CHAPTER 08

Calculus for Modelling

Use calculus to describe real systems, interpret rates and understand how mathematical models evolve.

\[ \text{assumptions} \rightarrow \text{rates} \rightarrow \text{equations} \rightarrow \text{model behaviour} \]

MATHEMATICAL MODELLING

From rates to mathematical models

Rates in Mathematical Models

Interpret rates as change per unit of another quantity, usually time.

Learn → 02

Variables and Parameters

Distinguish changing model variables from quantities treated as fixed parameters.

Learn → 03

Growth and Decay

Model systems whose rate of change is proportional to the current amount.

Learn → 04

Logistic Growth

Extend exponential growth by introducing limits caused by finite resources.

Learn → 05

Population Models

Use differential equations to describe births, deaths and population change.

Learn → 06

Epidemic Models

Describe susceptible, infectious and recovered populations using systems of equations.

Learn → 07

Compartment Models

Represent systems by dividing a population or quantity into interacting states.

Learn → 08

Equilibria in Models

Identify states where all model rates of change are zero.

Learn → 09

Thresholds

Understand parameter values that separate qualitatively different model behaviours.

Learn → 10

Stability in Models

Study whether small disturbances move a system back toward or away from equilibrium.

Learn → 11

Numerical Approximation

Approximate model solutions when exact analytical solutions are unavailable.

Learn → 12

Model Assumptions

Understand how mathematical simplifications define what a model can and cannot represent.

Learn → 13

Model Limitations

Interpret model outputs carefully by recognising uncertainty and simplification.

Learn →

The modelling idea

\[ \frac{dX}{dt} = \text{inflow rate} - \text{outflow rate} \]

Many mathematical models are built by describing the processes that increase or decrease a quantity. Calculus converts those assumptions into equations that describe how the system changes.