Mixing time for an epidemic model on graphs with external sources of infection

We study the mixing time of a Susceptible--Infected--Susceptible (SIS) model on graphs with external sources of infection, which we refer to as the noisy SIS model. Under suitable assumptions on the parameters of the dynamics, we show that the mixing time is of the order $\Theta(n\log n)$ with respect to the number of vertices $n$. We further investigate the model on random graph families, including Erd{\"o}s--R{\'e}nyi graphs, random regular multigraphs, and Galton--Watson trees. By identifying high-probability structural properties of these graphs and conditioning on typical realizations, we prove that the mixing time remains of order $\Theta(n\log n)$ with high probability.