Variables, parameters and states

Mathematical models describe biological systems using mathematical quantities. Three fundamental ideas are variables, parameters and states.

Variables

Variable. A variable is a quantity whose value can change within the model.

For example, population size may change with time:

\[N(t).\]

In an epidemic model, \(S(t)\), \(I(t)\) and \(R(t)\) may represent the numbers of susceptible, infectious and recovered individuals.

Parameters

Parameter. A parameter is a quantity that specifies or controls a characteristic of a model. It is usually treated as fixed during a particular model run.

A parameter has a numerical value, usually together with units. What that value represents depends on the parameter. A parameter may represent a rate, population capacity, probability, diffusion coefficient or another model characteristic.

For example, consider logistic population growth:

\[\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right).\]

Two different parameters.

\(r=0.2\ \text{day}^{-1}\) is a rate parameter. Its numerical value is \(0.2\), with units per day.

\(K=1000\ \text{individuals}\) is a capacity parameter. Its numerical value is \(1000\); it is not a rate.

Therefore, parameter, parameter value and rate are not the same idea. The parameter is the model quantity, its parameter value is the chosen numerical value, and a rate is one possible biological meaning of a parameter.

Term	Meaning	Example
Parameter	A quantity specifying a model characteristic	\(r\), \(K\)
Parameter value	The numerical value assigned to a parameter	\(r=0.2\), \(K=1000\)
Rate parameter	A parameter whose value represents a rate	\(r=0.2\ \text{day}^{-1}\)
Non-rate parameter	A parameter representing another quantity	\(K=1000\ \text{individuals}\)

States

State. The state of a model is the information needed to describe the system at a particular time.

For an SIR epidemic model, the state can be written

\[ \mathbf{X}(t)= \begin{pmatrix} S(t)\\ I(t)\\ R(t) \end{pmatrix}. \]

For example,

\[ \mathbf{X}(t)= \begin{pmatrix} 900\\ 80\\ 20 \end{pmatrix}. \]

means that at time \(t\) there are 900 susceptible, 80 infectious and 20 recovered individuals.

How the ideas differ

Concept	Role	Example
Variable	A quantity that can change	\(I(t)\)
Parameter	A quantity specifying a model characteristic	\(r\), \(K\)
State	Values describing the system at a given time	\((S(t),I(t),R(t))\)

Key distinction. A parameter always has a value, but it is not necessarily a rate. For example, \(r\) can be a rate parameter while \(K\) is a capacity parameter.