Epidemics in networks with nodal self-infection and the epidemic threshold

Since the Susceptible-Infected-Susceptible (SIS) epidemic threshold is not precisely defined in spite of its practical importance, the classical SIS epidemic process has been generalized to the ε-SIS model, where a node possesses a self-infection rate ε, in addition to a link infection rate β and a curing rate δ. The exact Markov equations are derived, from which the steady state can be computed. The major advantage of the ε-SIS model is that its steady state is different from the absorbing (or overall-healthy state) and approximates, for a certain range of small ε > 0, the in reality observed phase transition, also called the "metastable" state, that is characterized by the epidemic threshold. The exact steady-state analysis for the complete graph illustrates the effect of small ε and the quality of the first-order mean-field approximation, the N-intertwined model, proposed earlier. Apart from duality principles, often used in the mathematical literature, we present an exact recursion relation for the Markov infinitesimal generator.