Analytic prediction for the threshold of non-Markovian epidemic process on temporal networks

The transmission of pathogen between hosts and the interactions among hosts are two crucial factors for the spreading of epidemics. The former process is generally non-Markovian as the amount of the pathogen developed in hosts undergoes complicated biological process, while the latter one is time-varying due to the dynamic nature of modern society. Despite the abundant efforts working on the effects of the two aspects, a framework that integrates these two factors in a unified representation is still missing. In this paper, we develop a framework with tensorial description encoding non-Markovian process and temporal structure by introducing a super-matrix representation that incorporates multiple discrete time steps in a chronological order. Our proposed framework formulated with super-matrix representation allows a general analytical derivation of the epidemic threshold in terms of the spectral radius of the super-matrix. The accuracy of the approach is verified by different temporal network models. This framework could serve as an effective tool to offer novel understanding of integrated dynamics induced from non-Markovian individual processes and temporal interacting structures.