Semiclassical approximations of stochastic epidemiological processes towards parameter estimation using as prime example the SIS system with import

In this paper we investigate several schemes to approximate the stationary distribution of the stochastic SIS system with import. We begin by presenting the model and analytically computing its stationary distribution. We then approximate this distribution using Kramers–Moyal approximation, van Kampen's system size expansion, and a semiclassical scheme, also called WKB or eikonal approximation depending on its different applications in physics. For the semiclassical scheme, done in the context of the Hamilton–Jacobi formalism, two approaches are taken. In the first approach we assume a semiclassical ansatz for the generating function, while in the second the solution of the master equation is approximated directly. The different schemes are compared and the semiclassical approximation, which performs better, is then used to analyse the time dependent solution of stochastic systems for which no analytical expression is known. Stochastic epidemiological models are studied in order to investigate how far such semiclassical approximations can be used for parameter estimation.