  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.
