Mathematical Modeling: Representations and Classifications of Models

Mathematical models are widely used in everyday life. They are just abstractions of reality. Models are a representation of a particular thing, idea, or condition. Mathematical Models are simplified representations of some real world unit. They can be in equations or computer codes, and are intended to imitate essential features while leaving out the details that are unimportant. Mathematical models are characterized by assumptions about three things: variables (the things which change), parameters (the things which do not change), and functional forms (the relationship between the two).

A Mathematical model personifies a hypothesis about the study system, and allows one to compare that hypothesis with data gathered. A Mathematical model is often most useful when it fails to fit the data, because that says that some of the ideas about the study system are wrong and limits the amount of possibilities. Mathematical models are useful experimental tools for building and testing theories, assessing quantitative inferences, answering specific questions, determining sensitivities to changes in parameter values and estimating key parameters from data.

In some cases, Mathematical Modeling is the only way to come up with a solution to a problem. For example, experiments with infectious disease spread in human populations are often impossible, unethical or expensive without the use of Mathematical Modeling. Another example of this would be cases involving endangered species since they cannot be managed by trial and error. So in real world instances such as these, Mathematical Modeling becomes very useful.

There are many different classifications of Mathematical Modeling. Mathematical Modeling can be labeled as deterministic or stochastic. Deterministic models have no components that are essentially unclear. For fixed starting values, a deterministic model will always produce the same result. A stochastic model will yield multiple results depending on the actual values that the random variables take in each comprehension. Structured Models are based on age, size, etc. whereas individual models are not. Mathematical Modeling can also be defined as static or dynamic. Static models are at an equilibrium or steady state. Dynamic models change with respect to time. Statistical models, also known as empirical models, are typically based on reversion. They provide a quantitative summary of the observed relationships among a set of measured variables. A mechanistic model, also known as a scientific model, begins with a description of how nature might work, and proceeds from this description to a set of predictions relating the independent and dependent variables. Finally, Mathematical Models can be defined as qualitative or quantitative. Qualitative models lead to a detailed, numerical prediction about responses, while qualitative models lead to general descriptions about the responses.

Mathematical modeling can be used for forecasting disease or pest outbreaks, designing man-made systems, biological pest control, and bioengineering. Mathematical modeling can also be used for managing existing systems such as agriculture, as well as optimizing medical treatments. Mathematical modeling specifies the underlying assumptions about a collection of data. Mathematical modeling should be more widely recognized, especially when attempting to solve complex problems that would otherwise not have a solution.