Parasites on parasites: coupled fluctuations in stacked contact processes

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. Introduction. – The need to understand and control the dynamics of infections has motivated the development of a variety of statistical mechanical models. One of the most important of these is the contact process (CP) [1–4], which describes the dynamics of an infection in a spatially structured population. In the contact process, each site on a lattice represents a host organism which can be infected or susceptible (uninfected). Infection is transmitted to neighbouring susceptible host sites at rate b, and infected hosts recover (i.e. become susceptible) at rate d. The contact process provides a baseline model for many problems in ecology and epidemiology. It is also of fundamental importance in statistical physics. This is because it exhibits a non-equilibrium phase transition between an infected and a non-infected phase at a critical value of λ = b/d [2, 3]. The standard CP model for an infected population assumes that the host population is of fixed size, without turnover, i.e. host births and deaths. This is valid if the timescale on which the infection is gained and lost is much shorter than the lifespan of an individual. Some infections, however, are carried by individuals for long times, and may be transmitted to offspring upon reproduction (vertical transmission), as well as being transmitted horizontally— i.e. upon physical contact between individuals. Examples include plasmids carried by bacteria [5], some microsporidian parasites of insects [6] and pathogens including HIV and several hepatitis viruses. Here, we investigate the interplay between vertical and horizontal transmission in such populations. Other authors have determined the conditions for parasite persistence in mean-field models that lack spatial structure, in which the parasite may affect host fitness [7–9], and have suggested that these conditions may be affected by spatial structure [10]. Here, we show that spatial structure can produce a qualitatively new effect: a coupling between the dynamics of the infection and of the underlying host population, even when the infection does not affect the fitness of the host. We present a two level stacked contact process, in which the host population is represented by a CP on a lattice, and the parasite population is represented by a second CP which sits on top of the host CP. This model incorporates both vertical and horizontal transmission — if an infected host reproduces or dies, the parasite is reproduced or dies with it (vertical transmission), and a parasite-infected site can infect a neighbouring site if it is occupied by an uninfected host (horizontal transmission). We characterize the conditions for parasite persistence in the form of a phase diagram and discover an interesting phenomenon: although the steady-state properties of the host population depend only on the ratio of its birth and death rates, λ = b/d, the prevalence of the parasite can depend non-monotonically on the host population’s turnover rate. This phenomenon has its origins in the fluctuations