Fertility and mathematics

Summary: Individual fertility cycles can be mathematically predicted and national fertility rates are a useful statistical measure for analyzing population demographics.

The term “fertility” has been used historically in a variety of contexts, including the richness of croplands with respect to producing food, the creativity of the human mind and imagination, and the ability of people to have children. The term “fecundity” is often interchanged with fertility when discussing human reproduction. However, nineteenth-century physician Matthews Duncan, who researched birth statistics and fertility, differentiated the two terms by defining fecundity in an essentially binary fashion as the capability of bearing children or not, versus fertility, which he used to quantify the number of children a woman had borne. Demographers often use fertility rate as a standardized metric to describe the number of children borne per person, couple, or population and to make comparisons across populations. Many collections of global statistics, like the CIA World Factbook, include fertility rates, which have been connected by mathematical and statistical models to economic measures such as individual income or a country’s gross domestic product. Others study relationships to medical and social variables, such as the availability of birth control and assisted reproduction or attitudes about single parenting. Some rates adjust for women in specific age groups or other variables. At the start of the twenty-first century, organizations such as the United Nations also began to turn serious attention to the issue of population decline in many nations and its potential effects on national economies, workforces, and social security systems. Mathematicians, statisticians, demographers, and others continue to research the reciprocal relationships between fertility and other measures to attempt to determine causes and effects and to forecast future trends as well as to contribute to the development of technologies related to fertility and reproduction. Statistician Leslie Kish was awarded the American Statistical Association’s Samuel S. Wilks Award for his work on the World Fertility Survey, which “illustrates his impact as an international ambassador of statistics and a tireless advocate for scientific statistical methods.”

Fertility Rates

In the years immediately following World War II, many countries, especially the United States, Canada, Australia, and New Zealand, saw a marked increase in the number of babies borne. This “baby boom” generation has been widely researched and continues to have an impact on society and social policy. There are many ways to quantify fertility. For example, birth rate is typically the number of live births per thousand people per year for a given population. The total fertility rate of a population is an estimated measure based on observed age-related fertility rates during a given time period and assuming a woman lives throughout her entire likely reproductive span, or roughly to age 50. It is intended to represent the average number of live births per woman in a given population. However, since human reproduction requires genetic contributions from both males and females and social conventions typically restrict who may reproduce with whom, the male-female ratios in populations can affect actual fertility. Net reproduction rate quantifies the number of daughters borne to a woman, using statistical estimation methods similar to the total fertility rate. This statistic is often used in researching countries that exhibit strong preferences for one sex of child over another or that practice sex selection. Some other possible estimates include gross fertility rate, generational or cohort fertility rate, or completed family size.

In 2010, Russian president Vladimir Putin publicly addressed the growing concern of Russia’s declining population, which he attributed to both declining fertility and high death rates, calling it “the most acute problem of contemporary Russia.” Sub-replacement fertility rate is a threshold value of the total fertility rate where the number of births is not large enough to replace or maintain a given population at its current level. In theory, each couple must produce two children to replace themselves or, referring to net reproduction rate, each woman must have one daughter to replace herself. In reality, not all people pair and reproduce and early mortality and other factors affect population sizes. Mathematical and statistical models have been used to model average behavior and account for such variables. In the early twenty-first century, the global replacement fertility rate was about 2.3 children per woman: the theoretical value of two, plus a fractional value that adjusts for mortality and other factors. Anything below this value is sub-replacement, leading to a declining overall population. In developed countries, the value was about 2.1 children per woman, while in some developing nations, the replacement rate has been calculated to be as high as 3.3 children per woman. Leslie models, named after population biologist Patrick Leslie, often include fertility matrices based on age groups to model population growth. They are also related to Euler–Lotka equations of population dynamics, named for mathematical demography pioneer Alfred Lotka and mathematician Leonhard Euler.

Fertility Cycles

Individuals seeking to improve their own fertility often rely on various methods to either predict when a woman will be fertile, such as measuring and charting basal body temperature, or to study the viability and motility of male sperm. In the late nineteenth century, physician Mary Putnam Jacobi was among the first to observe biphasic patterns in basal body temperature during menstrual cycles, though the connection with ovulation was not made until the early twentieth century. Studies by many researchers throughout the twentieth century statistically determined patterns in ovulation and fertility, such as the frequency of ovulation, the most probable window of ovulation during the menstrual cycle, and associations between fertility and observable physical characteristics, such as temperature, pain, and mucosal secretions. Many of these studies were the basis for calendar-based methods of fertility planning, such as basal body temperature (BBT) graphs. Beginning in the mid-twentieth century, physicians and others mathematically analyzed and interpreted BBT charts, though some techniques required complete data over long periods, which was considered not to be practical for use by individual couples. In the 1960s, neurologist John Marshall proposed the “three over six” prediction method: a pattern of any three plotted daily temperatures higher than the previous six was a sign of likely ovulation. This method was still in common use at the start of the twenty-first century, though with advances in computing technology, mathematical algorithms for detecting patterns may be used. Alternatively, the Billings method, named for physicians John and Evelyn Billings, is a scoring or quantification system for rating and graphing characteristics of cervical mucus to predict ovulation.

Greater understanding of the biomechanics of conception resulted in new studies of the male role in fertility. Male fertility is often quantified by sperm count or sperm concentration, which is the number of sperm cells per unit fluid volume. The term “oligozoospermia” refers to a sperm count that falls below “normal” as compared to statistically derived reference standards set by the World Health Organization and other agencies. Sperm cells may also be analyzed for abnormal morphology or geometry, which is one of the factors that affects their motility (rate of motion). Mathematical analyses have been used to explore motility. For example, mathematicians David Smith and John Blake created a mathematical model of a swimming sperm cell that they used to explore the fluid dynamic forces between sperm cells and surfaces. Understanding normal sperm motility via such models may help correct motility problems in infertile men and suggest future clinical practices.

Bibliography

Brown, Robert. Introduction to the Mathematics of Demography. 3rd ed. Winstead, CT: Actex Publications, 1997.

Poston, Dudley, and Leon Bouvier. Population and Society: An Introduction to Demography. Cambridge, England: Cambridge University Press, 2010.