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2012 | Buch

Continuous Population Models Kapitel lesen Erstes Kapitel lesen

Mathematical Models in Population Biology and Epidemiology

verfasst von: Fred Brauer, Carlos Castillo-Chavez

Verlag: Springer New York

Buchreihe : Texts in Applied Mathematics

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

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Über dieses Buch

This textbook provides an introduction to the field of mathematical biology through the integration of classical applications in ecology with more recent applications to epidemiology, particularly in the context of spread of infectious diseases. It integrates modeling, mathematics, and applications in a semi-rigorous way, stating theoretical results and giving references but not necessarily giving detailed proofs, providing a solid introduction to the field to undergraduates (junior and senior level), graduate students in applied mathematics, ecology, epidemiology or evolutionary biology, sustainability scientists, and to researchers who must routinely read the practical and theoretical results that come from modeling in ecology and epidemiology.

This new edition has been updated throughout. In particular the chapters on epidemiology have been updated and extended considerably, and there is a new chapter on spatially structured populations that incorporates dispersal. The number of problems has been increased and the number of projects has more than doubled, in particular those stressing connections to data. In addition some examples, exercises, and projects include use of Maple and Matlab.

Review of first edition:

"A strength of the book is the large number of biologically-motivated problem sets. These and the references to the original biological papers would be valuable resources for an instructor." (UK Nonlinear News, 2001)

Inhaltsverzeichnis

Frontmatter

Simple Single Species Models

Frontmatter
Chapter 1. Continuous Population Models
Abstract
In this chapter we look at a population in which all individuals develop independently of one another while living in an unrestricted environment where no form of competition is possible. If the initial population size is small then a stochastic model is more appropriate, since the likelihood that the population becomes extinct due to chance must be considered. Deterministic models often provide useful ways of gaining sufficient understanding about the dynamics of populations whenever they are large enough. Furthermore, perturbations to large populations at equilibrium often generate over short time scales independent individual responses, which may be appropriately modeled by deterministic models. For example, the introduction of a single infected individual into a large disease-free population leads to the generation of secondary cases of infection, propagating a disease. The environment is free of interference competition, at least at the beginning of the outbreak, when a large population of susceptibles provides a virtually unlimited supply of hosts. The spread of disease in a large population of susceptibles may be thought of as an invasion process generated by independent contacts between a huge pool of susceptibles and a few infectious individuals.
Fred Brauer, Carlos Castillo-Chavez
Chapter 2. Discrete Population Models
Abstract
In this chapter we shall consider populations with a fixed interval between generations or possibly a fixed interval between measurements. Thus, we shall describe population size by a sequence {x n }, with x 0 denoting the initial population size, x 1 the population size at the next generation (at time t 1), x 2 the population size at the second generation (at time t 2), and so on. The underlying assumption will always be that population size at each stage is determined by the population sizes in past generations, but that intermediate population sizes between generations are not needed. Usually the time interval between generations is taken to be a constant.
Fred Brauer, Carlos Castillo-Chavez
Chapter 3. Continuous Single-Species Population Models with Delays
Abstract
Up to now in our study of continuous population models we have been assuming that x′(t), the growth rate of population size at time t, depends only on x(t), the population size at the same time t. However, there are situations in which the growth rate does not respond instantaneously to changes in population size. One of the first models incorporating a delay was proposed by Volterra (1926) to take into account the delay in response of a population’s death rate to changes in population density caused by an accumulation of pollutants in the past.
Fred Brauer, Carlos Castillo-Chavez

Models for Interacting Species

Frontmatter
Chapter 4. Introduction and Mathematical Preliminaries
Abstract
In the 1920’s Vito Volterra was asked whether it would be possible to explain the fluctuations that had been observed in the fish population of the Adriatic sea–fluctuations that were of great concern to fishermen in times of low fish populations. Volterra (1926) constructed the model that has become known as the Lotka-Volterra model (because A.J. Lotka (1925) constructed a similar model in a different context about the same time), based on the assumptions that fish and sharks were in a predator–prey relationship.
Fred Brauer, Carlos Castillo-Chavez
Chapter 5. Continuous Models for Two Interacting Populations
Abstract
In this chapter we will consider populations of two interacting species with population sizes x(t) and y(t), respectively, modeled by a system of two first order differential equations:
$$\begin{array}{cc}{x^{\prime}=F(x, y),} \\ {y^{\prime}=G(x, y).}\end{array}$$
Fred Brauer, Carlos Castillo-Chavez
Chapter 6. Harvesting in Two-species Models
Abstract
The topics in this chapter are part of the subject of natural resource management and bioeconomics. This is an important subject that is developing rapidly. The classical reference is the book by Clark (1990), where additional references may be found.
Fred Brauer, Carlos Castillo-Chavez

Structured Population Models

Frontmatter
Chapter 7. Models for Populations with Age Structure
Abstract
In the preceding chapters we studied mainly models in which all members were alike, so that birth and death rates depended on total population size. However, we gave a few examples of populations with two classes of members and a birth rate that depended on the size of only one of the two classes, for discrete models in Section 2.6 and for continuous models in Section 3.3. These are examples of structured populations. In this chapter we shall study models for populations structured by age. In practice, animal populations are often measured by size with age structure used as an approximation to size structure. The study of age-structured models is considerably simpler than the study of general size-structured models, primarily because age increases linearly with the passage of time while the linkage of size with time may be less predictable. Age-structured models may be either discrete or continuous. We begin with linear models, for which total population size generally either increases or decreases exponentially over time.
Fred Brauer, Carlos Castillo-Chavez
Chapter 8. Models for Populations with Spatial Structure
Abstract
Populations may be structured by spatial location. There are two common different ways to include spatial location in a population. One way is by means of metapopulations, that is, populations of populations, with links between them such as a collection of towns and cities connected by a transportation network. The air transport subnetwork includes connecting links between distant communities, and we may study the dynamics of populations of different cities as a function of the flow of people between them and their own local dynamics in this framework. A metapopulation may be divided into patches, with each patch corresponding to a separate location. The corresponding models may be systems of ordinary differential equations, with the population size of each species in each patch as a variable. Thus metapopulation models are often systems of ordinary differential equations of high dimension. Some basic references are Hanski (1999), Hanski and Gilpin (1997), Levin, Powell and Steele (1993), Neuhauser (2001).
Fred Brauer, Carlos Castillo-Chavez

Disease Transmission Models

Frontmatter
Chapter 9. Epidemic Models
Abstract
Communicable diseases such as measles, influenza, and tuberculosis are a fact of life. We will be concerned with both epidemics, which are sudden outbreaks of a disease, and endemic situations, in which a disease is always present. The AIDS epidemic, the recent SARS epidemic, recurring influenza pandemics, and outbursts of diseases such as the Ebola virus are events of concern and interest to many people. The prevalence and effects of many diseases in less-developed countries are probably not as well known but may be of even more importance. Every year millions, of people die of measles, respiratory infections, diarrhea, and other diseases that are easily treated and not considered dangerous in the Western world. Diseases such as malaria, typhus, cholera, schistosomiasis, and sleeping sickness are endemic in many parts of the world. The effects of high disease mortality on mean life span and of disease debilitation and mortality on the economy in afflicted countries are considerable.
Fred Brauer, Carlos Castillo-Chavez
Chapter 10. Models for Endemic Diseases
Abstract
We have been studying SIR models, in which the transitions are from susceptible to infective to removed, with the removal coming through recovery with full immunity (as in measles) or through death from the disease (as in plague, rabies, and many other animal diseases). Another type of model is an SIS model in which infectives return to the susceptible class on recovery because the disease confers no immunity against reinfection. Such models are appropriate for most diseases transmitted by bacterial or helminth agents, and most sexually transmitted diseases (including gonorrhea, but not such diseases as AIDS, from which there is no recovery). One important way in which SIS models differ from SIR models is that in the former there is a continuing flow of new susceptibles, namely recovered infectives. Later in this chapter we will study models that include demographic effects, namely births and deaths, another way in which a continuing flow of new susceptibles may arise.
Fred Brauer, Carlos Castillo-Chavez
Backmatter
Metadaten
Titel
Mathematical Models in Population Biology and Epidemiology
verfasst von
Fred Brauer
Carlos Castillo-Chavez
Copyright-Jahr
2012
Verlag
Springer New York
Electronic ISBN
978-1-4614-1686-9
Print ISBN
978-1-4614-1685-2
DOI
https://doi.org/10.1007/978-1-4614-1686-9

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