  We study the SIRS (Susceptible-Infected-Recovered-Susceptible) spreading
processes over complex networks, by considering its exact $3^n$-state Markov
chain model. The Markov chain model exhibits an interesting connection with its
$2n$-state nonlinear "mean-field" approximation and the latter's corresponding
linear approximation. We show that under the specific threshold where the
disease-free state is a globally stable fixed point of both the linear and
nonlinear models, the exact underlying Markov chain has an $O(\log n)$ mixing
time, which means the epidemic dies out quickly. In fact, the epidemic
eradication condition coincides for all the three models. Furthermore, when the
threshold condition is violated, which indicates that the linear model is not
stable, we show that there exists a unique second fixed point for the nonlinear
model, which corresponds to the endemic state. We also investigate the effect
of adding immunization to the SIRS epidemics by introducing two different
models, depending on the efficacy of the vaccine. Our results indicate that
immunization improves the threshold of epidemic eradication. Furthermore, the
common threshold for fast-mixing of the Markov chain and global stability of
the disease-free fixed point improves by the same factor for the
vaccination-dominant model.
