Laws of large numbers for epidemic models with countably many types

TL;DR

This paper establishes a general law of large numbers for stochastic epidemic models with infinitely many host types, providing a convergence rate and broad applicability beyond specific cases.

Contribution

It proves a general theorem for epidemic models with countably infinite types, extending previous case-specific results and including a convergence rate in the -norm.

Findings

Proves a general law of large numbers for infinite-type epidemic models

Provides a rate of convergence in the -norm

Extends previous case-by-case results to a broad, unified framework

Abstract

In modeling parasitic diseases, it is natural to distinguish hosts according to the number of parasites that they carry, leading to a countably infinite type space. Proving the analogue of the deterministic equations, used in models with finitely many types as a "law of large numbers" approximation to the underlying stochastic model, has previously either been done case by case, using some special structure, or else not attempted. In this paper we prove a general theorem of this sort, and complement it with a rate of convergence in the $\ell_1$-norm.