Only a tiny proportion of the global population – maybe as few as 2% or 3% – appear to have antibodies in the blood showing they have been infected with Covid-19, according to the World Health Organization, a finding that bodes ill for hopes that herd immunity will ease the exit from lockdown.

“Easing restrictions is not the end of the epidemic in any country,” said WHO director-general Dr Tedros Adhanom Ghebreyesus at a media briefing in Geneva on Monday. “So-called lockdowns can help to take the heat out of a country’s epidemic.”

But serological testing to find out how large a proportion of the population have had the infection and developed antibodies to it – which it is hoped will mean they have some level of immunity – suggests that the numbers are low.

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“Early data suggests that a relatively small percentage of the populations may have been infected,” Tedros said. “Not more than 2%-3%.”

Dr Maria Van Kerkhove, an American infectious diseases expert who is the WHO’s technical lead on Covid-19, said they had thought the number of people infected would be higher, but she stressed it was still too early to be sure. “Initially, we see a lower proportion of people with antibodies than we were expecting,” she said. “A lower number of people are infected.”

Quick guide What does the World Health Organization do, and why has Trump stopped supporting it? Show Hide What is the World Health Organization’s remit? The World Health Organization (WHO) was founded as the UN global health body in 1948 in the aftermath of the second world war with a mandate to promote global health, protect against infectious disease and to serve the vulnerable. Its current programme envisages expanding universal healthcare to a billion more people, protecting another billion from health emergencies and providing a further billion people with better health and wellbeing. What does that involve? The WHO acts as a clearing house for investigation, data and technical recommendations on emerging disease threats such as the coronavirus and Ebola. It also supports eradication of existing diseases such as malaria and polio and promotes global public health. While its role on emerging diseases is most familiar in the developed world, its practical involvement is far more marked in the global south, where it has been working to expand basic healthcare, support vaccination and sustain weak and often stressed health systems through its emergencies programmes. Why is the WHO under fire from Trump? Trump has presented withdrawing funding and cutting ties with WHO as a direct response to what he claims was its slow reaction in raising the alarm over the global threat from the coronavirus and being too “China-centric” in its response. The allegation that the WHO was slow to warn of the risk of human-to-human transmission, and that it failed to cross-examine Chinese transparency early on, is largely not borne out by the evidence. And the organisation’s funding was already in his sights on 7 February, when his administration was suggesting cutting the US contribution by half. The WHO, to whom the US theoretically contributes roughly 10-15% of its budget as its largest contributor, has been appealing for an extra $1bn to help fight the coronavirus. While the suspension of funding by the US for 60-90 days is relatively small – not least because the US is so far in arrears in its annual payments – a general US withdrawal from global health funding under the cover of this announcement would be very serious and felt most profoundly in places that need the most support. Peter Beaumont and Sarah Boseley

On Friday, a study carried out in Santa Clara, California by Stanford University and released as a “pre-print” without peer review, found that 50 to 85 times more people had been infected with the virus than official figures showed.

Santa Clara county had 1,094 confirmed cases of Covid-19 at the time the study was carried out, but antibody tests suggest that between 48,000 and 81,000 people had been infected by early April, most of whom did not develop symptoms.

But even those high figures mean that within the whole population of the county, only 3% have been infected and have antibodies to the virus. A study in the Netherlands of 7,000 blood donors also found that just 3% had antibodies.

Van Kerkhove said they needed to look carefully at the way the studies were being carried out. “A number of studies we are aware of in pre-print have suggested that small proportions of the population [have antibodies],” she said. These were “in single digits, up to 14% in Germany and France”. “It is really important to understand how the studies were done.”

That would include asking how they found the people to test. Was it at random or were they blood donors, who tend to be healthy adults? They would also need to look at how well the blood tests were performed.

“We are working with a number of countries carrying out these serology studies,” she added. The WHO-supported studies would use robust methods and the tests would be validated for accuracy.

The hope will be that people who have had Covid-19 will be able to resume their lives. But Van Kerkhove last week said that even if tests showed a person had antibodies, it did not prove that they were immune.

“There are a lot of countries that are suggesting using rapid diagnostic serological tests to be able to capture what they think will be a measure of immunity,” she said. “Right now, we have no evidence that the use of a serological test can show that an individual has immunity or is protected from reinfection.”

• The headline and opening paragraph of this article were amended on 20 April 2020 to clarify that the 2%-3% figure cited by the WHO referred to the population as a whole.