Stochastic dynamics for epidemics based on a compartmental scheme: An application to the AH1N1 influenza

Abstract

In this paper a stochastic formulation for describing the dynamics of an epidemic is established. We state our model in the language of a compartmental scheme of three sites (susceptible, infected and recovered) SIR, for which the parameters model are extended to be statistical distributions of time. It is established a master equation for governing the dynamics of the probability density P of finding the system in the state characterized by the values of the aleatory variables S, I, R and time t. Our stochastic formalism allow us to recover the associated deterministic model in terms of the expected values of S, I and R; whereas the second momenta of P provide us statistical standard deviations for these three variables which delimit the region in which most of the particular realizations are to be expected. We have applied the analysis developed here, for studying the specific case of the influenza AH1N1 that took place in Mexico in 2009. The reported data by the main Mexican Health institution are in good agreement with the predictions of our model for the standard deviation of the aleatory variables.