US-based management consulting firm McKinsey & Company predicts that the most likely time for the United States to obtain herd immunity is the third or fourth quarter of 2021.
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US to Achieve Herd Immunity by Fourth Quarter of 2021: McKinsey"
‘If T-cell cross-reactivity produces meaningful immunity, it will offer significant progress toward herd immunity against COVID-19.’
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Herd immunity in the United States could be reached as soon as the second quarter of 2021 if vaccines are very effective and launched smoothly or if significant cross-immunity is observed in the population, McKinsey said in a report. 
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The epidemiological end of the COVID-19 pandemic might not be reached until 2022 or later if the vaccine candidates have efficacy or safety issues or if their distribution and adoption are slow.
"At worst, we perceive a long-tail possibility that the United States could still be fighting coronavirus into 2023 and beyond if several factors like short duration of natural immunity and low efficacy of vaccines align against us," it said.
The second endpoint of the COVID-19 pandemic may be reached earlier than the first.
In the US, while the transition to normal might be achieved sooner, the epidemiological endpoint looks most likely to be reached in the second half of 2021. Other advanced economies are presumably on similar timetables.
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Studies reveal that those who have not contracted coronavirus have cross-reactivity in specific immune cells (T cells).
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The degree to which T-cell cross-reactivity actually immunizes people hasn't been proven. If T-cell cross-reactivity produces meaningful immunity, it will offer significant improvement toward herd immunity. The prevalence of cross-reactive immunity may differ substantially by region.
Source-Medindia