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

TL;DR

This paper introduces a Laplace's method tailored for asymptotic analysis of sharply peaked series, applying it to semiclassical analysis in epidemiological models, especially the SIS model, demonstrating the equivalence of two semiclassical approaches.

Contribution

It develops a novel Laplace's method for series and applies it to semiclassical analysis of stochastic population models, unifying two common approaches.

Findings

Established asymptotic expansions for series from discretized integrals.

Demonstrated the equivalence of semiclassical probability and generating function approaches.

Applied the method to the SIS epidemiological model.

Abstract

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.