The UK has reached the peak of the coronavirus epidemic, cabinet ministers have confirmed, despite warnings that the disease may still be rampant in care homes.

Dominic Raab, who is deputising for the prime minister, and Matt Hancock, the health secretary, both said the UK was now “at the peak” of the outbreak

While updating MPs in the Commons on the government’s strategy for testing the disease, Hancock said social distancing was working, adding: “It is making a difference. We are at the peak.”

His comment appeared to substantiate claims made several days ago by England’s chief medical officer, Prof Chris Whitty, who said the country was “probably reaching the peak overall”.

However, Hancock said relaxing lockdown restrictions was not imminent as tests needed to be scaled up, including a consistent fall in the daily death rate.

Hancock’s comments were echoed by Nicola Sturgeon, Scotland’s first minister, who said she was increasingly optimistic Scotland had passed the peak in cases after “really encouraging” data showed a continuing fall in hospital admissions and intensive care cases.

Sturgeon warned, however, it was too early to start relaxing the strict lockdown and social isolation rules. “The progress is definitely there but it’s fragile at this stage, so any easing up that will very quickly send all of that into reverse,” she said.

Public Health England announced 665 new hospital deaths of people who tested positive for Covid-19.

The number of coronavirus deaths at UK hospitals stands at 17,000, though there are estimates the death toll is far higher – potentially at 41,000 - if non-hospital deaths are included. Italy is believed to be beyond its peak with the country’s death toll falling consistently for the past two weeks.

The number of intensive care patients in Scotland fell for the seventh day running to 155, the lowest figures since 1 April and down from 221 on 12 April. Covid-19 hospital admissions by ambulance fell to 156 on Tuesday, the lowest level since 18 March and down from a peak of 363 on 6 April.

“These figures for hospital admissions and admissions to intensive care are really encouraging and they’re cause for optimism – still cautious optimism – but optimism nevertheless,” Sturgeon said.

The UK health secretary has been increasingly under fire for the level of testing, which has fallen behind other European countries, and for a lack of personal protective equipment for NHS and care home staff.

Hancock said the expansion of testing capacity was ahead of schedule but demand had been “lower than expected”.

He said: “We are therefore ramping up the availability of this testing and expanding who is eligible for testing and making it easier to access the tests. The tests are conducted in NHS hospitals, through our drive-through centres, mobile units and home deliveries.”

Contact tracing will also be introduced at a “large scale”, he told the Commons, and an NHS app for tracing was in development. Public Health England has been criticised for not deploying pre-existing environmental health volunteers who have the expertise to help with the outbreak.

Quick guide Will there be a second wave of coronavirus? Show Hide In recent days the UK has seen a sudden sharp increase in Covid-19 infection numbers, leading to fears that a second wave of cases is beginning. Epidemics of infectious diseases behave in different ways but the 1918 influenza pandemic that killed more than 50 million people is regarded as a key example of a pandemic that occurred in multiple waves, with the latter more severe than the first. It has been replicated – albeit more mildly – in subsequent flu pandemics. Until now that had been what was expected from Covid-19.

How and why multiple-wave outbreaks occur, and how subsequent waves of infection can be prevented, has become a staple of epidemiological modelling studies and pandemic preparation, which have looked at everything from social behaviour and health policy to vaccination and the buildup of community immunity, also known as herd immunity. Is there evidence of coronavirus coming back in a second wave? This is being watched very carefully. Without a vaccine, and with no widespread immunity to the new disease, one alarm is being sounded by the experience of Singapore, which has seen a sudden resurgence in infections despite being lauded for its early handling of the outbreak. Although Singapore instituted a strong contact tracing system for its general population, the disease re-emerged in cramped dormitory accommodation used by thousands of foreign workers with inadequate hygiene facilities and shared canteens. Singapore’s experience, although very specific, has demonstrated the ability of the disease to come back strongly in places where people are in close proximity and its ability to exploit any weakness in public health regimes set up to counter it. In June 2020, Beijing suffered from a new cluster of coronavirus cases which caused authorities to re-implement restrictions that China had previously been able to lift. In the UK, the city of Leicester was unable to come out of lockdown because of the development of a new spike of coronavirus cases. Clusters also emerged in Melbourne, requiring a re-imposition of lockdown conditions. What are experts worried about? Conventional wisdom among scientists suggests second waves of resistant infections occur after the capacity for treatment and isolation becomes exhausted. In this case the concern is that the social and political consensus supporting lockdowns is being overtaken by public frustration and the urgent need to reopen economies. However Linda Bauld, professor of public health at the University of Edinburgh, says “‘Second wave’ isn’t a term that we would use at the current time, as the virus hasn’t gone away, it’s in our population, it has spread to 188 countries so far, and what we are seeing now is essentially localised spikes or a localised return of a large number of cases.” The overall threat declines when susceptibility of the population to the disease falls below a certain threshold or when widespread vaccination becomes available. In general terms the ratio of susceptible and immune individuals in a population at the end of one wave determines the potential magnitude of a subsequent wave. The worry is that with a vaccine still many months away, and the real rate of infection only being guessed at, populations worldwide remain highly vulnerable to both resurgence and subsequent waves. Peter Beaumont, Emma Graham-Harrison and Martin Belam

Defending the government’s strategy on procuring PPE, Hancock said it had focused on large and credible offers first but was engaging with 1,000 companies that buy stock from abroad as well as 159 UK-based manufacturers.

His statement came as Labour claimed dozens of companies offering PPE had been “ignored” by the government.

Labour says 36 British companies have approached it to say their offers to help frontline staff “have not received a reply”.

This includes Issa Exchange Ltd in Birmingham, which said it offered a quarter of a million aprons and masks, Network Medical Products in Ripon, which said it could provide 100,000 face visors a week, and CQM Learning, which says it can provide 8,000 face shields a day.

In a letter to Michael Gove, the chancellor of the Duchy of Lancaster, the shadow Cabinet Office minister, Rachel Reeves, said: “You and your officials and those at the Department for Health and Social Care will be best placed to validate what capability and capacity these firms have, but as they have not received a reply after contacting the government. I wanted to ensure that the Cabinet Office was aware of them.

“Of these firms, if just one, five or 10 were able to contribute to the national effort of ensuring that our NHS and care workers – and indeed anybody who needs to use some form of personal protective equipment and clothing – could be better protected, or just one hospital or care home were able to access adequate supplies of the PPE they need, I know you will agree that that would go a long way and make a big difference.”

Hancock told the Commons the government had coordinated the delivery of more than 1bn items of PPE so far.

“We have a rigorous system of verifying the offers we receive because not all offers have been credible and it’s important to focus on the biggest, most credible offers first,” he said.