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## Über dieses Buch

Mathematical Demography, the study of population and its analysis through mathematical models, has received increased interest in the mathematical com­ munity in recent years. It was not until the twentieth century, however, that the study of population, predominantly human population, achieved its math­ ematical character. The subject of mathematical demography can be viewed from either a deterministic viewpoint or from a stochastic viewpoint. For the sake of brevity, stochastic models are not included in this work. It is, therefore, my intention to consider only established deterministic models in this discussion, starting with the life table as the earliest model, to a generalized matrix model which is developed in this treatise. These deterministic models provide sufficient de­ velopment and conclusions to formulate sound mathematical population analy­ sis and estimates of population projections. It should be noted that although the subject of mathematical demography focuses on human populations, the development and results may be applied to any population as long as the preconditions that make the model valid are maintained. Information concerning mathematical demography is at best fragmented.

## Inhaltsverzeichnis

### 1. The Development of Mathematical Demography

Abstract
Demography is the study of population, primarily human population, in terms of its growth and decay, fertility and mortality, and its relative mobility, together with its impact on the economic, political and sociological components of society. Interest in this subject can be traced back to ancient times. Oriental legends and biblical references indicate that an enumeration or census of a population by age and by locality was not uncommon. This was done primarily for military records and manpower, as well as for taxation. Governments also used such information for administrative aids and to establish the socio-economic character of its people.
John Impagliazzo

### 2. An Overview of the Stable Theory of Population

Abstract
At the turn of the twentieth century the life table was the dominant model that was used to represent a population. This model, however, was and still is only a picture of mortality of a given population at a given time. The life table does not take into account fertility, and as a result, is not an effective tool that can be used to thoroughly analyze and project a population. The mathematical attempts that had been made to project populations, such as the exponential and logistic curves, represented simple models based on conjectures concerning the behavior of a population.
John Impagliazzo

### 3. The Discrete Time Recurrence Model

Abstract
As previously mentioned, the discrete time recurrence model in the study of population is credited to Euler, and has received sporadic and incomplete attention since his time. Indeed, the only development of this topic known to the author is an unpublished work by Meyer [1]. It is, therefore, in the best interest of the field of mathematical demography that this topic receive further exposure.
John Impagliazzo

### 4. The Continuous Time Model

Abstract
In Chapters Two and Three, attention was given to the female population stratified in a finite number of w + 1 age intervals that were indexed by i for 0≤;i≤w. In addition, the female births were considered on discrete time intervals of the form (t — h, i]. This led to the discrete time population renewal equation (2.23) of the form
$$\text{B}_h \text{(}t\text{) = }\sum\limits_{i\text{ = 0}}^w {\phi _{i\text{ + 1}} } \text{B}_h \text{(t - (}i\text{ + 1)}h\text{)}h$$
(4.1)
and its attendant solution.
John Impagliazzo

### 5. The Discrete Time Matrix Model

Abstract
The discrete time recurrence model and the continuous time model which were discussed in Chapters Three and Four focused on the births that occurred at a particular time and their relationship to the births that occurred prior to this time. In each case the age distribution of the population was discussed almost as an adjunct to the model and not as an integral part of it. Perhaps this was unfortunate since it was the asymptotic age distribution of the model that gave meaning to the concept of a ‘stable’ population.
John Impagliazzo

### 6. Comparative Aspects of Stable Population Models

Abstract
In Chapter Two the fundamental premises of stable population theory, as well as the other foundations for respective stable models, were introduced. The models were of two basic types: discrete time models, and the continuous model. The two discrete time models illustrated were the recursive or finite difference model as discussed in Chapter Three, and the matrix model as discussed in Chapter Five. The continuous model was discussed in Chapter Four, and an alternate approach to this model is given as an appendix. It is now the intent to consolidate these models by highlighting their similarities and their differences.
John Impagliazzo

### 7. Extensions of Stable Population Theory

Abstract
The developments and results of the previous chapters can be summarized by the following statement: Given a population
(a)
that is closed to migration,

(b)
that has time invariant age-specific birth rates, and

(c)
that has time invariant age-specific death rates.

John Impagliazzo

### 8. The Kingdom of Denmark — A Demographic Example

Abstract
The three premises of classical stable theory of population (closure to migration, time independent age-specific birth and death rates), if staunchly upheld, would eliminate virtually every existing biological population as a subject for its use. One must concede, for example, that human population is exposed to migration, and that birth rates and death rates are not constant over time. In this light, one may erroneously conclude that stable theory, even in its classical sense, serves little or no purpose. This is far from true. It must be remembered that stable theory serves as a model for study of a given population, and not as an impeccable image of it. Viewed in this manner, the concepts and formulations of stable theory do have much to offer, both as a mathematical entity as well as an applicable tool to be used in population study.
John Impagliazzo

### Backmatter

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