Sufficient conditions of endemic threshold on metapopulation networks

TL;DR

This paper establishes a rigorous sufficient condition for the epidemic threshold in metapopulation networks with heterogeneous infection rates, highlighting the influence of network structure and population size dependence on disease spread.

Contribution

It proves that the mean-field derived threshold bound applies to arbitrary networks and introduces an improved condition emphasizing the role of rich-club connectivity.

Findings

Rich-club networks are more susceptible to epidemics.

Infection rate dependence on population size significantly affects threshold estimates.

Numerical simulations confirm theoretical bounds.

Abstract

In this paper, we focus on susceptible-infected-susceptible dynamics on metapopulation networks, where nodes represent subpopulations, and where agents diffuse and interact. Recent studies suggest that heterogeneous network structure between elements plays an important role in determining the threshold of infection rate at the onset of epidemics, a fundamental quantity governing the epidemic dynamics. We consider the general case in which the infection rate at each node depends on its population size, as shown in recent empirical observations. We first prove that a sufficient condition for the endemic threshold (i.e., its upper bound), previously derived based on a mean-field approximation of network structure, also holds true for arbitrary networks. We also derive an improved condition showing that networks with the rich-club property (i.e., high connectivity between nodes with a large number of links) are more prone to disease spreading. The dependency of infection rate on population size introduces a considerable difference between this upper bound and estimates based on mean-field approximations, even when degree-degree correlations are considered. We verify the theoretical results with numerical simulations.