A pressing concern in the SARS-CoV-2 epidemic and other viral outbreaks, is the extent to which the containment measures are halting the viral spread. A straightforward way to assess this is to tally the active cases and the recovered ones throughout the epidemic. Here, we show how epidemic control can be assessed with molecular information during a well characterized epidemic in Iceland. We demonstrate how the viral concentration decreased in those newly diagnosed as the epidemic transitioned from exponential growth phase to containment phase. The viral concentration in the cases identified in population screening decreased faster than in those symptomatic and considered at high risk and that were targeted by the healthcare system. The viral concentration persists in recovering individuals as we found that half of the cases are still positive after two weeks. We demonstrate that accumulation of mutations in SARS-CoV-2 genome can be exploited to track the rate of new viral generations throughout the different phases of the epidemic, where the accumulation of mutations decreases as the transmission rate decreases in the containment phase. Overall, the molecular signatures of SARS-CoV-2 infections contain valuable epidemiological information that can be used to assess the effectiveness of containment measures.