As a result of the COVID-19 pandemic, whether and when the world can reach herd immunity and return to normal life and a strategy for accelerating vaccination programmes constitute major concerns. We employed Metropolis–Hastings sampling and an epidemic model to design experiments based on the current vaccinations administered and a more equitable vaccine allocation scenario. The results show that most high-income countries can reach herd immunity in less than 1 year, whereas low-income countries should reach this state after more than 3 years. With a more equitable vaccine allocation strategy, global herd immunity can be reached in 2021. However, the spread of SARS-CoV-2 variants means that an additional 83 days will be needed to reach global herd immunity and that the number of cumulative cases will increase by 113.37% in 2021. With the more equitable vaccine allocation scenario, the number of cumulative cases will increase by only 5.70% without additional vaccine doses. As SARS-CoV-2 variants arise, herd immunity could be delayed to the point that a return to normal life is theoretically impossible in 2021. Nevertheless, a more equitable global vaccine allocation strategy, such as providing rapid vaccine assistance to low-income countries/regions, can improve the prevention of COVID-19 infection even though the virus could mutate.