A mathematical model is an abstract representation of a real-world system using mathematical concepts and language. Mathematical modeling is the process of creating such a model to make predictions or provide insight into a real-world scenario. Mathematical models are used in various fields, including applied mathematics, natural sciences (such as physics, biology, earth science, chemistry), engineering disciplines (such as computer science, electrical engineering), social sciences (such as economics, psychology, sociology, political science), music, linguistics, and philosophy.

The process of mathematical modeling can be thought of as an iterative process made up of several components:

• Make Assumptions: Identify the key ingredients of the problem and make assumptions about how they interact.
• Formulate the Model: Write down the relevant equations, simplifying as much as possible.
• Solve the Model: Solve the equations to obtain a solution.
• Analyze the Solution: Consider the results and insights gained from the model and ask if the answer makes sense.
• Iterate: Refine the model by repeating the process, adjusting as needed to improve the solution.
• Communicate: Create a clear report on the model and its implementation to make the model understandable to others.

Mathematical models can take many forms, including dynamical systems, statistical models, differential equations, or game theoretic models. The quality of a scientific field often depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments.