Childhood diseases such as rubella, measles, mumps, and pertussis can pose serious threats to both children and adults. Years of diligent vaccination campaigns in the U.S. have resulted in high levels of immunity among the population, but these diseases have not yet been eradicated. It is a commonly accepted hypothesis that in large cities, the less-vaccinated immigrant population carries the diseases. We develop two compartmental models that describe the disease dynamics in New York City, specifically tracking the cases among immigrants. We derive thresholds that determine which vaccination rates result in the die-out or persistence of the disease. The analysis is applicable to any communicable disease that fails into the SIR criterion.