Mathematical Modelling Basics

Mathematical Modelling Basics

Mathematical modelling is a method of representing real-world problems using mathematical concepts and language. It involves creating a mathematical description of a situation to predict or understand its behaviour. This process typically includes defining the problem, making assumptions, developing a model, and validating that model against real-world data.

Understanding Mathematical Models

A mathematical model is a simplified representation of a system or process. It uses mathematical expressions to describe relationships between different variables. Models can vary in complexity, from simple equations to elaborate simulations.

Types of Mathematical Models

There are several types of mathematical models. The most common types include:

• Deterministic Models: These models assume that the outcomes are precisely determined by the input values. There is no randomness involved.
• Stochastic Models: These models incorporate randomness and uncertainty. They are used when the outcomes are not predictable.
• Static Models: These models do not change over time. They represent a snapshot of a situation at a specific moment.
• Dynamic Models: These models change over time and can represent processes that evolve.

Steps in Mathematical Modelling

The process of mathematical modelling involves several key steps:

1. Define the Problem: Clearly articulate the issue you want to model. Identify the key variables and relationships involved.
2. Make Assumptions: Simplify the real-world situation by making assumptions. These should be reasonable and based on available data.
3. Develop the Model: Create mathematical equations or expressions that represent the relationships between the variables.
4. Validate the Model: Compare the model's predictions with real-world data. Adjust the model as necessary to improve accuracy.
5. Use the Model: Apply the model to make predictions or inform decisions.

Example of a Simple Mathematical Model

Consider a business that wants to predict its revenue based on the number of units sold. Let:

• R: Revenue
• P: Price per unit
• Q: Quantity sold

The relationship can be expressed as:

R = P × Q

Suppose the price per unit is R50, and the business expects to sell 200 units. We can substitute these values into the equation:

R = 50 × 200

Calculating this gives:

R = 10000

The predicted revenue is R10,000.

Common Mistakes in Mathematical Modelling

Using Graphs in Mathematical Modelling

Graphs are useful tools in mathematical modelling. They visually represent the relationships between variables. For example, if you plot revenue against the quantity sold, you can see how changes in quantity affect revenue.

Example of Graphical Representation

Using the previous example, if we plot the revenue for different quantities sold at a price of R50, we can create a table:

This data can be plotted on a graph where the x-axis represents quantity sold and the y-axis represents revenue. The resulting graph will show a straight line, indicating a linear relationship between quantity sold and revenue.

Real-World Applications of Mathematical Models

Mathematical models are widely used in various fields, including:

• Economics: To model market behaviour and predict economic trends.
• Engineering: To design structures and systems.
• Environmental Science: To predict changes in ecosystems and assess the impact of human activities.
• Healthcare: To model disease spread and treatment outcomes.

Conclusion

Mathematical modelling is a powerful tool for understanding and predicting real-world phenomena. By following the steps of defining a problem, making assumptions, developing a model, validating it, and using it, you can create effective models for various applications.

Check your understanding

• What are the key steps in the mathematical modelling process?
• Define deterministic and stochastic models.
• Provide an example of a simple mathematical model and explain its components.
• Why is it important to validate a mathematical model?