CALCULUS FOR MODELLING 01

Rates in Mathematical Models

A rate describes how quickly one quantity changes relative to another, usually time.

The main idea

If a quantity \(X(t)\) changes with time, its instantaneous rate of change is

\[\boxed{\frac{dX}{dt}}\]

This is the basic language of many dynamical models.

1. Finite change and average rate

Over a time interval \(\Delta t\), the change is

\[\Delta X=X(t+\Delta t)-X(t).\]

The average rate is

\[\boxed{\frac{\Delta X}{\Delta t}}.\]

It measures change per unit time over the whole interval.

2. Instantaneous rate

The derivative is obtained by shrinking the interval:

\[\boxed{\frac{dX}{dt}=\lim_{\Delta t\to0}\frac{\Delta X}{\Delta t}}.\]

It describes the local rate at a particular time.

3. Units matter

If \(X\) is measured in people and \(t\) in days, then \(dX/dt\) has units people per day. A model equation must be dimensionally consistent.

4. Positive and negative rates

If \(dX/dt>0\), the quantity is increasing. If \(dX/dt<0\), it is decreasing. If \(dX/dt=0\), it is momentarily unchanged.

5. Rates built from mechanisms

Models usually construct rates from processes. A population may gain individuals through births and lose them through deaths:

\[\boxed{\frac{dP}{dt}=\text{birth rate}-\text{death rate}}\]

6. Per-capita rates

If each individual contributes an average rate \(r\), then the total rate may be proportional to population size:

\[\frac{dP}{dt}=rP.\]

Here \(r\) has units of inverse time.

7. Interaction rates

Some rates depend on interactions between two groups. In an epidemic model, a common infection term is

\[\beta\frac{SI}{N}.\]

The rate increases when either the susceptible population \(S\) or infectious population \(I\) increases.

8. Flow interpretation

A useful modelling principle is

\[\boxed{\text{rate of change}=\text{inflow}-\text{outflow}}\]

This applies to populations, compartments, chemical quantities, finance, fluid systems and many other models.

9. Rate versus probability

A continuous-time event rate is not itself a probability. For a sufficiently small interval \(\Delta t\), an event occurring at rate \(\lambda\) has probability approximately

\[P(\text{event in }\Delta t)\approx\lambda\Delta t.\]

This distinction becomes important later in stochastic modelling.

10. The central idea

\[\boxed{\text{mechanism}\rightarrow\text{rate}\rightarrow\text{differential equation}}\]

Calculus allows a verbal description of change to become a mathematical law of evolution.