Bacterial plasmids are extra-chromosomal genetic elements that code for a wide variety of phenotypes in their bacterial hosts and are maintained in bacterial communities through both vertical and horizontal transfer. Current mathematical models of plasmid-bacteria dynamics, based almost exclusively on mass-action differential equations that describe these interactions in completely mixed environments, fail to adequately explain phenomena such as the long-term persistence of plasmids in natural and clinical bacterial communities. This failure is at least in part due to the absence of any spatial structure in these models, whereas most bacterial populations are spatially structured in microcolonies and biofilms. To help bridge the gap between theoretical predictions and observed patterns of plasmid spread and persistence, we present here an individual-based lattice model (interacting particle system) that provides a predictive framework for understanding the dynamics of plasmid-bacteria interactions in spatially structured populations. To assess the accuracy and flexibility of the model, we performed a series of experiments that monitored plasmid loss, transconjugant formation, and cost of the IncP-1b plasmid pB10::rfp in E. coli K12 and other bacterial populations grown on agar surfaces. The model-based visual patterns of plasmid loss and spread, as well as quantitative measurements showing the effects of different initial densities of parental strains and incubation time on densities of transconjugants formed on a 2-dimensional grid, were in agreement with our experiments and with previously published empirical data. These results include features of spatially structured populations that are not predicted by mass-action differential equation models.