We will discuss recent developments, and some challenges, in models and predictions of epidemics. Among others, we will present results on parameter identifiability in ODE models, the role of spatial and other heterogeneities in the host population, the role of behavior in creating bistabilities in the dynamics, and finally trade-offs, which appear as nonlinearities in various system responses. We will use techniques ranging from differential algebra and bifurcation theory, to algorithms borrowed from transportation science.