Since the first official report of the spread of SARS-CoV-2 infections in the city of Qom in mid-February, Iran has become the country most affected by the COVID-19 epidemic in the Middle East. All Iranian provinces are now affected, although to a different extent. The aim of the present study was to evaluate whether the distance from the epicenter of the infection (Qom) or demographic factors such as population density and the ratio of the elderly population are associated with the incidence of COVID-19 in different Iranian provinces. For the purpose of determining whether the distance from the virus epicenter could be associated with the spread of infection, linear regression analysis was performed using STATA 12.0 software. The association of the incidence of COVID-19 with the population density and the ratio of the population over 65 years old in 31 Iranian provinces was also evaluated. According to our results, a strong association was found between the incidence of COVID-19 in Iranian provinces and their respective distance from Qom (p < 0.001; C = -0.68). The incidence of COVID-19 in Iranian provinces was also positively associated with the ratio of the population over 65 years old (p = 0.002; C = 0.53), while no significant association with population density was found (p = 0.39; C = 0.16). These results suggest that the implementation of travel restrictions from highly affected areas to other provinces could considerably reduce the rate of transmission of the disease throughout the country. Also, provinces with a higher proportion of elderly people (over 65) were identified as particularly at risk for the spread of SARS-CoV-2 infections. These results will contribute to better management of the COVID-19 outbreak in Iran, taking into account demographic and geographic characteristics of different provinces.