The contact process is an interacting particle system which models the spread of an infection in a population. In this talk I will focus on the evolution of this process in the supercritcal regime within a partial (and finite) subspace of the population. In particular, I will present some recent results on the loss of memory property for such partially observed processes and discuss their continuity properties with respect to conditioning (in the sense of g-measures). Several motivations for studying such projections, and some open questions, will be discussed during the talk.