Edge removal in random contact networks and the basic reproduction number.

Understanding the effect of edge removal on the basic reproduction number R0 for disease spread on contact networks is important for disease management. The formula for the basic reproduction number R0 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 R0. 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 R0. The discrepancy implies that any increase in R0 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 R0. Our models also allow an estimation of the number of edges that need to be removed in order to curtail an epidemic.