A class of stochastic epidemic models and its deterministic counterpart

A general stochastic model for the spread of an epidemic developing in a closed population is introduced. Each model consisting of a discrete-time Markov chain involves a deterministic counterpart represented by an ordinary differential equation. Our framework involves various epidemic models such as a stochastic version of the Kermack and McKendrick model and the SIS epidemic model. We prove the asymptotic consistency of the stochastic model regarding a deterministic model; this means that for a large population both modelings are similar. Moreover, a Central Limit Theorem for the fluctuations of the stochastic modeling regarding the deterministic model is also proved.