Reproduction numbers for epidemics on networks using pair approximation

One way to describe the spread of an infection on a network is by using the method of pair approximation. This method is a deterministic pair-based variant of the usual methods used to describe the progress of an epidemic in randomly mixing populations. However, although the ideas of pair approximation are intuitively clear, it is not straightforward to make all assumptions used explicit. Furthermore, in literature problems arise in defining basic quantities like the basic reproduction number R0 and the real-time epidemic growth rate parameter r. We formulate the pair approximations and the needed assumptions explicitly. We discuss problems inherent to this method. Furthermore, we define a new reproduction number, similar to R0 and a new real-time growth rate parameter similar to r. We illustrate the methods of the paper by an example for which we can compare the approximation of the reproduction number with exact results.