Here is the rate equation:

* \\(S' = - \beta S I\\)

* \\(I' = \beta S I\\)

Assuming a constant population size \\(S + I = P\\), we can substitute to get a differential equation in one variable:

* \\(I' = \beta I (P - I)\\)

This an instance of the **logistic equation**.

* \\(S' = - \beta S I\\)

* \\(I' = \beta S I\\)

Assuming a constant population size \\(S + I = P\\), we can substitute to get a differential equation in one variable:

* \\(I' = \beta I (P - I)\\)

This an instance of the **logistic equation**.