Effect of coupling on the epidemic threshold in interconnected complex networks: A spectral analysis

In epidemic spreading models, if the infection strength is higher than a certain critical value - which we define as the epidemic threshold - then the epidemic spreads through the population. For a single arbitrary graph representing the contact network of the population under consideration, the epidemic threshold turns out to be equal to the inverse of the spectral radius of the contact graph. However, in a real world scenario, it is not possible to isolate a population completely: there is always some interconnection with another network, which partially overlaps with the contact network. In this paper, we study the spreading process of a susceptible-infected-susceptible (SIS) epidemic model in an interconnected network of two generic graphs with generic interconnection and different epidemic-related parameters. Using bifurcation theory and spectral graph theory, we find the epidemic threshold of one network as a function of the infection strength of the other coupled network and adjacency matrices of each graph and their interconnection, and provide a quantitative measure to distinguish weak and strong interconnection topology. These results have implications for the broad field of epidemic modeling and control.