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

Abstract

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 ??SIS model, where a node possesses a self-infection rate ?, in addition to a link infection rate ? and a curing rate ?. The exact Markov equations are derived, from which the steady state can be computed. The major advantage of the ??SIS model is that its steady state is different from the absorbing (or overall-healthy state) and approximates, for a certain range of small ?>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 ? 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.