Population dynamics: Poisson approximation and its relation to the Langevin process

We discuss how to simulate a stochastic evolution process in terms of difference equations with Poisson distributions of independent events when the problem is naturally described by discrete variables. For large populations the Poisson approximation becomes a discrete integration of the Langevin approximation [T.G. Kurtz, J. Appl. Prob. 7. 49 (1970), 8, 344 (1971)]. We analyze when the latter gives a reasonable representation of the original evolution for finite size systems. A simple example of an epidemic process is used to organize the discussion and to perform statistical tests that underline the goodness of the proposed method.