Introduction to Mathematical Modelling

Question 1

What is a model?

Answer 1

• A simplified representation of reality
• When systems are qualitatively intuitive a mathematical description can help you determine the magnitude of effects
• When systems are not qualitatively intuitive the qualitative behaviour emerges from the model

Question 2

Types of models

Answer 2

• Conceptual
• Experimental
• Analytical
• Numerical

Question 3

Stages of model evolution

Answer 3

1. Model development
2. Experimental data
3. Model analysis/validation
4. Predictions

Question 4

How do you solve a system of ODEs?

Answer 4

Steady-state solution:
- Set the equations to zero and rearrange them for an analytical solution
- Set equations to zero and solve as a linear system
- Numerically integrate them from an arbitrary starting position until a steady state is reached

When dynamics are required:
- Integrate mathematically to determine an analytical solution
- Numerically integrate from a known starting position and observe changes over time

Question 5

Numerical integration

Answer 5

• Loop through space and ts
• Loop through x (and y and z if multiple dimensions) to incorporate interactions with neighbouring cells then loop through it
• Vectorise and use ODE solvers in Matlab, R, Python
• PDE solvers

Question 6

Agent-based models (ABMs)

Answer 6

• ABMs simulate individuals. They allow randomness in decision-making by individuals to manifest as variability at the population level
• Very simple rules can be given to each agent and emergent dynamics can be observed
• Each agent has properties including a spatial coordinate
• Loop through agents rather than spatial grids and then through ts