# Population Mathematics

According to the U.S. Census Bureau, the world's population reached 6 billion people on June 19, 1999. The Bureau operates a population "clock" that shows the population of the world changing second by second. How is this possible? How can anyone possibly know the exact population of the world at any given second? Of course, the answer is that no one can know the world population with the degree of accuracy implied by the population clock.

And yet accurate projections of future population growth are indispensable to world leaders who must plan and set policies for health care, education, housing, energy, transportation, food and water delivery, environmental protection, disaster preparedness, and national defense. The collection and continual updating of data from the nations of the world and the use of sophisticated mathematical models provide projections that are as accurate as is currently possible.

## Making Projections

The 6 billion mentioned in the opening paragraph is an estimate created by a mathematical model using data from every country in the world. The data used to make the current estimate are actually about 2 years old. So thePage 136 | Top of Article number on the clock is really a mathematical projection from the 2-year-old data. The following example is greatly oversimplified, but it illustrates the principle involved in such projections.

Suppose that you had high-quality data showing that the population of a certain city was one million at the beginning of the year 2000 and that this city's population had historically grown at about 2 percent per year. If you assume that there are no extraordinary events that would change this growth rate, you could calculate as follows. Two percent of 1 million is 20,000, so at the beginning of 2001, you would project the city's population to be 1,020,000. Then 2 percent of 1,020,000 is 20,400, which would give 1,040,400 at the beginning of 2002, and so on.

To make your projections as accurate as possible, you would want to continue to collect data about this city and watch for changes that might effect the growth rate. For example, suppose that the city's largest employer decided to move its business to another city, causing many people to lose their jobs and move to other locations to find work. This could negatively impact the city's growth rate and necessitate a modification in the mathematical model. The keepers of the world population clock must do a similar sort of continual updating of data and assumptions, only on a much larger scale.

**Types of Models.** The mathematical model used in the simple example above was an exponential model, one of the most basic models used in the study of population growth. Exponential functions can be effective models for populations that grow essentially unchecked in a straightforward way for relatively short periods of time. However, human population growth over time is constrained by environmental, social, political, and economic factors. To account for such limitations, more sophisticated mathematical models must be developed.

A fairly simple alternative to the exponential model is the logistic model, which projects short-term growth as exponential-like, but which includes a factor that models a slowing of the growth rate as the population moves closer to its upper limit, or carrying capacity. The logistic model can be effective with relatively isolated animal and human populations, but, by itself, it is not sophisticated enough to handle the kind of complex dynamics that occur in a system with a number of different populations, each requiring its own assumptions about its rate of growth.

This is the situation facing the scientists who make projections about the world population. Their models must be a great deal more complex to account for the multitude of variables that go into modeling the growth of the entire population of the world. In fact, such models are so complex that they take the form of sophisticated computer programs that essentially simulate the population growth for all of the countries of the world and then aggregate the results to get an estimate of the total population.

## Projecting Population Growth

The first step in making projections for population growth is to obtain a base population for each country. This is usually done by using data from the most recent census. It is not sufficient just to know how many people are in a country. The population must also be broken down by gender and age, since different genders and different age groups have varying life expectancies and varying rates of producing offspring. It is of utmost importance to make sure that the counting of young children is as accurate as possible, because a serious miscount of this age group will have a ripple effect over generations. The three major components that go into making projections from some base population are fertility, mortality, and international migration.

**Fertility.** Fertility refers to the frequency of childbearing of women in different age groups. This can vary widely from country to country or even within the same country. Some societies encourage childbearing for women in their teens, and others have strong proscriptions against it. Poorer nations tend not to provide easy access to birth-control measures, whereas wealthier nations do. Some countries, worried about the future consequences of ballooning populations, penalize parents for having too many children. Others may be dominated by religions that place a high value on childbearing and regard contraception as unacceptable. All of these factors must be taken into account when making estimates of the fertility rate of a given country.

**Mortality.** Mortality refers to the life-expectancy of a given age group. Here, again, this can vary dramatically in different countries. Wealthy nations tend to have good healthcare for their citizens, while developing nations do not. This results in lower infant mortality and longer life spans for the citizens of the economically developed countries. Lack of research and development of new drugs and inadequate levels of education for medical personnel in the developing nations lead to higher susceptibility to diseases and epidemics among the populations of those countries.

An example of this is the devastation being inflicted on certain African nations by the AIDS epidemic. If left unchecked, this will have serious repercussionsPage 138 | Top of Article on the populations of those countries for generations to come. The U.S. Census Bureau maintains an HIV/AIDS Surveillance Database which contains data for each country giving the percent of that country's population who are HIV positive.

These data are collected from more than 3,800 scientific publications on HIV/AIDS. The database is updated twice a year and provides the criteria for selecting countries whose population models will include an AIDS mortality component for use in projections. As of spring, 2001, twenty-six countries have such AIDS mortality components in their projections. Twenty-one of these countries are in Africa.

For each of these countries a special mathematical modeling tool is used to project the effect of a high impact, medium impact, and low impact AIDS epidemic on future population growth. This model uses nonlinear partial differential equations to simulate the epidemics. The three scenarios generated by the model are then combined with actual data points to interpolate a "most likely" projection of the effect of HIV/AIDS on the population growth.

For many of these African nations, it is estimated that the AIDS epidemic will peak around 2010, and that population projections for these countries will not return to their pre-AIDS normalcy until near the middle of the twenty-first century.

**International Migration.** The third major component in projecting population growth is international migration. This is the component that is subject to the highest relative error in the modeling process. The factors that influence migration among countries include changing economic conditions, political unrest, natural disasters, and other extreme and unfavorable conditions in the original homeland. These types of events tend to be highly unpredictable, making the modeler's development of accurate assumptions difficult.

Fortunately, while migration projections are the least accurate of the three components of population growth, the effect of such errors is generally much less than for errors in fertility or mortality. In the future, however, migration may become the deciding factor in whether populations in the more developed nations continue to grow or begin to decline. This is because fertility rates in these countries have been declining since the seventies, whereas in the less developed nations fertility rates have remained steady or have declined at a much slower rate. Ninety-six percent of the world's population increase now occurs in the less developed regions of Africa, Asia, and Latin America. By the year 2020, according to projections, it will be virtually 100 percent.

**SEE ALSO** CENSUS
; COMPUTER SIMULATIONS
; PREDICTIONS
.

*Stephen Robinson*

## Bibliography

McDevitt, Thomas M. *World Population Profile—1998.* Washington, D.C.: U.S. Agency for International Development-Bureau for Global Programs, Field Support, and Research, Office of Population. U.S. Department of Commerce-Economics and Statistics Administration, Bureau of the Census, 1999.

Hollmann, Frederick W., Tammany J. Mulder, Jeffrey E. Kallan. *Methodology and Assumptions for the Population Projections of the United States: 1999 to 2100.* 13 January, 2000. U.S. Census Bureau, Population Division, Population Projections Branch.

### Internet Resources

*World POPClock Projection.* March, 2001. U.S. Census Bureau. <http://www.census.gov/cgi-bin/ipc/popclockw
>

### Source Citation

**Gale Document Number: **
GALE|CX3407500235