The NHS is to join forces with the private health sector this week in an emergency plan to combat the coronavirus crisis, as fears grow that publicly funded hospitals will soon be unable to cope with the number of patients suffering from the virus.

News of the unprecedented partnership came as 245 scientists and mathematicians denounced the government plan to achieve “herd immunity” by delaying measures to prevent the virus spreading, saying acting now would save “thousands of lives”.

The number of deaths in the UK rose on Saturday by 10 in just 24 hours, almost doubling the total from 11 to 21. The government’s chief medical adviser, Professor Chris Whitty, said all were patients in “at-risk” groups.

The number of confirmed cases in the UK reached 1,140 – up from 798 on Friday. But in a sign of how fast the virus can spread officials in Italy said its number of cases had jumped 20% to more than 21,000 since Friday. The death toll rose by 175.

Around the world countries began to shut themselves off. Denmark and Lithuania said they would close their borders, Morocco cut flights from 25 countries, Norway closed its ports, Spain announced a national lockdown and France shut its cafes and restaurants, as well as most shops and cinemas from midnight on Saturday.

After Donald Trump extended a ban on certain foreign nationals entering the US to include Britons, Boris Johnson called the president to explain the British government’s stance, but he faced growing questions over whether he was doing enough to protect the public. The open letter from the academics said the UK was seeing infections grow at the same rate as in Italy, Spain, France and Germany, which meant Britain’s infected would “be in the order of thousands within a few days”.

“Under unconstrained growth, this outbreak will affect millions of people in the next few weeks,” the letter states, adding the NHS would be “at serious risk” of not being able to cope.

“Going for ‘herd immunity’ at this point does not seem a viable option, as this will put the NHS at an even stronger level of stress, risking many more lives than necessary. By putting in place social distancing measures now, the growth can be slowed down dramatically, and thousands of lives can be spared. We consider the social distancing measures taken as of today as insufficient.”

A Department of Health spokesman responded: “Herd immunity is not part of our action plan, but is a natural by-product of an epidemic.” He said every measure was “based on the best scientific evidence”. “Our awareness of the likely levels of immunity in the country over the coming months will ensure our planning and response is as accurate and effective as possible.”

Earlier, Whitty defended the government’s approach: “I understand this increase in the number of deaths will be a cause for concern. The public should know every measure we are taking is seeking to save lives and protect the most vulnerable.”

From left: chief medical officer for England, Chris Whitty; Boris Johnson; and chief scientific adviser Sir Patrick Vallance attend a news conference. Photograph: Simon Dawson/Reuters

He and the chief scientific adviser, Patrick Vallance, said countries were working together to ease the crisis: “We are dealing with a very fast-moving epidemic with emerging data from many disciplines and many complex decisions. We will publish our models and welcome input from the scientific community.

“Scientists across the world are helping each other, governments and society to deal with this emergency.”

On Saturday night the government said it was in talks with manufacturers to increase the supply of ventilators, which will be needed to treat patients with the most severe symptoms of Covid-19. NHS England will this week issue new guidance to hospitals and clinical commissioning groups on working with the private sector to carry out more non-urgent operations on NHS patients to free up beds.

“We need all parts of the health sector to step up and play their part given what’s coming our way,” said one senior NHS official. “They have the things we’re going to need: space on wards, beds and people.”

NHS bosses expect the cost of expanding the independent sector’s role to be met from the government’s £5bn Covid-19 response fund, announced by the chancellor, Rishi Sunak, in his first budget last week.

Spire Healthcare, which runs 39 hospitals across England, Scotland and Wales, has offered to help the NHS. However, a spokesman said “the exact nature, extent and the timing of this support is yet to be determined”.

This week the government will launch a massive public information campaign, with advertisements on television, radio and press featuring Whitty. A key message will be that anyone with a continual cough, however mild, or a temperature, should self-isolate for seven days.

Scientists, the World Health Organization (WHO) and opposition politicians have all questioned the government’s decision to limit testing to patients who have been admitted to hospital. There were also growing calls for clarity on whether schools will have to close and for how long.

In a warning of how dramatic and lasting the impact of school shutdowns could be, Northern Ireland’s first minister, Arlene Foster, told parents to prepare for schools to shut for “at least 16 weeks”.

A petition calling for schools and colleges in the UK to be shut had gathered more than half a million signatures ,on Saturday, while the National Education Union – the biggest teaching union – wrote to the prime minister asking for details on why schools in the UK were being kept open while they had been shut elsewhere.

The WHO raised concerns over the government’s strategy to develop “herd immunity” – where a large proportion of the population catch, recover and therefore develop immunity from the virus. Margaret Harris, of WHO, said: “We don’t know enough about the science of this virus, it hasn’t been in our population for long enough to know what it does in immunological terms. Every virus functions differently in your body and stimulates a different immunological profile.”

Labour’s health spokesman, Jonathan Ashworth, told the Observer on Saturdaythat public confidence in the government was fragile. “Delaying the spread of this virus has to be a priority so as not to overwhelm our overstretched NHS. But people are increasingly questioning if the UK is following international best practice.

“Why are we no longer testing those with symptoms who are advised to stay at home when the WHO recommends continued testing, for example?”

Jeremy Corbyn wrote to the prime minister urging him to allow mortgage payment holidays and rent deferrals, and demanded sight of emergency legislation that would give the government powers to detain virus victims, keep schools open and allow people to keep their jobs while volunteering for the NHS.

But other health experts backed the government. “The UK approach is driven by evidence that is appropriate for the local context,” said Michael Head, senior research fellow at Southampton University. “What other countries are doing with their populations, with subtle or markedly different cultural and social expectations and habits, does not mean the UK is doing the wrong thing.”