This paper is concerned with the approximation of a class of open population epidemic models by time-inhomogeneous birth-and-death processes. In particular, we consider models in which the population of susceptibles behaves in the absence of infection as a general branching process. It is shown that for a large initial number of susceptibles, the process of infectives behaves approximately as a time-inhomogeneous birth-and-death process. Strong convergence results are obtained over an increasing sequence of time intervals [0,tN], where N is the initial number of susceptibles and tN→∞ as N→∞.
Clancy, D., & O'Neill, P. D. (1998). Approximation of epidemics by inhomogeneous birth-and-death processes. Stochastic Processes and their Applications, 73, 233-245. https://doi.org/10.1016/S0304-4149(97)00092-6