There could be a "very significant expansion" of the number of cases of coronavirus in the UK, Prime Minister Boris Johnson has warned.
Speaking to BBC political editor Laura Kuenssberg, Mr Johnson said the possibility of further transmission of the virus was "clearly on the cards".
It comes as the number of cases in the UK rose to 39 on Monday.
Meanwhile, the EU has raised the coronavirus risk level in member states to "moderate to high".
There were four new UK cases announced on Monday, all of whom had travelled to Italy - which has seen the largest outbreak in Europe.
They are from Hertfordshire, Devon and Kent and tracing their contacts has started, England's chief medical officer Prof Chris Whitty said.
It was earlier reported that this took the total to 40. However, a hospital in Middlesex where a doctor was thought to have contracted the virus on Sunday said further tests have proved negative.
What do I need to know about the coronavirus?
The prime minister said: "The most important thing now is that we prepare against a possible very significant expansion of coronavirus in the UK population."
It is "much more likely than not that we will face a challenge in the weeks, months ahead," he said, adding that the country was "well prepared with a fantastic NHS".
Mr Johnson said that "further protective measures" against the spread of the disease would be announced "as and when that spread happens".
"Don't forget the importance of washing your hands," he added.
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The prime minister was speaking after an emergency Cobra meeting where ministers discussed plans to tackle the spread of coronavirus in the UK - due to be published on Tuesday.
Mr Johnson said plans would include a "range of calibrated responses to the spread of coronavirus".
He said the issue with measures such as closing schools and cancelling major public events would be "when and how and with what logic to deploy them".
Education Secretary Gavin Williamson later confirmed that schools "should stay open unless they're advised otherwise by Public Health England".
The Budget will go ahead as planned on 11 March, despite the impact of the outbreak on the UK and global economy, Downing Street said.
All Six Nations rugby matches currently scheduled are set to go ahead, including Saturday's England v Wales fixture, it has been confirmed.
Elsewhere, Scotland's first minister Nicola Sturgeon has warned the nation should prepare for a "significant outbreak" of coronavirus.
There has so far been one confirmed case in Scotland, involving a patient who had recently travelled to Italy. Ms Sturgeon said that future cases were likely in the "days to come".
Earlier, Public Health England (PHE) said widespread transmission of coronavirus in the UK was now "highly likely".
Medical director Prof Paul Cosford said the increase in cases in the UK and abroad meant the UK must be prepared.
If the outbreak worsens, the government has said measures could include asking newly retired doctors and nurses to return to the NHS.
People could also be urged to work from home - and closing schools and cancelling major public events have also not been ruled out.
Legislation allowing the government to use extra powers to help control the spread of the virus is expected to go through Parliament by the end of March.
Meanwhile, British holidaymakers who were quarantined at a hotel in Tenerife are returning home after testing negative for the virus.
More than 700 guests, including hundreds of Britons, were told to isolate at the H10 Costa Adeje Palace last week after a group of Italians contracted the virus.
A Jet2 plane believed to be carrying about 100 passengers from the hotel landed at Manchester Airport on Monday evening before continuing to Belfast.
Once home, the group must isolate themselves until the original quarantine period ends on 10 March.
Travel firm Tui confirmed 17 of its customers were flown back to the UK on two separate flights to Bristol and Newcastle on Sunday night.
The company's remaining 19 guests will be flown home on Tuesday, subject to their test results, a spokeswoman said.
In other developments:
- British Airways has cancelled hundreds of flights between 6 March and 28 March over the outbreak, including flights from London Heathrow, Gatwick and London City airports
- Ryanair has also reduced the number of flights on some routes, in particular to and from Italy, by up to 25% due to a drop in demand
- London hospitality body UK Hospitality, which represents hotels, venues and restaurants has told BBC London that its members in the capital are currently seeing a 10-15% reduction in bookings
- Northern Ireland pubs have lost hundreds of thousands of pounds while feeling the "pain" of the coronavirus crisis, an industry chief has claimed
- Amazon has pulled out of London Book Fair, which is scheduled to begin on 10 March
Where are the latest UK cases?
There are now 36 cases of coronavirus in England, and one case each in Northern Ireland, Wales and Scotland.
On Monday, London's Guildhall School of Music and Drama announced that a member of its teaching staff was being treated at the Royal Free Hospital after testing positive for the virus.
The school, in the City of London, said he came in contact with a "limited" number of students, and added that it would be closed for up to 14 days.
North East London NHS Foundation Trust, which occupies a space at Vinters Business Park - home to Maidstone Studios - also said a worker had been diagnosed with the virus.
The trust said its staff member did not come into contact with patients, and its employees are working from home.
ITV Meridian, which uses Maidstone Studios, has also sent home staff who were at the site last week.
A pupil at a secondary school in Torbay, Devon, has also been diagnosed.
In Hertfordshire, Davenport House Surgery in Harpenden said a patient has been diagnosed with the virus, while a school in Stevenage said a parent has tested positive.
Some 13 new patients in the UK were diagnosed on Sunday, including a member of staff at Wimbledon College.
Three cases in England were linked to a man from Surrey, who was the first patient not to have been abroad recently and was instead infected within the UK.
Meanwhile, a clinician at Mount Vernon Cancer Centre in Northwood, Middlesex, had been confirmed positive but has since tested negative, East and North Hertfordshire NHS Trust said.
As of 09:00 GMT on Monday, the Department of Health said a total of 13,525 people had been tested in the UK, of which 13,485 were negative.
Globally, about 86,000 people have been infected, with cases in more than 50 countries. More than 3,000 people have died - the vast majority in China's Hubei province, where the outbreak originated in December.