RNA viruses have extremely high mutation rates and therefore exist as a large population of mutants. However, their replication often occurs from infections which initiate with only a single infectious particle. We took a systems-level view and quantitative approaches to describe how individual poliovirus infections are affected by stochastic effects and to infer the mode of replication use by the virus within single-cell infections. Temporal, quantitative measurements of positive-sense genomes, negative-sense templates, virions, and infectious particles were the major data source. Stochastic mathematical modeling was used to bridge a gap between wet lab science and computational biology. We find that poliovirus is sensitive to both kinetic stochastic effects and spatial resource variabilities in individual infections. We also infer that poliovirus replicates with a geometric growth mode, with progeny resulting from a single infection being on average 5 genomic replication cycles away from the infecting parent. This replication mode not only allows the opportunity for significant amounts of intracellular selection but also creates the potential for expansive population structures.