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Networked epidemic models have been widely adopted to describe propagationphenomena. The endemic equilibrium of these models is of great significance inthe field of viral marketing, innovation dissemination, and informationdiffusion. However, its stability conditions have not been fully explored. Inthis paper we study the stability of the endemic equilibrium of a networkedSusceptible-Infected-Susceptible (SIS) epidemic model with heterogeneoustransition rates in a discrete-time manner. We show that the endemicequilibrium, if it exists, is asymptotically stable for any nontrivial initialcondition. Under mild assumptions on initial conditions, we further prove thatduring the spreading process there exists no overshoot with respect to theendemic equilibrium. Finally, we conduct numerical experiments on real-worldnetworks to demonstrate our results.

On the Stability of the Endemic Equilibrium of A Discrete-Time Networked Epidemic Model