Overview#

What is a Mathematical Model?#

A mathematical model is a mathematical representation of a system used to make predictions and provide insight about a real-world scenario, and mathematical modelling is the process of constructing, simulating and evaluating mathematical models.

Why do we construct mathematical models? It can often be costly (or impossible!) to conduct experiments to study a real-world problem and so a mathematical model is a way to describe the behaviour of a system and predict outcomes using mathematical equations and computer simulations.

See also

Check out the following resources to get started with mathematical modelling:

Outline of the Modelling Process#

Mathematical modelling involves observing some real-world phenomenon and formulating a mathematical representation of the system. But how do we even know where to start? Or how to find a solution? The modelling process is a systematic approach:

  1. Clearly state the problem

  2. Identify variables and parameters

  3. Make assumptions and identify constraints

  4. Build solutions

  5. Analyze and assess

  6. Report the results

Models can have a wide range of complexity! More complex does not necessarily mean better and we can sometimes work with more simplistic models to achieve good results. In many instances, we often start with a simple model and then build-up the complexity by iterating through the steps in modelling process until the model accurately describes the real-world application.

See also

Check out Math Modeling: Getting Started and Getting Solutions to read more about the modelling process.

Types of Models#

There are many different types of mathematical models! In this course we focus on the following:

Deterministic models predict future based on current information and do not include randomness. These kinds of models often take the from of systems of differential equations which describe the evolution of a system over time.

Stochastic models include randomness and are based on probability distributions and stochastic processes.

Data-driven models look for patterns in observed data to predict the output of a system. These kinds of models often take the form of functions with parameters computed to fit observed data.