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2023 | Buch

Modeling in Biology Kapitel lesen Erstes Kapitel lesen

Mathematical Modeling for Epidemiology and Ecology

verfasst von: Glenn Ledder

Verlag: Springer International Publishing

Buchreihe : Springer Undergraduate Texts in Mathematics and Technology

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Mathematical Modeling for Epidemiology and Ecology provides readers with the mathematical tools needed to understand and use mathematical models and read advanced mathematical biology books. It presents mathematics in biological contexts, focusing on the central mathematical ideas and the biological implications, with detailed explanations. The author assumes no mathematics background beyond elementary differential calculus.

An introductory chapter on basic principles of mathematical modeling is followed by chapters on empirical modeling and mechanistic modeling. These chapters contain a thorough treatment of key ideas and techniques that are often neglected in mathematics books, such as the Akaike Information Criterion. The second half of the book focuses on analysis of dynamical systems, emphasizing tools to simplify analysis, such as the Routh-Hurwitz conditions and asymptotic analysis. Courses can be focused on either half of the book or thematically chosen material from both halves, such as a course on mathematical epidemiology.

The biological content is self-contained and includes many topics in epidemiology and ecology. Some of this material appears in case studies that focus on a single detailed example, and some is based on recent research by the author on vaccination modeling and scenarios from the COVID-19 pandemic.

The problem sets feature linked problems where one biological setting appears in multi-step problems that are sorted into the appropriate section, allowing readers to gradually develop complete investigations of topics such as HIV immunology and harvesting of natural resources. Some problems use programs written by the author for Matlab or Octave; these combine with more traditional mathematical exercises to give students a full set of tools for model analysis. Each chapter contains additional case studies in the form of projects with detailed directions. New appendices contain mathematical details on optimization, numerical solution of differential equations, scaling, linearization, and sophisticated use of elementary algebra to simplify problems.

Inhaltsverzeichnis

Frontmatter
7. Correction to: Mathematical Modeling for Epidemiology and Ecology
Glenn Ledder

Mathematical Modeling

Frontmatter
Chapter 1. Modeling in Biology
Abstract
This chapter presents the most fundamental concepts of mathematical modeling. There is an initial section that introduces parameters, followed by a section that discusses biological data and a section on modeling concepts. The final two sections present an example of an agent-based model and the basic ideas of distributions of sample means.
Glenn Ledder
Chapter 2. Empirical Modeling
Abstract
Simulations require values for the model parameters, which raises the question of how parameter values should be determined. Occasionally, they can be measured directly, but more often they can only be inferred from their effects. This is done by collecting experimental data for the independent and dependent variables of a model and then using a mathematical procedure to determine the parameter values that give the best fit for the data. This is the subject of empirical modeling, which encompasses two questions.
Glenn Ledder
Chapter 3. Mechanistic Modeling
Abstract
This chapter complements the chapter on empirical modeling by providing a broad introduction to mechanistic modeling. The first two sections present models for transition and interaction processes. Section 3 presents the basic ideas of compartment analysis, illustrated by the SEIR epidemic model. The mathematical properties of the SEIR model are developed in Section 4. The next section presents some models for COVID-19 scenarios. Section 6 introduces scaling, and then the chapter concludes with three case studies, on lead poisoning, biochemical kinetics, and endemic disease models. Five projects allow students to investigate a variety of questions in epidemiology.
Glenn Ledder

Dynamical Systems

Frontmatter
Chapter 4. Dynamics of Single Populations
Abstract
In this chapter, we use ecological scenarios as settings in which to develop and study models for the change of populations over time. We restrict ourselves for now to models that require careful monitoring of only one population. There are two main categories of dynamic models, discrete and continuous, differing in the assumption made about how to mark time. Discrete dynamic models assume that time can be broken up into distinct uniform intervals. The length of the interval depends on the life history of the organism being modeled. Salmon have yearly spawning periods, so a time interval of 1 year is chosen for a discrete salmon model. Continuous dynamic models assume that time flows continuously from one moment to the next. The assumption of continuity in time is relative to the overall duration of the population. For example, it is common to ignore the diurnal variation of temperature and sunlight in a model that tracks a population of plants over a complete growing season.
Glenn Ledder
Chapter 5. Discrete Linear Systems
Abstract
In Chap. 4, we considered the dynamics of single quantities changing in either discrete or continuous time. Here we consider the dynamics of systems of several related quantities changing in discrete time. This chapter deals exclusively with linear systems, which are used to represent dynamics of structured populations divided into classes by age, size, or developmental stage. Discrete linear systems are a major tool in conservation biology modeling, where the primary goal is to determine the effects of parameters on the growth rate of a population. Nonlinear discrete systems appear in Sect. 6.​5. We begin in Sect. 5.1 with an introduction to the dynamics of structured populations using scalar notation. The models we obtain are analogous to exponential growth models for single quantities, except that the possibility of various distributions of population among the classes makes the exponential growth rate difficult to determine. However, we can demonstrate that such systems do eventually tend toward exponential growth (or decay); for problems with a limited number of classes, we can prescribe an intuitive method for determining both the eventual growth rate and the stable distribution of the population.
Glenn Ledder
Chapter 6. Nonlinear Dynamical Systems
Abstract
In Chap. 4, we studied the dynamics of a single variable. Now we look at the dynamics of nonlinear systems. The methods available to us are analogous to the methods we used for single-variable dynamics, but with some important differences. The phase line for one variable scales up to the phase plane for two variables; however, there is no graphical method for discrete systems. The analytical method for determining stability with the derivative scales up to higher dimensions, but with technical complications that rapidly increase as the number of variables increases. When possible, we can greatly simplify the analysis of a model by using an appropriate approximation to reduce the number of dynamic equations.
Glenn Ledder
Backmatter
Metadaten
Titel
Mathematical Modeling for Epidemiology and Ecology
verfasst von
Glenn Ledder
Copyright-Jahr
2023
Electronic ISBN
978-3-031-09454-5
Print ISBN
978-3-031-09453-8
DOI
https://doi.org/10.1007/978-3-031-09454-5

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