Zika virus (ZIKV) has caused large, brief outbreaks in isolated populations, however ZIKV can also persist at low levels over multiple years. The reasons for these diverse transmission dynamics remain poorly understood. In Fiji, which has experienced multiple large single-season dengue epidemics, there was evidence of multi-year transmission of ZIKV between 2013 and 2017. To identify factors that could explain these differences in dynamics between closely related mosquito-borne flaviviruses, we jointly fit a transmission dynamic model to surveillance, serological and molecular data. We estimate that the observed dynamics of ZIKV were the result of two key factors: strong seasonal effects, which created an ecologically optimal time of year for outbreaks; and introduction of ZIKV after this optimal time, which allowed ZIKV transmission to persist over multiple seasons. The ability to jointly fit to multiple data sources could help identify a similar range of possible outbreak dynamics in other settings.