A mathematical model is an abstract model that uses mathematical language to describe the behavior of a system.
Mathematical models are used particularly in the natural sciences and engineering disciplines (such as physics, biology, and electrical engineering) but also in the social sciences (such as economics, sociology and political science); physicists, engineers, computer scientists, and economists use mathematical models most extensively.
Eykhoff (1974) defined a mathematical model as a representation of the essential aspects of an existing system (or a system to be constructed) which presents knowledge of that system in usable form.
Mathematical models can take many forms, including but not limited to dynamical systems, statistical models, differential equations, or game theoretic models.
These and other types of models can overlap, with a given model involving a variety of abstract structures.
There are six basic groups of variables: decision variables, input variables, state variables, exogenous variables, random variables, and output variables.
Since there can be many variables of each type, the variables are generally represented by vectors.
Mathematical modelling problems are often classified into black box or white box models, according to how much a prior information is available of the system.
A black-box model is a system of which there is no a prior information available.
A white-box model (also called glass box or clear box) is a system where all necessary information is available.
Practically all systems are somewhere between the black-box and white-box models, so this concept only works as an intuitive guide for approach.
Usually it is preferable to use as much a prior information as possible to make the model more accurate.