Edge removal in random contact networks and the basic reproduction number

Abstract

Understanding the effect of edge removal on the basic reproduction number \({\mathcal{R}_0}\) for disease spread on contact networks is important for disease management. The formula for the basic reproduction number \({\mathcal{R}_0}\) in random network SIR models of configuration type suggests that for degree distributions with large variance, a reduction of the average degree may actually increase \({\mathcal{R}_0}\). To understand this phenomenon, we develop a dynamical model for the evolution of the degree distribution under random edge removal, and show that truly random removal always reduces \({\mathcal{R}_0}\). The discrepancy implies that any increase in \({\mathcal{R}_0}\) must result from edge removal changing the network type, invalidating the use of the basic reproduction number formula for a random contact network. We further develop an epidemic model incorporating a contact network consisting of two groups of nodes with random intra- and inter-group connections, and derive its basic reproduction number. We then prove that random edge removal within either group, and between groups, always decreases the appropriately defined \({\mathcal{R}_0}\). Our models also allow an estimation of the number of edges that need to be removed in order to curtail an epidemic.