Stochastic SIS Model with Dynamic Node Activity: Epidemic Threshold and Stationary States

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This paper develops and evaluates a stochastic model of infection spreading on a graph with dynamic node activity. Each node is characterized by an epidemiological state (susceptible or infected) and an activity state (enabled or disabled). Only enabled nodes participate in transmission, while the network topology remains fixed. The infection probability of an enabled susceptible node is proportional to its degree and to the fraction of infected nodes among the enabled ones. The continuum analogue of the model is derived in the mean-field approximation and its stationary states are analyzed. Both the epidemic threshold and the stationary infection level coincide with the classical SIS results and are independent of the activity parameter in the symmetric case. The activity parameter therefore affects only the transient dynamics: both the decay of the infection in the subcritical regime and the convergence to the stationary state in the supercritical regime slow down as the activity rate increases. This slowdown is explained by the fact that recovery is possible only for enabled nodes, so that infected nodes temporarily excluded from transmission act as a reservoir that sustains the infection. The results extend the classical mean-field SIS model to the case of dynamic node availability.