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An outbreak of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), epi-centred in Hubei Province of the People’s Republic of China, quickly spread worldwide and caused COVID-19 pandemic. It infected hundreds of millions of people and caused millions of deaths. In this paper, we develop a compartmental ODE model of COVID-19 transmission. We consider a possibility of breakthrough infections after the vaccination and account for both symptomatic and asymptomatic infections and transmissions. We also incorporate game theory to study the optimal vaccination decisions from the individuals’ perspective. We show that vaccination alone is unlikely to eliminate COVID-19. To achieve herd immunity, the individuals would have to receive a dose of a vaccine more frequently than once every 3 months. It is therefore crucial to adhere to various guidelines, such as quarantine, isolate and wear a mask if tested positive for COVID-19.