Dynamical interplay between epidemics and cascades in complex networks

Epidemics and cascading failure are extensively investigated. Traditionally, they are independently studied, but in practice, there are many cases where these two dynamics interact with each other and neither of their effects can be ignored. For example, consider that a digital virus is spreading in a communication network, which is transferring data in the meantime. We build a model based on the epidemiological SIR model and a local load sharing cascading failure model to study the interplay between these two dynamics. In this model, when the dynamical process stops at equilibrium, the nodes both uninfected and unfailed form several clusters. We consider the relative size of the largest one, i.e. the giant component. A phenomenon is observed in both Erdős-Rényi (ER) random networks and Barabási-Albert (BA) scale-free networks that when the infection probability is over some critical value, a giant component forms only if the tolerance parameter α is within some interval $(\alpha_l,\alpha_u)$ . In this interval, the size of the remained giant component first increases and then decreases. After analyzing the cause of this phenomenon, we then present in ER random networks a theoretical solution of the key values of $\alpha_l$ and $\alpha_u$ , which are very important when we evaluate the robustness of the network. Finally, our theory is verified by numerical simulations