Lineage of the SEIRS model and its critical thresholds on social networks

One of the most natural processes occurring on quenched social networks is the transmission of infectious diseases, accompanied by nonequilibrium continuous phase transitions. However, there have been long-standing theoretical challenges in accurately predicting disease outbreaks, such as the multiplicity of disease stages, the mode of disease transmission, and, in particular, dynamic correlation caused by quenched connections. Theoretical progress in solving the dynamical correlation has been made only for simple epidemic models, and a comprehensive understanding of the general spreading dynamics on quenched networks is quite involved and still lacking. We study the most general compartmental framework, the Susceptible-Exposed-Infected-Recovered-Susceptible (SEIRS) model, and two typical transmission modes: infection of all connected neighbors and random infection of a single neighbor, corresponding to a traditional SEIRS model and a novel SEIRS model. We derive critical thresholds of both models by introducing fundamental conditions that the number of each type of two connected nodes should satisfy in the stationary state and by taking into account the dynamic correlation. The derived thresholds provide the thresholds for all the possible derivatives induced from both models in appropriate limits of related rates. The predicted thresholds are verified numerically with high accuracy on quenched random and scale-free networks.