Steady states and critical behavior of epidemic spreading on complex networks

In this paper, we derive the interaction Markov chains (IMC) mean-field equations for dynamics of the epidemic spreading susceptible-infected-recovery (SIR) model that takes place on top of complex homogeneous and heterogeneous networks. These equations are solved numerically by means of a stochastic approach. We use these equations to examine the threshold behaviour and dynamics of the model on small-world and scale-free networks. Theoretical and numerically analysis show that in homogeneous networks the model exhibits a critical threshold in the epidemic spreading rate below which it cannot diffuse in the system. In the case of SF networks, on the other hand, this threshold becomes vanishingly small in the limit of infinite system size. The results indicate the critical threshold value for SIR epidemic is greatly decided by the network topology and scale-free networks are prone to the spreading epidemics. We present analytical and Monte Carlo calculations for both small-world and scale-free networks and compare the results with those obtained by the numerical method, which shows stochastic numerical approach (SNA) can save memory and get the fast exploration. For small-world networks, we find different reconnection probability values p will affect the epidemic spreading speed but no influence for the final statement.