Necessary and sufficient conditions for exact closures of epidemic equations on configuration model networks

TL;DR

This paper identifies specific degree distributions (Poisson, Binomial, Negative Binomial) on which the exact closure of SIR epidemic equations is possible on configuration model networks, linking pairwise, DSA, and Volz models.

Contribution

It establishes the necessary and sufficient conditions for exact closures of epidemic models, connecting pairwise, DSA, and Volz models for certain degree distributions.

Findings

Exact closure possible only for Poisson, Binomial, Negative Binomial distributions.

Equivalence of pairwise, DSA, and Volz models under these distributions.

Reductions to a single susceptible-based equation with statistical interpretation.

Abstract

We prove that the exact closure of SIR pairwise epidemic equations on a configuration model network is possible if and only if the degree distribution is Poisson, Binomial, or Negative Binomial. The proof relies on establishing, for these specific degree distributions, the equivalence of the closed pairwise model and the so-called dynamical survival analysis (DSA) edge-based model which was previously shown to be exact. Indeed, as we show here, the DSA model is equivalent to the well-known edge-based Volz model. We use this result to provide reductions of the closed pairwise and Volz models to the same single equation involving only susceptibles, which has a useful statistical interpretation in terms of the times to infection. We illustrate our findings with some numerical examples.