There is no doubt that getting vaccinated protects the recipient. Still, several infectious-disease researchers contacted by The New York Times cautioned that it would be months before enough people in the United States will have gotten the shots to allow for normal life to begin again.

Only then will the number of people with immunity — those who have had the disease and recovered, plus those who have been vaccinated — be large enough to take the wind out of the pandemic, said Jeffrey Shaman, an epidemiologist at Columbia who shared his team’s modeling calculations.

Dr. Shaman estimates that more than 105 million people have already been infected across the U.S., well above the number of cases that have been reported. And his projections show millions more infections are yet to come as the vaccine rolls out.