Threshold behavior of epidemics in regular networks

Current research is interested in identifying how topology impacts epidemics in networks. In this paper, we model SIS (susceptible-infected-susceptible) epidemics as a continuous-time Markov process and for which we can obtain a closed form description of the equilibrium distribution. Such distribution describes the long-run behavior of the epidemics. The adjacency matrix of the network topology is reflected explicitly in the formulation of the equilibrium distribution. Secondly, we are interested in analyzing the model in the regime where the topology dependent infection process opposes the topology independent healing process. Specifically, how will network topology affect the most probable long-run network state? We show that for k-regular graph topologies, the most probable network state transitions from the state where everyone is healthy to one where everyone is infected at a threshold that depends on k but not on the size of the graph.