Discrete-Time Non-Markovian SEIS Model on Complex Networks

We study a discrete-time variant of the non-Markovian SEIS (Susceptible-Exposed-Infectious-Susceptible) model, occurring on complex networks. The model assumes for an arbitrary form of the Discrete Temporal Probability Functions (DTPFs), that govern the transitions from Exposed to Infectious state (incubation period) and Exposed/Infectious back to Susceptible state (recovery period); this enables the model to address a wide range of real-world spreading phenomena. Theoretical analysis, based on methods from systems theory, leads to an expression that defines the epidemic threshold, for the analyzed model, as a critical relation between the DTPF's, infection rate and the network topology (the largest eigenvalue of the networks adjacency matrix), in a form that extends the result for the Markovian case. Validity of the suggested model, and the obtained theoretical result are confirmed by the numerical analyzes. We argue that the approach used in the paper, may be further extended to describe a wide variety of model variants and sub-models, occurring both on natural, as well as technological (engineering) networks.