Epidemic Spreading On Complex Networks

Epidemic spreading on complex networks has attracted wide attentions from the field of mathematics, physical and biology. The quench mean-field theory is one of the most important theoretical method to analyze epidemic spreading. In this paper, we focus on the validity of quench mean-field theory and its applications to non-Markovian epidemic spreading.Firstly, by using the quench mean-field theory, we investigate the validation of threshold for SIR epidemic spreading models on ER networks, SF networks, and real networks. It is found that, when the eigenvector of the leading eigenvalue of the adjacent matrix is delocalized, the threshold predicated by this theoretical approach can basically fit its real threshold, and the theoretical threshold is lower than its actual threshold when the eigenvector corresponding to the largest eigenvalue is localized. Results can help us understand the quench mean-field theory further as well as develop more accurate method in studying epidemic spreading dynamics.Secondly, we generalize the quench mean-field theory to analyzing the non-Markovian epidemic spreading. In the process of analysis, we propose a model with heterogeneous waiting time. In this model, each node’s waiting time is determined by its degree, and every node tries to transmit the epidemic to its neighbors after the assigned waiting time. With a large number of numerical simulations and theoretical analysis, we find that the epidemic outbreak threshold is dependent on both the network’s topology and the distribution of the waiting time.For a given network, we find an optimal waiting-time distribution under which the epidemic is heavily suppressed.