CALCULUS FOR MODELLING 12
Model Assumptions
A mathematical model is built from assumptions. Those assumptions determine both its equations and the meaning of its conclusions.
The main idea
1. Why assumptions are necessary
Real systems contain enormous detail. A model deliberately keeps some mechanisms and ignores or simplifies others so that the question can be studied mathematically.
2. Structural assumptions
These determine how the system is represented: which variables exist, which compartments are used, and which interactions are allowed.
3. Parameter assumptions
A model may assume that parameters are constant, known exactly, or shared across a population. Each choice simplifies reality.
4. Homogeneous mixing
In a simple epidemic model, homogeneous mixing assumes that individuals interact without explicit network, spatial or demographic structure.
5. Continuous states
ODE population models often treat population variables as continuous even though individuals are discrete. This approximation is often more reasonable for large populations than very small ones.
6. Closed populations
An equation such as
\[S+I+R=N\]usually assumes no births, deaths unrelated to the model, immigration or emigration during the modelled period.
7. Time-independent parameters
Using a constant \(\beta\) assumes the transmission parameter does not change with behaviour, season, interventions or other factors. Piecewise or time-varying parameters can represent changes when needed.
8. Assumptions should match the question
A simplification is not automatically a weakness. It becomes problematic when it removes a mechanism essential to the scientific question.
9. State assumptions explicitly
Clear assumptions make a model interpretable, reproducible and criticisable. They also make extensions easier to design.
10. Test assumptions where possible
Compare predictions with data, examine residual patterns, vary parameters and structures, and ask whether conclusions remain robust when assumptions change.