Burst of virus infection and a possibly largest epidemic threshold of non-Markovian susceptible-infected-susceptible processes on networks.

Since a real epidemic process is not necessarily Markovian, the epidemic threshold obtained under the Markovian assumption may be not realistic. To understand general non-Markovian epidemic processes on networks, we study the Weibullian susceptible-infected-susceptible (SIS) process in which the infection process is a renewal process with a Weibull time distribution. We find that, if the infection rate exceeds 1/ln(λ_{1}+1), where λ_{1} is the largest eigenvalue of the network's adjacency matrix, then the infection will persist on the network under the mean-field approximation. Thus, 1/ln(λ_{1}+1) is possibly the largest epidemic threshold for a general non-Markovian SIS process with a Poisson curing process under the mean-field approximation. Furthermore, non-Markovian SIS processes may result in a multimodal prevalence. As a byproduct, we show that a limiting Weibullian SIS process has the potential to model bursts of a synchronized infection.