  Networked epidemic models have been widely adopted to describe propagation
phenomena. The endemic equilibrium of these models is of great significance in
the field of viral marketing, innovation dissemination, and information
diffusion. However, its stability conditions have not been fully explored. In
this paper we study the stability of the endemic equilibrium of a networked
Susceptible-Infected-Susceptible (SIS) epidemic model with heterogeneous
transition rates in a discrete-time manner. We show that the endemic
equilibrium, if it exists, is asymptotically stable for any nontrivial initial
condition. Under mild assumptions on initial conditions, we further prove that
during the spreading process there exists no overshoot with respect to the
endemic equilibrium. Finally, we conduct numerical experiments on real-world
networks to demonstrate our results.
