We consider the problem of modeling point-level spatial count data with a large number of zeros. We develop a model that is compatible with scientific assumptions about the underlying data-generating process. We utilize a two-stage spatial generalized linear mixed model framework for the counts, modeling incidence, resulting in 0–1 outcomes, and prevalence, resulting in positive counts, as separate but dependent processes and utilize a Gaussian process model for characterizing the underlying spatial dependence. We describe a Bayesian approach and study several variants of our two-stage model. We fit the models via Markov chain Monte Carlo methods. We study several Markov chain Monte Carlo algorithms, including a version of the Langevin–Hastings algorithm, for exploring the complicated posterior distribution efficiently and recommend an algorithm that is fairly efficient. Finally, we demonstrate the application of our modeling and computational approach on both simulated data and real data from an ecological field survey. Copyright © 2011 John Wiley & Sons, Ltd.