Other evidence supports this model. A recent peer-reviewed study led by a team from Erasmus University, in the Netherlands, published data from 12 patients showing that they had developed antibodies after infection with SARS-CoV-2. Several of my colleagues and students and I have statistically analyzed thousands of seasonal coronavirus cases in the United States and used a mathematical model to infer that immunity over a year or so is likely for the two seasonal coronaviruses most closely related to SARS-CoV-2 — an indication perhaps of how immunity to SARS-CoV-2 itself might also behave.

If it is true that infection creates immunity in most or all individuals and that the protection lasts a year or more, then the infection of increasing numbers of people in any given population will lead to the buildup of so-called herd immunity. As more and more people become immune to the virus, an infected individual has less and less chance of coming into contact with a person susceptible to infection. Eventually, herd immunity becomes pervasive enough that an infected person on average infects less than one other person; at that point, the number of cases starts to go down. If herd immunity is widespread enough, then even in the absence of measures designed to slow transmission, the virus will be contained — at least until immunity wanes or enough new people susceptible to infection are born.

At the moment, cases of Covid-19 have been undercounted because of limited testing — perhaps by a factor of 10 in some places, like Italy as of late last month. If the undercounting is around this level in other countries as well, then a majority of the population in much (if not all) of the world still is susceptible to infection, and herd immunity is a minor phenomenon right now. The long-term control of the virus depends on getting a majority of people to become immune, through infection and recovery or through vaccination — how large a majority depends on yet other parameters of the infection that remain unknown.

One concern has to do with the possibility of reinfection. South Korea’s Centers for Disease Control and Prevention recently reported that 91 patients who had been infected with SARS-CoV-2 and then tested negative for the virus later tested positive again. If some of these cases were indeed reinfections, they would cast doubt on the strength of the immunity the patients had developed.

An alternative possibility, which many scientists think is more likely, is that these patients had a false negative test in the middle of an ongoing infection, or that the infection had temporarily subsided and then re-emerged. South Korea’s C.D.C. is now working to assess the merit of all these explanations. As with other diseases for which it can be difficult to distinguish a new infection from a new flare-up of an old infection — like tuberculosis — the issue might be resolved by comparing the viral genome sequence from the first and the second periods of infection.

For now, it is reasonable to assume that only a minority of the world’s population is immune to SARS-CoV-2, even in hard-hit areas. How could this tentative picture evolve as better data come in? Early hints suggest that it could change in either direction.

It is possible that many more cases of Covid-19 have occurred than have been reported, even after accounting for limited testing. One recent study (not yet peer-reviewed) suggests that rather than, say, 10 times the number of detected cases, the United States may really have more like 100, or even 1,000, times the official number. This estimate is an indirect inference from statistical correlations. In emergencies, such indirect assessments can be early evidence of an important finding — or statistical flukes. But if this one is correct, then herd immunity to SARS-CoV-2 could be building faster than the commonly reported figures suggest.