In this talk, I will present some recent results regarding the modeling of epidemic spreading on graphs. I will mainly focus on the case of epidemic models set on (infinite) homogeneous trees and completely characterize the spreading properties of the model as a function of the degree of the tree, the intrinsic basic reproduction number and the diffusivity of the population of infected individuals. If time allows, I will also discuss a new class of epidemic models set on connected graphs which consist in PDE-ODE systems with Robin like boundary conditions. Some general properties of such systems will be described and a link with the first part of the talk will be formally explained. This is joint work with Christophe Besse.