We consider a bivariate Markov counting process with transition probabilities having a particular structure, which includes a number of useful population processes. Using a suitable random time-scale transformation, we derive some probability statements about the process and some asymptotic results. These asymptotic results are also derived using martingale methods. Further, it is shown that these methods and results can be used for inference on the rate parameters for the process. The general epidemic model and the square law conflict model are used as illustrative examples.

We consider the standard epidemic model and several extensions of this model, including Downton's carrier-borne epidemic model. A random time-scale transformation is used to obtain equations for the size distribution and to derive asymptotic approximations for the size distribution for each of the models

In this paper it is shown that a random time-scale transformation leads to a simple derivation of some asymptotic results describing the progress of a major outbreak in the standard epidemic model. These results find application in approximation of the size distribution and in estimation of the threshold parameter.