Age-Structured Population Dynamics in Demography and Epidemiology PDF
5.63 MB PDF
Preface
The main purpose of this book was to explain deterministic, age-structured population dynamics models that appear in demography and epidemiology. The first part of this book addresses pure demographic models, while the second part considers epidemic models for infectious diseases.
It should be of essential importance to human beings to know the laws governing their own reproduction and associated results. In fact, the major problems with which we have been confronted relate to population dynamics, as was clearly recognized by Thomas Robert Malthus in the eighteenth century. From the beginning, the main concerns of demography have extended beyond the variations in population size and distribution to the age structure dynamics of the population. The starting point of modern demography was the life table technique developed to measure human life expectancy in the seventeenth century. In the eighteenth century, Euler developed a difference equation model for human populations to show that an age-structured population with constant fertility and mortality will grow geometrically. Moreover, Euler derived relations among demographic indices under this geometrical growth and suggested that these relations could be used to estimate incomplete data. Some 150 years later, Euler’s idea was rediscovered by Sharpe and Lotka at 1911, and modern demography was born. As the human vital rates are under conscious control and strongly depend on historical, social, and economic environmental variations, mathematical studies of human populations have developed on the boundary between the social sciences and life sciences under problematic concerns that differ from those of population biology. It should be, however, noted that the basic mathematical tools used in demography (such as life tables, renewal equations, and the basic reproduction number) are universally applicable to the description of any self-renewing aggregates.
In Chap. 1, we introduce the most basic age-dependent population model, called the stable population model. We then explain many resulting facets of the Fundamental Theorem of Demography, because the stable population model is a central tenet in the development of modern demography. Indeed, its revision using functional analysis was the prelude to new, more general developments for structured population dynamics from the end of the 1970s. A crucial point of demographic applications is that the basic demographic indices cannot be interpreted without the stable population model. Although the operator semigroup theory is a powerful tool for studying structured population models, we here mainly adopt the classical integral equation approach, because the scalar integral equation is more elementary and is a most natural expression for the self-reproduction process of any population. Through the renewal equation, we can define the basic reproduction number, which enables the essential relation between individual vital rates and the Malthusian parameter to be established at the population level. The semigroup approach is, however, briefly introduced in Chap. 10.
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