Bounds for the expected value of one-step processes

TL;DR

This paper derives explicit bounds for the expected value of one-step Markov processes, such as birth-death processes, using mean-field models and simple ODEs, under certain polynomial rate conditions.

Contribution

It introduces a method to bound the expected value of one-step processes with polynomial transition rates, enhancing approximation accuracy.

Findings

Bounds are tight for SIS epidemic and voter models.

The approach applies to processes with density-dependent polynomial rates.

Explicit bounds improve understanding of process expectations.

Abstract

Mean-field models are often used to approximate Markov processes with large state-spaces. One-step processes, also known as birth-death processes, are an important class of such processes and are processes with state space $\{0,1,\ldots,N\}$ and where each transition is of size one. We derive explicit bounds on the expected value of such a process, bracketing it between the mean-field model and another simple ODE. Our bounds require that the Markov transition rates are density dependent polynomials that satisfy a sign condition. We illustrate the tightness of our bounds on the SIS epidemic process and the voter model.