We establish a compartmental model to study the transmission of Zika virus disease including spread through sexual contacts and the role of asymptomatic carriers. To incorporate the impact of the seasonality of weather on the spread of Zika, we apply a nonautonomous model with time-dependent mosquito birth rate and biting rate, which allows us to explain the differing outcome of the epidemic in different countries of South America: using Latin Hypercube Sampling for fitting, we were able to reproduce the different outcomes of the disease in various countries. Sensitivity analysis shows that, although the most important factors in Zika transmission are the birth rate of mosquitoes and the transmission rate from mosquitoes to humans, spread through sexual contacts also highly contributes to the transmission of Zika virus: our study suggests that the practice of safe sex among those who have possibly contracted the disease, can significantly reduce the number of Zika cases.