Multiscale Modelling of Birth-Death Processes

TL;DR

This paper introduces a new framework for selecting thresholds in hybrid stochastic-deterministic models of biological systems, improving accuracy in extinction probability predictions while maintaining computational efficiency.

Contribution

It formalizes the Jump-Switch-Flow algorithm as a piecewise-deterministic Markov process and develops a heuristic for optimal threshold selection based on error bounds.

Findings

Heuristic reliably upper-bounds errors in extinction probability

Threshold selection enables efficient multiscale modeling

Framework applicable to stochastic biological systems

Abstract

Many biological systems exhibit multiscale dynamics, where some species occur in high copy numbers while others remain rare. This heterogeneity necessitates hybrid modelling approaches: deterministic models are computationally efficient but inaccurate for low-count species, while fully stochastic simulations are accurate but prohibitively expensive. Hybrid methods like the Jump-Switch-Flow (JSF) algorithm address this by simulating low-count species stochastically and high-count species deterministically. However, selecting regime-switching thresholds to control errors for specific observables remains an open challenge. We develop a principled framework for threshold selection targeting extinction probability. We formalise JSF as a piecewise-deterministic Markov process and derive backward equations for extinction under exact and hybrid dynamics. Near extinction boundaries, complex nonlinear dynamics reduce to tractable time-inhomogeneous linear birth-death processes. This structure yields a rigorous error decomposition based on early and late excursions. Isolating the dominant error term motivates a fast, actionable heuristic. We demonstrate via Monte Carlo studies on a stochastic Lotka-Volterra model that our heuristic reliably upper-bounds empirical errors in extinction probability. This enables users to select the smallest threshold that satisfies a target error tolerance. This work paves the way for principled, efficient multiscale modelling and simulation in stochastic biological systems.