Society for Industrial and Applied Mathematics, 1975, -85 pp.
Mathematical theories of populations have been derived and effectively used in many contexts in the last two hundred years. They have appeared both implicitly and explicitly in many important studies of populations: human populations as well as populations of animals, cells and viruses.
Several features of populations can be analyzed. First, growth and age structure can be studied by considering birth and death forces acting to change them. In particular, the long time state of the population and its sensitivity to changes in birth and death schedules can be determined.
Another population phenomenon which is amenable to analysis is the way various individual traits are propagated from one generation to the next. An extensive theory for this has evolved from the first observations of inheritance by Gregor Mendel.
In addition, the spread of a contagious phenomenon, such as disease, rumor, fad or information, can be studied by means of mathematical analysis. Of particular importance in this area are studies of the dependence of contagion on parameters such as contact and quarantine rates.
Finally, the dynamics of several interacting populations can be analyzed. Theories of interaction have become useful with recent studies of ecological systems and economic and social structures.
The most direct approach to the study of populations is the collection and analysis of data. However, serious questions arise about how this should be done. For one thing, facilities can be swamped by even simple manipulation and analysis of vast amounts of data. In addition, there are the questions: Which data should be collected? Which data adequately describe the phenomenon, in particular which are sensitive indicators for detecting the presence of a phenomenon?
A study of the population's underlying structure is essential for answering these questions, and mathematical theories provide a systematic way for doing this. Many techniques for analyzing complicated physical problems can be applied to population problems. In addition, many new techniques peculiar to these problems must be developed. Several population problems will be analyzed here which illustrate these methods and techniques.
The monograph begins with a study of population age structure. A basic model is derived first, and it reappears frequently throughout the remainder. Various extensions and modifications of the basic model are then applied to several population phenomena, such as stable age distributions, self-limiting effects and two-sex populations.
The second part is devoted to population genetics, and it contains a summary of some of the most successful applications of mathematics in the biological sciences. Attention is focused on the derivation and analysis of a model for a one locus, two-allele trait in a large randomly mating population. Then extensions of the system are considered which account for more complicated social structure (assortative mating and migration) and for age structure. This part ends with a description of Fisher's model for the propagation of a gene in a spatially distributed population, and stable gene waves are shown to exist. A reason for the success of mathematical theories in genetics has been the wealth of precise data which can be collected. Unfortunately, this is not the case in the topics discuss in Parts I and III.
The final part, Part III, is conceed with the dynamics of contagious phenomena in a population. These are studied in the context of epidemic diseases, but the same methods can be used to describe other phenomena such as rumors, fads and information as well as models for two interacting systems. Several classic examples are discussed first, then a general age dependent theory is formulated. However, the emphasis in Part III is placed on studies of qualitative properties of several typical models. First, a threshold theorem is derived for an age dependent epidemic, and then the long time behavior of solutions to a relapse-recovery model is determined. Finally, models for the spatial spread of contagion are derived and extensively discussed.
Ecological systems and other complicated interacting population phenomena are not discussed in the monograph. This is primarily because the scope of these applications is too broad for the present study. Therefore, the only work mentioned in this direction is that relating to the interaction of two populations, such as in the Volterra-Lotka theory, and this only because it is equivalent to a susceptible infective interaction model.
All of the theories discussed in this monograph are deterministic. This restriction has been made so that a much broader range of population phenomena can be discussed than only those for which stochastic theories (i.e., theories which account for random fluctuations in population size, parameters, etc.) have been derived and studied. Many interesting and important studies have been made with the stochastic analogues to some of the models developed here. While these are not reported here, references are frequently made to appropriate literature. An introduction to these studies and recent work is given by Ludwig (1974) (see Part III).
The Equations of Population Dynamics
Deterministic Models in Genetics
Theories of Epidemics
Mathematical theories of populations have been derived and effectively used in many contexts in the last two hundred years. They have appeared both implicitly and explicitly in many important studies of populations: human populations as well as populations of animals, cells and viruses.
Several features of populations can be analyzed. First, growth and age structure can be studied by considering birth and death forces acting to change them. In particular, the long time state of the population and its sensitivity to changes in birth and death schedules can be determined.
Another population phenomenon which is amenable to analysis is the way various individual traits are propagated from one generation to the next. An extensive theory for this has evolved from the first observations of inheritance by Gregor Mendel.
In addition, the spread of a contagious phenomenon, such as disease, rumor, fad or information, can be studied by means of mathematical analysis. Of particular importance in this area are studies of the dependence of contagion on parameters such as contact and quarantine rates.
Finally, the dynamics of several interacting populations can be analyzed. Theories of interaction have become useful with recent studies of ecological systems and economic and social structures.
The most direct approach to the study of populations is the collection and analysis of data. However, serious questions arise about how this should be done. For one thing, facilities can be swamped by even simple manipulation and analysis of vast amounts of data. In addition, there are the questions: Which data should be collected? Which data adequately describe the phenomenon, in particular which are sensitive indicators for detecting the presence of a phenomenon?
A study of the population's underlying structure is essential for answering these questions, and mathematical theories provide a systematic way for doing this. Many techniques for analyzing complicated physical problems can be applied to population problems. In addition, many new techniques peculiar to these problems must be developed. Several population problems will be analyzed here which illustrate these methods and techniques.
The monograph begins with a study of population age structure. A basic model is derived first, and it reappears frequently throughout the remainder. Various extensions and modifications of the basic model are then applied to several population phenomena, such as stable age distributions, self-limiting effects and two-sex populations.
The second part is devoted to population genetics, and it contains a summary of some of the most successful applications of mathematics in the biological sciences. Attention is focused on the derivation and analysis of a model for a one locus, two-allele trait in a large randomly mating population. Then extensions of the system are considered which account for more complicated social structure (assortative mating and migration) and for age structure. This part ends with a description of Fisher's model for the propagation of a gene in a spatially distributed population, and stable gene waves are shown to exist. A reason for the success of mathematical theories in genetics has been the wealth of precise data which can be collected. Unfortunately, this is not the case in the topics discuss in Parts I and III.
The final part, Part III, is conceed with the dynamics of contagious phenomena in a population. These are studied in the context of epidemic diseases, but the same methods can be used to describe other phenomena such as rumors, fads and information as well as models for two interacting systems. Several classic examples are discussed first, then a general age dependent theory is formulated. However, the emphasis in Part III is placed on studies of qualitative properties of several typical models. First, a threshold theorem is derived for an age dependent epidemic, and then the long time behavior of solutions to a relapse-recovery model is determined. Finally, models for the spatial spread of contagion are derived and extensively discussed.
Ecological systems and other complicated interacting population phenomena are not discussed in the monograph. This is primarily because the scope of these applications is too broad for the present study. Therefore, the only work mentioned in this direction is that relating to the interaction of two populations, such as in the Volterra-Lotka theory, and this only because it is equivalent to a susceptible infective interaction model.
All of the theories discussed in this monograph are deterministic. This restriction has been made so that a much broader range of population phenomena can be discussed than only those for which stochastic theories (i.e., theories which account for random fluctuations in population size, parameters, etc.) have been derived and studied. Many interesting and important studies have been made with the stochastic analogues to some of the models developed here. While these are not reported here, references are frequently made to appropriate literature. An introduction to these studies and recent work is given by Ludwig (1974) (see Part III).
The Equations of Population Dynamics
Deterministic Models in Genetics
Theories of Epidemics