Understanding the spread of diseasesthrough complex networks is of great interest where realistic, heterogeneouscontact patterns play a crucial role in the spread. So far, most works havefocused on mean-field behavior quantifying how contact patterns affect theemergence and stability of long-lived endemic states in networks. On the otherhand, long-time dynamics and rare events in such systems have been given farless attention. In this talk, we will explore the dynamics of thesusceptible-infected-susceptible (SIS) model on heterogeneous, assortativenetworks. In particular, we will demonstrate how assortativity (the degreecorrelation between neighboring nodes), demographic stochasticity, and contactheterogeneity combine to influence large fluctuations and spontaneous infectionclearance, i.e., disease extinction. Employing various perturbation schemes weshow that in undirected networks, assortativity and heterogeneity areinterchangeable, both reducing disease prevalence. Conversely, in directed networks,in some cases heterogeneity can also hinder extinction events and increase thedisease lifetime. We corroborate all our analytical findings using highlyefficient and robust weighted-ensemble network simulations, which can predictthe occurrence of rare events with very high accuracy. This numericalcapability allows us to mimic the long-time epidemic dynamics on very largenetworks with an arbitrary connectivity matrix, in parameter regimes which wereinaccessible using standard numerical schemes. We will conclude by discussingthe SIS dynamics with time-dependent transition rates, and how these affect theprobability of disease extinction.