Poisson approximation to density dependent stochastic processes . A numerical implementation and test

We implement the recently introduced Poisson approximation on a density dependent jump process corresponding to a simple, but not exactly solvable, epidemiological model. Statistics produced with 10 independent realizations of the Feller process are compared with equivalent statistics produced with Poisson, proper Gaussian and diffusion realizations. While in the Poisson approximation the statistical deviation can be reduced to a satisfactory level by reducing the time step of the simulation, and the Gauss approximation will only be rejected in the most demanding situations, the statistical deviation of the diffusion approximation can easily be perceived in a city with mean population of ∼ 10 individuals. This deviation cannot be reduced decreasing the time step of the simulation.