Stochastic processes with distributed delays: chemical Langevin equation and linear-noise approximation

TL;DR

This paper introduces a new systematic approach to approximate stochastic reaction systems with distributed delays, using a dynamical generating functional instead of the traditional master equation, applicable to non-Markovian systems.

Contribution

It develops a formalism for the linear-noise approximation that does not depend on the master equation, broadening analysis of delayed stochastic systems.

Findings

Derived general expressions for the chemical Langevin equation with distributed delays.

Applied the formalism to gene regulation with delayed auto-inhibition.

Applied the formalism to epidemic models with delayed recovery.

Abstract

We develop a systematic approach to the linear-noise approximation for stochastic reaction systems with distributed delays. Unlike most existing work our formalism does not rely on a master equation, instead it is based upon a dynamical generating functional describing the probability measure over all possible paths of the dynamics. We derive general expressions for the chemical Langevin equation for a broad class of non-Markovian systems with distributed delay. Exemplars of a model of gene regulation with delayed auto-inhibition and a model of epidemic spread with delayed recovery provide evidence of the applicability of our results.