  We present a model for host-parasite dynamics which incorporates both
vertical and horizontal transmission as well as spatial structure. Our model
consists of stacked contact processes (CP), where the dynamics of the host is a
simple CP on a lattice while the dynamics of the parasite is a secondary CP
which sits on top of the host-occupied sites. In the simplest case, where
infection does not incur any cost, we uncover a novel effect: a nonmonotonic
dependence of parasite prevalence on host turnover. Inspired by natural
examples of hyperparasitism, we extend our model to multiple levels of
parasites and identify a transition between the maintenance of a finite and
infinite number of levels, which we conjecture is connected to a roughening
transition in models of surface-growth.
