Rates and biological processes

Biological systems change because processes occur: individuals are born or die, infections occur, cells divide, chemicals react, and organisms move. Mathematics often represents these processes through rates.

What a rate means

Rate. A rate measures how much a quantity changes per unit of another quantity, commonly per unit time.

If \(N(t)\) is population size, its instantaneous rate of change is written

\[\frac{dN}{dt}.\]

If \(N\) is measured in individuals and \(t\) in days, then \(dN/dt\) has units of individuals per day.

Per-capita rates

A rate may apply to each individual. If every individual contributes births at per-capita rate \(b\), then a population of size \(N\) has total birth rate

\[bN.\]

Similarly, if \(d\) is a per-capita death rate, the total death rate is \(dN\). The net population change can therefore be represented as

\[\frac{dN}{dt}=bN-dN=(b-d)N.\]

Rates can depend on the state

Biological rates need not be constant. They may depend on population density, resource availability, numbers of infectious individuals, temperature, age, or other state quantities.

Example. In an epidemic, the rate of new infections may depend on both susceptible and infectious populations. A commonly used form is \(\beta SI/N\). Its biological interpretation depends on how \(\beta\) and the population variables are defined.

Rate and probability are different

A rate is not itself a probability. In stochastic models, an event occurring at rate \(\lambda\) is often related, over a sufficiently short interval \(\Delta t\), to

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

The detailed stochastic interpretation is developed later in the stochastic-process sections.

Key idea. Rates provide the mathematical link between biological mechanisms and change through time. Their units and biological interpretation must always be stated clearly.