Mathematical modeling is used to solve real problems in a variety of fields
In mathematical modeling, you write down equations to describe how a real world system behaves. The "system" might be drawn from many different fields. For example, you might look at several populations living off the same food supply and try to predict how the populations grow and shrink. Or you might look at how an epidemic spreads and see how people move among the groups of "immune," "infected," and "at risk to get the disease". In physics or engineering you might be interested in how heat is dissipated through the heat shield of a space vehicle, or how different planets exert gravitational pulls on each other. Or you might want to apply the laws of fluid dynamics to describe how blood flows in veins and what happens when blood pressure is increased. In economics you might want to predict how a strike in the steel industry will affect other parts of the economy, such as the unemployment rate and the gross national product.
Building a mathematical model is usually a multi-stage process: you write down the equations, use them to predict what will happen, see if your predictions agree with experiments, modify your equations if necessary, make new predictions, and so on.
These models may use one of more of these kinds of mathematics: linear algebra (systems of simultaneous equations, as solved in high school algebra ), calculus (which deals with predicting a function's values from knowing the rate at which the function is changing), and statistics.
The model may be solved exactly (you may be able to write down a function that tells you the values you want to know), or you may have to approximate the values because they can't be found exactly, or you may have to simulate the model on a computer -- i.e., let the computer imitate the real system to see what happens as you change some of the conditions, such as the amount of food available or how contagious the epidemic disease is.
One advantage of using a mathematical model of a "real world system" is that one can investigate the effects of changes in the system without altering the real world. For example, it is clearly more economical to discover from a computer simulation that a new two lane tunnel under Boston Harbor will be inadequate in three years than from the experience of actually building the tunnel.
Another advantage of making a mathematical model of a "real world system" is that you can use well-developed mathematical theory to build up theory in the system (rather than having to build it up from scratch), and to make predictions about the system. As usual, the power of mathematics comes from its ability to handle general abstract problems (such as solving simultaneous equations) and then to apply these general methods to an enormous variety of problems.
Let's look at a mathematical model that may have affected you without your knowing it.
Have you been vaccinated against smallpox (the vaccination looks like a thumb-sized area on your upper arm)? Perhaps you have an older sibling who was vaccinated or a younger one who wasn't? The decision to stop innoculating against smallpox was based on a mathematical model.
At the time the decision was made, about ten years ago, there was no naturally occurring smallpox in the United States, but there were children who died every year as a result of the smallpox vaccination program. (Typically the child who died was not the one who was vaccinated, but a sibling with eczema or some other skin problem who touched the vaccination then his/her own sores and got a massive case of smallpox.) It was very well known how many children died from the smallpox program.
Since there was no smallpox in the United States, it was fair to assume that any smallpox cases that occurred here would be introduced through a major port such as New York or Washington or Los Angeles. Estimates were made of how long it would take before such a case was detected. At that point, you would begin vaccinating in expanding circles around the city where the case appeared. Some people who came in contact with the infected person might go to another city, and cases would begin to appear elsewhere. You would then start to vaccinate around these other cities. Finally, it is possible to predict the expected number of deaths if all this happened, which turned out to be fewer than the known number of deaths under the vaccination program. So the decision was made to stop vaccinating. (Smallpox has since been declared a dead disease.)
There are other important models that deal with a person or thing that can move among a finite number of situations -- for example, a person can be employed or unemployed, an electron can be in one of a finite number of orbits, a person can have cancer or not, a piece of tire on Route 128 can lie on the left shoulder, in one of the eight lanes, or on the right shoulder, a person can use Crest, or Colgate, or Brand X, or not brush his/her teeth at all. A simple but powerful model (based on high school algebra) can be used to predict what fraction of the population will end up in each state in the long run, and whether or not everything will eventually end up in one state (as tires do on the highway shoulders).
As you can see from some of the examples at the start of this section, mathematical modeling is a very important area of mathematics. It is the kind of work that can be done for its own sake, or in combination with another field, -- biology, chemistry, economics, business, engineering, sociology, psychology, computer science, urban planning, or medicine. It is an area in which you can draw from several different branches of mathematics. It can be as abstract or as "hands-on" as you like.
Leontieff and Samuelson have separately won Nobel Prizes in economics for their mathematical models of the economy. (Leontieff's model used linear algebra to predict how one segment of the economy affects other segments. Samuelson's model was based on calculus.)