Variables, parameters and states

Mathematical models describe biological systems using mathematical quantities. Three fundamental ideas are variables, parameters and states. They have different roles in a model and should be distinguished clearly.

Variables

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

In a population model, the population size may change with time. We can write it as

\[ N(t), \]

where \(N\) is the population size and \(t\) is time. Writing \(N(t)\) emphasises that the value of \(N\) may be different at different times.

A model can contain several variables. In an epidemic model, for example,

\[ S(t),\qquad I(t),\qquad R(t) \]

may represent the numbers of susceptible, infectious and recovered individuals at time \(t\).

Parameters

Parameter. A parameter is a quantity used to specify a characteristic of the model. In a particular model run or analysis, it is usually treated as fixed unless the model explicitly allows it to vary.

Consider the simple population equation

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

Here \(N(t)\) is a variable, while \(r\) is a parameter controlling the growth or decline represented by the model.

Example. If \(r=0.05\ \text{day}^{-1}\), the chosen parameter value remains \(0.05\ \text{day}^{-1}\) during that model run, while \(N(t)\) changes with time.

Parameters often represent biological characteristics such as transmission, recovery, birth or death characteristics. Their precise interpretation depends on the model in which they appear.

States

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

If population size is the only variable, the state at time \(t\) can simply be \(N(t)\). If several variables are required, the state consists of their values together.

For an SIR epidemic model, the state can be written as the vector

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

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. These values describe the epidemic state at that time.

How the ideas differ

Concept	Role	Example
Variable	A quantity that changes	\(I(t)\)
Parameter	A quantity specifying a model characteristic	\(r\)
State	The collection of values describing the system at a given time	\((S(t),I(t),R(t))\)

Key distinction. Variables describe quantities that can change. Parameters specify characteristics of the model. The state records the current values of the variables needed to describe the system.