Message passing approach for social contagion and epidemic spreading in multiplex networks

In this paper, we set up a double-layer multiplex network to elucidate the interaction of social contagion and epidemic spreading. The two types of transmission occur at layer 1 and layer 2 respectively, which are correspondingly described by the threshold model and classical SIR model. The assumption that the transmission and recovery probabilities are constants in unit time implies that the time intervals of transmission and recovery follow the exponential distribution. However, the time interval distributions are not only limited to be exponential in most practical cases. To address the general scenario, we take arbitrary distributions of transmission and recovery intervals into consideration. We formulate the interactional transmission in a finite multiplex network and large scale random multiplex networks via the message passing approach. Finally, concrete numerical examples demonstrate the consistency of the theoretical results with Monte Carlo simulations, and specifically, we investigate the effects of parameters (the threshold and inhibiting factor) on the two types of transmission.