biomedicine is a rapidly developing interdisciplinary field of research that connects the natural and exact sciences in an attempt to respond to the modeling and simulation challenges raised by biology and medicine. There exist a large number of mathematical methods and procedures that can be brought in to meet these challenges and this book presents a palette of such tools ranging from discrete cellular automata to cell population based models described by ordinary differential equations to nonlinear partial differential equations representing complex time- and space-dependent continuous processes. Both stochastic and deterministic methods are employed to analyze biological phenomena in various temporal and spatial settings. This book illustrates the breadth and depth of research opportunities that exist in the general field of mathematical biomedicine by highlighting some of the fascinating interactions that continue to develop between the mathematical and biomedical sciences. It consists of five parts that can be read independently, but are arranged to give the reader a broader picture of specific research topics and the mathematical tools that are being applied in its modeling and analysis. The main areas covered include immune system modeling, blood vessel dynamics, cancer modeling and treatment, and epidemiology. The chapters address topics that are at the forefront of current biomedical research such as cancer stem cells, immunodominance and viral epitopes, aggressive forms of brain cancer, or gene therapy. The presentations highlight how mathematical modeling can enhance biomedical understanding and will be of interest to both the mathematical and the biomedical communities including researchers already working in the field as well as those who might consider entering it. Much of the material is presented in a way that gives graduate students and young researchers a starting point for their own work.

​Waves in Neural Media: From Single Neurons to Neural Fields surveys mathematical models of traveling waves in the brain, ranging from intracellular waves in single neurons to waves of activity in large-scale brain networks. The work provides a pedagogical account of analytical methods for finding traveling wave solutions of the variety of nonlinear differential equations that arise in such models. These include regular and singular perturbation methods, weakly nonlinear analysis, Evans functions and wave stability, homogenization theory and averaging, and stochastic processes. Also covered in the text are exact methods of solution where applicable. Historically speaking, the propagation of action potentials has inspired new mathematics, particularly with regard to the PDE theory of waves in excitable media. More recently, continuum neural field models of large-scale brain networks have generated a new set of interesting mathematical questions with regard to the solution of nonlocal integro-differential equations.  Advanced graduates, postdoctoral researchers and faculty working in mathematical biology, theoretical neuroscience, or applied nonlinear dynamics will find this book to be a valuable resource. The main prerequisites are an introductory graduate course on ordinary differential equations or partial differential equations, making this an accessible and unique contribution to the field of mathematical biology.

This monograph introduces some current developments in the modelling of the spread of tick-borne diseases. Effective modelling requires the integration of multiple frameworks. Here, particular attention is given to the previously neglected issues of tick developmental and behavioral diapause, tick-borne pathogen co-feeding transmission, and their interactions. An introduction to the required basics of structured population formulations and delay differential equations is given, and topics for future study are suggested. The described techniques will also be useful in the study of other vector-borne diseases. The ultimate aim of this project is to develop a general qualitative framework leading to tick-borne disease risk predictive tools and a decision support system.The target audience is mathematical biologists interested in modelling tick population dynamics and tick-borne disease transmission, and developing computational tools for disease prevention and control.

This book discusses the mathematical simulation of biological systems, with a focus on the modeling of gene expression, gene regulatory networks and stem cell regeneration. The diffusion of morphogens is addressed by introducing various reaction-diffusion equations based on different hypotheses concerning the process of morphogen gradient formation. The robustness of steady-state gradients is also covered through boundary value problems. The introduction gives an overview of the relevant biological concepts (cells, DNA, organism development) and provides the requisite mathematical preliminaries on continuous dynamics and stochastic modeling. A basic understanding of calculus is assumed. The techniques described in this book encompass a wide range of mechanisms, from molecular behavior to population dynamics, and the inclusion of recent developments in the literature together with first-hand results make it an ideal reference for both new students and experienced researchers in the field of systems biology and applied mathematics.

This book presents current meta-ecosystem models and their derivation from classical ecosystem and metapopulation theories. Specifically, it reviews recent modelling efforts that have emphasized the role of nonlinear dynamics on spatial and food web networks, and which have cast their implications within the context of spatial synchrony and ecological stoichiometry. It suggests that these recent advances naturally lead to a generalization of meta-ecosystem theories to spatial fluxes of matter that have both a trophic and non-trophic impact on species.Ecosystem dynamics refers to the cycling of matter and energy across ecological compartments through processes such as consumption and recycling. Spatial dynamics established its ecological roots with metapopulation theories and focuses on scaling up local ecological processes through the limited movement of individuals and matter. Over the last 15 years, theories integrating ecosystem and spatial dynamics have quickly coalesced into meta-ecosystem theories, the focus of this book.The book will be of interest to graduate students and researchers who wish to learn more about the synthesis of ecosystem and spatial dynamics, which form the foundation of the theory of meta-ecosystems.

This book demonstrates how delay differential equations (DDEs) can be used to compliment the laboratory investigation of human balancing tasks. This approach is made accessible to non-specialists by comparing mathematical predictions and experimental observations. For example, the observation that a longer pole is easier to balance on a fingertip than a shorter one demonstrates the essential role played by a time delay in the balance control mechanism. Another balancing task considered is postural sway during quiet standing. With the inverted pendulum as the driver and the feedback control depending on state variables or on an internal model, the feedback can be identified by determining a critical pendulum length and/or a critical delay. This approach is used to identify the nature of the feedback for the pole balancing and postural sway examples. Motivated by the question of how the nervous system deals with these feedback control challenges, there is a discussion of ‘’microchaotic’’ fluctuations in balance control and how robust control can be achieved in the face of uncertainties in the estimation of control parameters. The final chapter suggests some topics for future research.Each chapter includes an abstract and a point-by-point summary of the main concepts that have been established. A particularly useful numerical integration method for the DDEs that arise in balance control is semi-discretization. This method is described and a MATLAB template is provided.This book will be a useful source for anyone studying balance in humans, other bipedal organisms and humanoid robots. Much of the material has been used by the authors to teach senior undergraduates in computational neuroscience and students in bio-systems, biomedical, mechanical and neural engineering.

This book provides an introduction to the analysis of stochastic dynamic models in biology and medicine. The main aim is to offer a coherent set of probabilistic techniques and mathematical tools which can be used for the simulation and analysis of various biological phenomena. These tools are illustrated on a number of examples. For each example, the biological background is described, and mathematical models are developed following a unified set of principles. These models are then analyzed and, finally, the biological implications of the mathematical results are interpreted. The biological topics covered include gene expression, biochemistry, cellular regulation, and cancer biology. The book will be accessible to graduate students who have a strong background in differential equations, the theory of nonlinear dynamical systems, Markovian stochastic processes, and both discrete and continuous state spaces, and who are familiar with the basic concepts of probability theory.

This book presents the basic theoretical concepts of dynamical systems with applications in population dynamics. Existence, uniqueness and stability of solutions, global attractors, bifurcations, center manifold and normal form theories are discussed with cutting-edge applications, including a Holling's predator-prey model with handling and searching predators and projecting the epidemic forward with varying level of public health interventions for COVID-19.As an interdisciplinary text, this book aims at bridging the gap between mathematics, biology and medicine by integrating relevant concepts from these subject areas, making it self-sufficient for the reader. It will be a valuable resource to graduate and advance undergraduate students for interdisciplinary research in the area of mathematics and population dynamics.

This book introduces some basic mathematical tools in reaction-diffusion models, with applications to spatial ecology and evolutionary biology. It is divided into four parts.The first part is an introduction to the maximum principle, the theory of principal eigenvalues for elliptic and periodic-parabolic equations and systems, and the theory of principal Floquet bundles.The second part concerns the applications in spatial ecology. We discuss the dynamics of a single species and two competing species, as well as some recent progress on N competing species in bounded domains. Some related results on stream populations and phytoplankton populations are also included. We also discuss the spreading properties of a single species in an unbounded spatial domain, as modeled by the Fisher-KPP equation.The third part concerns the applications in evolutionary biology. We describe the basic notions of adaptive dynamics, such as evolutionarily stable strategies and evolutionarybranching points, in the context of a competition model of stream populations. We also discuss a class of selection-mutation models describing a population structured along a continuous phenotypical trait.The fourth part consists of several appendices, which present a self-contained treatment of some basic abstract theories in functional analysis and dynamical systems. Topics include the Krein-Rutman theorem for linear and nonlinear operators, as well as some elements of monotone dynamical systems and abstract competition systems.Most of the book is self-contained and it is aimed at graduate students and researchers who are interested in the theory and applications of reaction-diffusion equations.

This book provides a self-contained introduction to a new class of stochastic models for systems of spiking neurons. These systems have a large number of interacting components, each one evolving as a stochastic process with a memory of variable length. Several mathematical tools are put to use, such as Markov chains, stochastic chains having memory of variable length, point processes having stochastic intensity, Hawkes processes, random graphs, mean field limits, perfect sampling algorithms, the Context algorithm, and statistical model selection.The book’s focus on mathematically tractable objects distinguishes it from other texts on theoretical neuroscience. The biological complexity of neurons is not ignored, but reduced to some of its main features, such as the intrinsic randomness of neuronal dynamics. This reduction in complexity aims at explaining and reproducing statistical regularities and collective phenomena that are observed in experimental data, an approach that leads to mathematically rigorous results. With an emphasis on a constructive and algorithmic point of view, this book is directed towards mathematicians interested in learning about stochastic network models and their neurobiological underpinning, and neuroscientists interested in learning how to build and prove results with mathematical models that relate to actual experimental settings.

The book provides a self-contained and complete description of the long time evolution of the solutions to a class of one-dimensional reaction–diffusion equations, in which the diffusion is given by an integral operator. The underlying motivation is the mathematical analysis of models for biological invasions. The model under study, while simple looking, is of current use in real-life situations. Interestingly, it arises in totally different contexts, such as the study of branching random walks in probability theory. While the model has attracted a lot of attention, and while many partial results about the time-asymptotic behaviour of its solutions have been proved over the last decades, some basic questions on the sharp asymptotics have remained unanswered. One ambition of this monograph is to close these gaps. In some of the situations that we envisage, the level sets organise themselves into an invasion front that is asymptotically linear in time, up to a correction that converges exponentially in time to a constant. In other situations that constitute the main and newest part of the work, the correction is asymptotically logarithmic in time. Despite these apparently different behaviours, there is an underlying common way of thinking that is underlined. At the end of each chapter, a long set of problems is proposed, many of them rather elaborate and suitable for master's projects or even the first question in a PhD thesis. Open questions are also discussed. The ideas presented in the book apply to more elaborate systems modelling biological invasions or the spatial propagation of epidemics. The models themselves may be multidimensional, but they all have in common a mechanism imposing the propagation in a given direction; examples are presented in the problems that conclude each chapter. These ideas should also be useful in the treatment of further models that we are not able to envisage for the time being. The book is suitable for graduate or PhD students as well as researchers.

This book provides an overview of recent mathematical models for dynamics in cellular systems and offers a unique vision of the field by prominent experts.It covers, among others, the regulatory basis of oscillations in biological systems; ergodic and chaotic properties in biological models with the example of maturity distribution of precursors of blood cells; time-delayed feedbacks; mathematical models of cell division and heterogeneous stem cell regeneration; quantitative mathematical modeling of glucose regulation; data-driven models of chemotherapy-induced neutropenia; and effects of irradiation and antioxidants in Alzheimer’s disease.This book is directed towards mathematicians interested in learning about modeling in cellular systems and is accessible also to theoreticians in biology and medicine.

This textbook introduces mathematical modelling in life sciences in an interdisciplinary setting for students whose backgrounds range from medicine to mathematics, focusing on a set of topics that introduce mathematical thinking through biological questions. Each chapter begins with a clear biological question and then introduces the mathematical tools required to analyze it. This iterative approach is a proven success in the classrom. Starting with population dynamics, because its concepts are intuitive to students from the exact sciences, the text then broadens the canvas to topics such as electrophysiology and chemical kinetics, demonstrating the wide applicability of the same mathematical techniques. This structure is intended to build intuition alongside technical skill—students learn why a model is useful before learning how to build it.Also included are short historical perspectives and vignettes about the development of key ideas. These historical contexts motivate students and help them better grasp why particular concepts matter and how they arose in response to real scientific problems.Whether you are a student seeking a first rigorous introduction to mathematical biology or an instructor looking for a compact course text, this book is designed to be accessible, adaptable, and practical: you will encounter clear problems, worked mathematical tools, and exercises aimed at building modeling confidence and insight.

This book brings theories in nonlinear dynamics, stochastic processes, irreversible thermodynamics, physical chemistry and biochemistry together in an introductory but formal and comprehensive manner. Coupled with examples, the theories are developed stepwise, starting with the simplest concepts and building upon them into a more general framework. Furthermore, each new mathematical derivation is immediately applied to one or more biological systems. The last chapters focus on applying mathematical and physical techniques to study systems such as: gene regulatory networks and molecular motors.The target audience of this book are mainly final year undergraduate and graduate students with a solid mathematical background (physicists, mathematicians and engineers), as well as with basic notions of biochemistry and cellular biology. This book can also be useful to students with a biological background who are interested in mathematical modeling and have a working knowledge of calculus, differential equations and a basic understanding of probability theory.

This book on mathematical modeling of biological processes includes a wide selection of biological topics that demonstrate the power of mathematics and computational codes in setting up biological processes with a rigorous and predictive framework.  Topics include: enzyme dynamics, spread of disease, harvesting bacteria, competition among live species, neuronal oscillations, transport of neurofilaments in axon, cancer and cancer therapy, and granulomas. Complete with a description of the biological background and biological question that requires the use of mathematics, this book is developed for graduate students and advanced undergraduate students with only basic knowledge of ordinary differential equations and partial differential equations; background in biology is not required. Students will gain knowledge on how to program with MATLAB without previous programming experience and how to use codes in order to test biological hypothesis.

This book presents several fundamental questions in mathematical biology such as Turing instability, pattern formation, reaction-diffusion systems, invasion waves and Fokker-Planck equations. These are classical modeling tools for mathematical biology with applications to ecology and population dynamics, the neurosciences, enzymatic reactions, chemotaxis, invasion waves etc. The book presents these aspects from a mathematical perspective, with the aim of identifying those qualitative properties of the models that are relevant for biological applications. To do so, it uncovers the mechanisms at work behind Turing instability, pattern formation and invasion waves. This involves several mathematical tools, such as stability and instability analysis, blow-up in finite time, asymptotic methods and relative entropy properties. Given the content presented, the book is well suited as a textbook for master-level coursework.

This book is a “How To” guide for modeling population dynamics using Integral Projection Models (IPM) starting from observational data. It is written by a leading research team in this area and includes code in the R language (in the text and online) to carry out all computations. The intended audience are ecologists, evolutionary biologists, and mathematical biologists interested in developing data-driven models for animal and plant populations. IPMs may seem hard as they involve integrals. The aim of this book is to demystify IPMs, so they become the model of choice for populations structured by size or other continuously varying traits. The book uses real examples of increasing complexity to show how the life-cycle of the study organism naturally leads to the appropriate statistical analysis, which leads directly to the IPM itself. A wide range of model types and analyses are presented, including model construction, computational methods, and the underlying theory, with the more technical material in Boxes and Appendices. Self-contained R code which replicates all of the figures and calculations within the text is available to readers on GitHub.Stephen P. Ellner is Horace White Professor of Ecology and Evolutionary Biology at Cornell University, USA; Dylan Z. Childs is Lecturer and NERC Postdoctoral Fellow in the Department of Animal and Plant Sciences at The University of Sheffield, UK; Mark Rees is Professor in the Department of Animal and Plant Sciences at The University of Sheffield, UK.

This is a short and self-contained introduction to the field of mathematical modeling of gene-networks in bacteria. As an entry point to the field, we focus on the analysis of simple gene-network dynamics. The notes commence with an introduction to the deterministic modeling of gene-networks, with extensive reference to applicable results coming from dynamical systems theory. The second part of the notes treats extensively several approaches to the study of gene-network dynamics in the presence of noise—either arising from low numbers of molecules involved, or due to noise external to the regulatory process.  The third and final part of the notes gives a detailed treatment of three well studied and concrete examples of gene-network dynamics by considering the lactose operon, the tryptophan operon, and the lysis-lysogeny switch.  The notes contain an index for easy location of particular topics as well as an extensive bibliography of the current literature. The target audience of these notes are mainly graduates students and young researchers with a solid mathematical background (calculus, ordinary differential equations, and probability theory at a minimum), as well as with basic notions of biochemistry, cell biology, and molecular biology. They are meant to serve as a readable and brief entry point into a field that is currently highly active, and will allow the reader to grasp the current state of research and so prepare them for defining and tackling new research problems.

This book analyzes the impact of quiescent phases on biological models. Quiescence arises, for example, when moving individuals stop moving, hunting predators take a rest, infected individuals are isolated, or cells enter the quiescent compartment of the cell cycle. In the first chapter of Topics in Mathematical Biology general principles about coupled and quiescent systems are derived, including results on shrinking periodic orbits and stabilization of oscillations via quiescence. In subsequent chapters classical biological models are presented in detail and challenged by the introduction of quiescence. These models include delay equations, demographic models, age structured models, Lotka-Volterra systems, replicator systems, genetic models, game theory, Nash equilibria, evolutionary stable strategies, ecological models, epidemiological models, random walks and reaction-diffusion models. In each case we find new and interesting results such as stability of fixed points and/or periodic orbits, excitability of steady states, epidemic outbreaks, survival of the fittest, and speeds of invading fronts. The textbook is intended for graduate students and researchers in mathematical biology who have a solid background in linear algebra, differential equations and dynamical systems. Readers can find gems of unexpected beauty within these pages, and those who knew K.P. (as he was often called) well will likely feel his presence and hear him speaking to them as they read.

This book developed from classes in mathematical biology taught by the authors over several years at the Technische Universität München. The main themes are modeling principles, mathematical principles for the analysis of these models and model-based analysis of data. The key topics of modern biomathematics are covered: ecology, epidemiology, biochemistry, regulatory networks, neuronal networks and population genetics. A variety of mathematical methods are introduced, ranging from ordinary and partial differential equations to stochastic graph theory and  branching processes. A special emphasis is placed on the interplay between stochastic and deterministic models.

With the availability of high speed computers and advances in computational techniques, the application of mathematical modeling to biological systems is expanding. This comprehensive and richly illustrated volume provides up-to-date, wide-ranging material on the mathematical modeling of kidney physiology, including clinical data analysis and practice exercises. Basic concepts and modeling techniques introduced in this volume can be applied to other areas (or organs) of physiology.The models presented describe the main homeostatic functions performed by the kidney, including blood filtration, excretion of water and salt, maintenance of electrolyte balance and regulation of blood pressure. Each chapter includes an introduction to the basic relevant physiology, a derivation of the essential conservation equations and then a discussion of a series of mathematical models, with increasing level of complexity.This volume will be of interest to biological and mathematical scientists, as well as physiologists and nephrologists, who would like an introduction to mathematical techniques that can be applied to renal transport and function. The material is written for students who have had college-level calculus, but can be used in modeling courses in applied mathematics at all levels through early graduate courses.

This book provides an introduction to age-structured population modeling which emphasizes the connection between mathematical theory and underlying biological assumptions.Through the rigorous development of the linear theory and the nonlinear theory alongside numerics, the authors explore classical equations that describe the dynamics of certain ecological systems. Modeling aspects are discussed to show how relevant problems in the fields of demography, ecology and epidemiology can be formulated and treated within the theory. In particular, the book presents extensions of age-structured modeling to the spread of diseases and epidemics while also addressing the issue of regularity of solutions, the asymptotic behavior of solutions, and numerical approximation. With sections on transmission models, non-autonomous models and global dynamics, this book fills a gap in the literature on theoretical population dynamics.The Basic Approach toAge-Structured Population Dynamics will appeal to graduate students and researchers in mathematical biology, epidemiology and demography who are interested in the systematic presentation of relevant models and mathematical methods.