Many models of parasitic infections lead to an approximately Poisson distribution of parasites among hosts, in stark contrast to the highly over-dispersed distributions that are usually encountered in practice. In this paper, a model is analyzed which, while assuming all individuals to be alike, can still lead to a very heterogeneous distribution of parasites among the host population. The model can be viewed as a very simple mean field interacting particle system, with the particles corresponding to the individual hosts, which behaves like an associated deterministic system when the number of hosts is large. The deterministic system describes the evolution over time of the proportions of the population with different parasite loads, and its equilibria are interpreted as typical distributions of parasites among hosts. Despite its simplicity, the model is complicated enough mathematically to leave a number of open problems.