A Laplace's method for series and the semiclassical analysis of epidemiological models

We develop a Laplace's method to compute the asymptotic expansions of sums of sharply peaked sequences. These series arise as discretizations (Riemann sums) of sharply-peaked integrals, whose asymptotic behavior can be computed by the standard Laplace's method. We apply the Laplace's method for series to the WKB (i.e. semiclassical) analysis of stochastic models of population biology, with special focus on the SIS model. In particular we show that two different and widely-used approaches to the semiclassical limit, i.e. either considering a semiclassical probability distribution or a semiclassical generating function, are equivalent.