Society for Industrial and Applied Mathematics, 2005, -192 pp.
For thousands of years humans have been occupied, and even preoccupied, with counting. As the species evolved, an interest grew in knowing and forecasting the size of various populations, such as animal herds that humans hunted for their food and clothing or crops their diet depended on.
The first mathematical model for populations is attributed to Leonardo Pisano, better known as Fibonacci, who described in a publication dated 1208 the cumulative size of a population of rabbits after n successive generations through his celebrated sequence
s0=1, s1=1, sn+1=sn+sn-1 (n?1).
Among many interesting properties the sequence has, it is nice to note that the ratio of consecutive terms approaches the golden ratio (1+sqrt(5))/2?
1.618. This leads to an exponential growth of the population at an asymptotic rate of 1.618 per generation.
Most of the models proposed since then are conceed with a population that may be unstructured, or may be structured according to one or more important features such as age, sex, race, size, and economic status.
For human populations it is quite useful to have models that are structured at least by sex and age, because many health care, education, and social security issues, for example, depend on the sex and age structure of the population. Yet few studies of the mathematical properties of such models have been done and, in particular, a major gap still exists in the study and modeling of marriage functions.
Among all the processes intervening in the dynamics of human populations, perhaps migration is the least understood and most difficult to model. In this book we shall ignore it altogether and assume that the population studied is closed in the sense that individuals arrive into it only by birth and leave it only by death. Processes related to births and deaths are among those we can understand and model with the greatest ease and, when reliable data are available, the models lead to fairly accurate predictions of the evolution of the population—even when modeled Unearly. In terms of a sex-structured model, this means that the equations used for the dynamics of the age structure of females and males can be deterministic, and the error stemming from real-life randomness can be neglected.
Equations that model couples' dynamics become necessary in order to have some closed form for the birth functions of females and males. Ignoring couples and applying linear extrapolation of known birth data leads to significant errors by the tenth year—when a new census is taken and new data are thus acquired to compare prediction and reality. This creates a need for modeling couples separately, allowing the nearly linear birth processes to be described in terms of the age distribution of couples and their fertility, thus resulting in more accurate long-term projections.
If we choose to model couples using differential equations, these must include terms corresponding to divorces, separations, and marriages. The dynamics of divorces and separations is fairly well understood and is actually modeled very similarly to the death process, linearly, without the introduction of large errors. Marriages are much more complex. Assuming that they are constant leads to very poor estimation of births from married couples and, therefore, should be avoided. In fact it is always expected that the mating process is nonlinear in terms of available "singles."
In this book we concentrate our attention on deterministic models for monogamous populations, in which the past history of marriages and divorces plays no role in future behavior with respect to these processes.
The problem of existence of a marriage function and its form began being widely discussed sometime in the early 1970s. It was interesting both from the point of view of the two-sex model, because it was expected to help make sense of male and female marriage rates, and for the possibility of forecasting marriages, for different purposes— mainly business related. As demographers define it, a marriage function is a function for predicting the number of marriages that will occur during a unit of time between males in particular age categories and females in particular age categories, from knowledge of the numbers of available singles in the various categories.
Marriage is a complex socioeconomic process influenced by many factors, just some of which are dominating perceptions and rituals, religion, health, economy, educational status, racial and ethnic interactions, and age and sex composition. As can be seen from the definition of the marriage function, it is assumed that age and sex composition are the only essential properties that should be explicitly taken into account in modeling marriages.
A general perception in demography is that the marriage system represents a market and is ruled by laws of competition. At a personal level marriage is an act expected to bring more comfort in life—health and/or premarital childbearing are some of the factors that can decrease a person's chance of getting married.
Comparative studies relative to actual population data involve few of the known candidates for marriage functions and we frequently find a weighted harmonic mean as the function of choice in mathematical models, even though demographers know that it is not a good choice.
We try to present in this book a brief historical perspective of deterministic modeling of human populations and then focus on pair formation (marriage) and two-sex models. We describe several models, derive theoretical results that show these are well posed, and try to elucidate which marriage function might make a better choice for a particular population—in our example, that of the United States. We describe numerical methods to approximate the solution of the differential models, which is equivalent to creating discrete simulators.
We present comparisons of simulation results with actual demographic data in the hope that they will help the believer better understand some of the difficulties conceing the availability of data and demographic modeling, and that they might convince the skeptic that mathematical demography does provide reasonable qualitative and quantitative estimates.
Historical Perspective of Mathematical Demography
Gender Structure and the Problem of Modeling Marriages
Well-Posedness of the Fredrickson-Hoppensteadt Two-Sex Model
Numerical Methods
Age Profiles and Exponential Growth
The Main Algorithm
For thousands of years humans have been occupied, and even preoccupied, with counting. As the species evolved, an interest grew in knowing and forecasting the size of various populations, such as animal herds that humans hunted for their food and clothing or crops their diet depended on.
The first mathematical model for populations is attributed to Leonardo Pisano, better known as Fibonacci, who described in a publication dated 1208 the cumulative size of a population of rabbits after n successive generations through his celebrated sequence
s0=1, s1=1, sn+1=sn+sn-1 (n?1).
Among many interesting properties the sequence has, it is nice to note that the ratio of consecutive terms approaches the golden ratio (1+sqrt(5))/2?
1.618. This leads to an exponential growth of the population at an asymptotic rate of 1.618 per generation.
Most of the models proposed since then are conceed with a population that may be unstructured, or may be structured according to one or more important features such as age, sex, race, size, and economic status.
For human populations it is quite useful to have models that are structured at least by sex and age, because many health care, education, and social security issues, for example, depend on the sex and age structure of the population. Yet few studies of the mathematical properties of such models have been done and, in particular, a major gap still exists in the study and modeling of marriage functions.
Among all the processes intervening in the dynamics of human populations, perhaps migration is the least understood and most difficult to model. In this book we shall ignore it altogether and assume that the population studied is closed in the sense that individuals arrive into it only by birth and leave it only by death. Processes related to births and deaths are among those we can understand and model with the greatest ease and, when reliable data are available, the models lead to fairly accurate predictions of the evolution of the population—even when modeled Unearly. In terms of a sex-structured model, this means that the equations used for the dynamics of the age structure of females and males can be deterministic, and the error stemming from real-life randomness can be neglected.
Equations that model couples' dynamics become necessary in order to have some closed form for the birth functions of females and males. Ignoring couples and applying linear extrapolation of known birth data leads to significant errors by the tenth year—when a new census is taken and new data are thus acquired to compare prediction and reality. This creates a need for modeling couples separately, allowing the nearly linear birth processes to be described in terms of the age distribution of couples and their fertility, thus resulting in more accurate long-term projections.
If we choose to model couples using differential equations, these must include terms corresponding to divorces, separations, and marriages. The dynamics of divorces and separations is fairly well understood and is actually modeled very similarly to the death process, linearly, without the introduction of large errors. Marriages are much more complex. Assuming that they are constant leads to very poor estimation of births from married couples and, therefore, should be avoided. In fact it is always expected that the mating process is nonlinear in terms of available "singles."
In this book we concentrate our attention on deterministic models for monogamous populations, in which the past history of marriages and divorces plays no role in future behavior with respect to these processes.
The problem of existence of a marriage function and its form began being widely discussed sometime in the early 1970s. It was interesting both from the point of view of the two-sex model, because it was expected to help make sense of male and female marriage rates, and for the possibility of forecasting marriages, for different purposes— mainly business related. As demographers define it, a marriage function is a function for predicting the number of marriages that will occur during a unit of time between males in particular age categories and females in particular age categories, from knowledge of the numbers of available singles in the various categories.
Marriage is a complex socioeconomic process influenced by many factors, just some of which are dominating perceptions and rituals, religion, health, economy, educational status, racial and ethnic interactions, and age and sex composition. As can be seen from the definition of the marriage function, it is assumed that age and sex composition are the only essential properties that should be explicitly taken into account in modeling marriages.
A general perception in demography is that the marriage system represents a market and is ruled by laws of competition. At a personal level marriage is an act expected to bring more comfort in life—health and/or premarital childbearing are some of the factors that can decrease a person's chance of getting married.
Comparative studies relative to actual population data involve few of the known candidates for marriage functions and we frequently find a weighted harmonic mean as the function of choice in mathematical models, even though demographers know that it is not a good choice.
We try to present in this book a brief historical perspective of deterministic modeling of human populations and then focus on pair formation (marriage) and two-sex models. We describe several models, derive theoretical results that show these are well posed, and try to elucidate which marriage function might make a better choice for a particular population—in our example, that of the United States. We describe numerical methods to approximate the solution of the differential models, which is equivalent to creating discrete simulators.
We present comparisons of simulation results with actual demographic data in the hope that they will help the believer better understand some of the difficulties conceing the availability of data and demographic modeling, and that they might convince the skeptic that mathematical demography does provide reasonable qualitative and quantitative estimates.
Historical Perspective of Mathematical Demography
Gender Structure and the Problem of Modeling Marriages
Well-Posedness of the Fredrickson-Hoppensteadt Two-Sex Model
Numerical Methods
Age Profiles and Exponential Growth
The Main Algorithm