Mathematical modeling

Summary: Modeling reformulates scenarios to mathematical elements for analysis and problem solving.

Mathematical modeling has been in use since prehistory and was likely the first kind of mathematics ever employed. Mathematical modeling can be thought of as the activity involved in finding a solution to a real-life problem by working with a mathematical structure that captures the important characteristics of the situation. In the twenty-first century, mathematical modeling is found in many areas of mathematics, engineering, science, social science, and business and has often resulted in the formation of recognized “subdisciplines” within these fields. Research and applications occur in a diverse range of theoretical and real-world problems, and modeling is used in schools starting in the primary grades to help students visualize and solve problems, create alternative representations of various concepts, and make connections between different areas of mathematics. The advent of computers has facilitated mathematical modeling and allowed researchers to conduct simulations or find numerical solutions to problems that may be difficult to solve analytically. However, there are those who argue against overuse of mathematical models, citing issues of faulty data, unwarranted extrapolation, and the inherent error of attempting to quantify many complex or qualitative real-world phenomena. These types of criticisms have been applied to models associated with the financial and housing crises of the early twenty-first century and evidence on both sides of the global warming debate.

Anyone who has ever attempted to solve a story problem has dabbled in modeling. Consider the following story problem:

I asked my dad for some money. He gave me 24 coins with three times as many dimes as quarters, for a total of $3.30. How many of each coin do I have?

To solve the problem, one converts the verbal statements into equations. The set of equations is the mathematical model, which can be solved to determine an answer. When formulating the mathematical equations, assumptions would be made, such as which denominations of coins to include in the model.

Process

Modern treatises on the modeling process often portray the steps involved in modeling using a diagram similar to Figure 1.

Starting with a problem statement, the first action is to determine the assumptions that should be made, information in the problem that is extraneous and can be neglected, quantities that are known (parameters) and unknown (variables), and relationships between the quantities. This work may entail using a variety of strategies, including developing or using existing physical laws, proportionality arguments, equations from the current experts in the field, or equations empirically determined from experimental data.

That first step will lead to a mathematical representation of the real-world situation. The mathematics may take the form of equations, inequalities, recursive relations, matrices, graphs, integrals, differential equations, geometric structures, or other mathematical objects.

The next action is to “solve” the mathematics, leading to an answer. That answer may be an exact solution, a simulation, or an approximation. The answer must then be interpreted in the context of the problem, and any approximations must be checked (validated) to see if the solution is correct. Lastly, the explanation of the answer in the context of the situation should be used to verify or predict the solution to the problem.

In practice for complex real-world problems, the modeling process is really a cycle that is traversed repeatedly as the model is refined to produce more realistic behavior. It is common to find that these processes involve multidisciplinary teams of professionals, including mathematicians and scientists, who participate in a dialogue to clarify and refine the assumptions based on the success of the last step in the process: analyzing the mathematics in the context of the problem.

History

There is archeological evidence from more than 10,000 years ago of simple mathematical ideas being developed to solve problems related to counting objects, measuring land area and distance, and recording time. More complex mathematical problem solving appears around 3000 b.c.e., when the cultures of Asia, the Middle East, and North Africa began using arithmetic, algebra, and geometry to solve problems in astronomy, building construction, and financial situations, such as taxation. The design and construction of complex pyramids and temples and the development of sophisticated astronomical calendars in Central and South America in the first century c.e. point to the development of mathematical ideas to solve problems. It can be argued, in fact, that most of the mathematics developed before 1800 was conceived to help model a situation in the real world.

Before the mid-nineteenth century, many of the real-world problems that were approached using mathematics would be classified today as astronomy, physics, or engineering. For example, Archimedes (287–212 b.c.e.) was instrumental in modeling physical tools, such as levers, and in the development of models for hydrostatics (the properties of water at rest, such as pressure). Eratosthenes (276–194 b.c.e.) used a geometric model and his knowledge of how the sun casts shadows to determine the circumference of the Earth. Abu Ali Hasan Al-Haitham, known more commonly as Alhazan (965–1040), developed the first principles of optics for spherical and parabolic lenses. Blaise Pascal (1623–1662) developed the ideas fundamental to probability while helping a gambling friend by modeling a dice-rolling game. Isaac Newton (1642–1727) is perhaps the best-known “mathematical modeler” who ever lived, famous for his ground-breaking work on the classical laws of motion and gravitation. Building on equations of fluid flow developed by Leonhard Euler (1707–1783), Claude Henry Navier (1785–1836), and George Stokes (1819–1903) produced the Navier–Stokes equations, which model velocity, pressure, temperature, and density of a moving fluid. The Navier–Stokes equations, a set of nonlinear partial differential equations, were truly understood only after the advent of modern digital computers in the 1960s.

The nineteenth century saw an expansion into biological and social science modeling. Thomas Malthus (1766–1834) wrote about population growth and the familiar exponential model for population growth is named after him. Pierre Verhulst (1804–1859) took Malthus’s ideas and developed the logistic, limited growth model (see Figure 2).

Late Nineteenth Century Through Twentieth Century

From the late nineteenth century forward, mathematicians have become more concerned with the development of theoretical—sometimes called “pure”—mathematics: abstract structures derived from fundamental axioms and built through proving theorems following logical precepts. However, this interest in mathematics for its own sake did not slow down the development and use of mathematics as a tool to model the real world. The application areas have become increasingly diverse, and the twentieth century saw the process of mathematical modeling adopted in many fields outside physics and engineering.

In the first two decades of the twentieth century, Albert Einstein (1879–1955) developed his theories of relativity, mathematical models that predict gravitational processes on the planetary scale more accurately than Newton’s—now called “classical”—mechanics. Alfred Lotka (1880–1949) and Vito Volterra (1860–1940) worked in the 1920s on models of the interaction between predator and prey species, each arriving at the same model using different assumptions and arguments about how variables interact. Population models continue to be explored and refined through the present day. George Danzig (1914–2005) developed the simplex algorithm in 1947 to solve the mixing, supply chain, and other logistical problems that arose in World War II; these problems could be modeled with the well-understood linear programming approach, but the problems had so many variables and constraints that they were too complex to solve without computers. Linear programming is arguably the mathematical model most used in business and agriculture today. Edward Lorenz (1917–2008) developed one of the first nonlinear models for the atmosphere in the early 1960s, a precursor to the sophisticated climate models of today. His model displayed a very interesting sensitivity to initial conditions, and the study of this and similar models led to the field of chaos theory.

Twenty-First Century

In the twenty-first century, the use of mathematical modeling is ubiquitous across many research areas and academic disciplines. The Society for Mathematical Psychology publishes research in mathematical models used to examine psychological problems in neurology and cognition. The Society for Mathematical Biology concerns itself with applications of mathematics to modeling complex ecological systems, genetics, medicine, and cell biology. The Journal of Mathematical Chemistry is published by Springer-Verlag to provide a venue for researchers to share results from mathematical models of molecular behavior and chemical reactions. The American Sociological Association Section for Mathematical Sociology meets regularly to share research that uses “the language of mathematics to describe the structure, explain the events, and predict the dynamics of the social world.” The American Institute of Physics publishes the Journal of Mathematical Physics, which focuses on applications of mathematical modeling to classical mechanics and quantum physics. The field of operations research, also called “management science,” has evolved with the goal of solving mathematical models to determine the best business decision (often the maximum profit or minimum cost) given a situation in which there are limited resources. The Institute for Operations Research and the Management Sciences (INFORMS) is one of many organizations that publish results from this discipline.

The research presented through these venues tackles a diverse range of real-world problems. In medicine, mathematical models for physical principles of flow and pressure have been adapted and expanded to model the heart as a double-chambered pump. The flow of blood through the vessels can be examined and the parameters for flexibility of the vessels can be changed to investigate health conditions, such as hardening of the arteries brought on by aging. The ideal gas law and equations governing transport have been used to model how the lungs function to transport oxygen from the air inhaled to the blood in the aveoli in exchange for carbon dioxide. In the social sciences, Markov processes (mathematical matrices of transition probabilities) have been used to model social mobility and vacancy chains (the notion that a vacancy in a company or a house causes a chain reaction as others move in to fill the vacancy) as well as recidivism (the likelihood that a criminal will become a repeat offender). Generalizations of the Navier–Stokes and Lorenz models have been used to model the Earth’s atmosphere, including sources of pollution and other greenhouse gases in an effort to prove and to predict the presence (or absence) of global warming. Scientists at NASA’s Goddard Institute for Space Studies conduct research in three-dimensional atmospheric circulation models and in coupled atmosphere-ocean models in an effort to understand climate sensitivity.

With the advent and development of computers, increasingly sophisticated situations can be modeled and approximate solutions or simulations obtained using numerical algorithms. With modern computers to do the heavy computational work solving or simulating the mathematics, the most challenging step in the process is often the formulation of the mathematical model.

Bibliography

Beltrami, E. Mathematical Models in the Social and Biological Sciences. Boston: Jones and Bartlett Publishers, 1993.

Friedman, A., and W. Littman. Industrial Mathematics: A Course in Solving Real-World Problems. Philadelphia: Society for Industrial and Applied Mathematics, 1994.

Hadlock, C. Mathematical Modeling in the Environment. Reston, VA: Mathematical Association of America, 1998.

Hoppensteadt, F., and C. Peskin. Modeling and Simulation in Medicine and the Life Sciences. 2nd ed. New York: Springer-Verlag, 2002.

Klamkin, M., ed. Problems in Applied Mathematics: Selections from SIAM Review. Philadelphia: Society for Industrial and Applied Mathematics, 1990.

NASA Goddard Institute for Space Sciences. “GISS Research: Global Climate Modeling” (2010). http://www.giss.nasa.gov/research/modeling.

Smith, S. Agnesi to Zeno: Over 100 Vignettes From the History of Math. Berkeley, CA: Key Curriculum Press, 1996.

Swetz, F., ed. From Five Fingers to Infinity: A Journey Through the History of Mathematics. Chicago: Open Court Publishing, 1994.