The ongoing COVID-19 outbreak that originated in the city of Wuhan, China, has caused a significant damage to the world population and the global economy. It has claimed more than 0.8 million lives worldwide, and more than 27 million people have been infected as of 07th September 2020. In Sri Lanka, the first case of COVID-19 was reported late January 2020 which was a Chinese national and the first local case was identified in the second week of March. Since then, the government of Sri Lanka introduced various sequential measures to improve social distancing such as closure of schools and education institutes, introducing work from home model to reduce the public gathering, introducing travel bans to international arrivals, and more drastically, imposed island wide curfew expecting to minimize the burden of the disease to the Sri Lankan health system and the entire community. Currently, there are 3123 cases with 12 fatalities and also, it was reported that 2925 patients have recovered and are discharged from hospitals, according to the Ministry of Health, Sri Lanka. In this study, we use the SEIR conceptual model and its modified version by decomposing infected patients into two classes: patients who show mild symptoms and patients who tend to face severe respiratory problems and are required to be treated in intensive care units. We numerically simulate the models for about a five-month period reflecting the early stage of the epidemic in the country, considering three critical parameters of COVID-19 transmission mainly in the Sri Lankan context: efficacy of control measures, rate of overseas imported cases, and time to introduce social distancing measures by the respective authorities.