A law of large numbers approximation for Markov population processes with countably many types

TL;DR

This paper establishes a law of large numbers for complex Markov population models with infinitely many types, providing convergence rates and broad applicability to ecological and epidemiological systems.

Contribution

It extends the law of large numbers to systems with countably infinite types and offers sharp convergence bounds in weighted $ ext{ell}_1$ norms.

Findings

Proves a law of large numbers for infinite-type Markov processes.

Provides explicit convergence rate bounds.

Applicable to ecological and disease transmission models.

Abstract

When modelling metapopulation dynamics, the influence of a single patch on the metapopulation depends on the number of individuals in the patch. Since the population size has no natural upper limit, this leads to systems in which there are countably infinitely many possible types of individual. Analogous considerations apply in the transmission of parasitic diseases. In this paper, we prove a law of large numbers for rather general systems of this kind, together with a rather sharp bound on the rate of convergence in an appropriately chosen weighted $\ell_1$ norm.