Britons returning from northern Italy are being told to self-isolate in the UK if they show coronavirus symptoms.
Health Secretary Matt Hancock said people with flu-like symptoms who have been north of Pisa are asked to stay at home for 14 days.
The advice also applies to anyone who recently returned from Italy's quarantined towns even if they have no symptoms, he said.
Italy has put 11 towns in Lombardy and Veneto into lockdown.
Police are manning checkpoints around the towns in quarantine, after 229 people have tested positive for the virus and seven died.
Some travellers suggested UK authorities were too slow to respond to the outbreak in Italy, where the number of confirmed cases spiked over four days.
Milly Phillips, a GP from Oxford, told the BBC she developed a flu-like illness on Saturday after returning from Venice two days earlier - but NHS 111 had said Italy was not "on the list" of affected places.
She said: "I've worked for the NHS for 20 years and it's the first time I've been frustrated by the bureaucracy."
Several schools - including Brine Leas School in Cheshire, Salendine Nook High School in Yorkshire, Newquay Tretherras and Penair School in Cornwall - advised staff and students to stay home, after they returned from skiing trips in northern Italy.
And Cransley School, a private school in Cheshire, closed after 29 pupils and five members of staff returned from a Lombardy ski resort.
Headmaster Richard Pollock told parents in an email that he was going further than the advice from Public Health England "to completely minimise" the potential spread of infection and to carry out a deep clean of the school.
Stephen Paskins, who is due to fly back to Bristol from Venice on Wednesday, told the BBC there were fewer tourists in the city, more police and more people wearing face masks.
"The only way to get about is via public transport so keeping away from people is impossible. Everyone is still visiting the landmarks even though you can't enter," he said.
Meanwhile, a hotel in Tenerife has been locked down with up to 1,000 guests inside after a visiting Italian doctor tested positive for the coronavirus.
Mr Hancock said the travel advice to Italy, which attracts about three million British visitors each year, has not changed and there were no plans to stop flights from the country.
"If you look at Italy, they stopped all flights from China and they're now the worst affected country in Europe," he said.
Major events in northern Italy have been cancelled, cut short or rescheduled, including the Venice Carnival and Bologna book fair.
The quarantined towns in Italy are Bertonico, Casalpusterlengo, Castelgerundo, Castiglione d'Adda, Codogno, Fombio, Maleo, San Fiorano, Somaglia, Terranova dei Passerini - all in Lodi province in Lombardy - and Vo' Euganeo in Padova province, Veneto.
The updated advice means people returning from these Italian towns, or Iran, parts of South Korea and Hubei province in China - where the outbreak began - are being asked to call NHS 111, stay indoors and avoid contact with others, even if they do not have symptoms.
Anyone coming back from other parts of northern Italy and several Asian countries affected by the virus is advised to self-isolate and call NHS 111 if they develop symptoms.
The NHS says the symptoms of Covid-19, the illness caused by the coronavirus, are a cough, a high temperature and shortness of breath - but they could also indicate other common illnesses such as a cold or flu.
Mr Hancock said he was not aware of any British citizens currently in the Italian quarantine area, but he said anyone there should contact the embassy in Rome.
'We expect more cases'
After the World Health Organization warned that countries should do more to prepare for a possible pandemic, the health secretary said that the UK had already done "a huge amount of work".
"Once it gets to that point there isn't a way to stop it coming into the UK in a big way," Mr Hancock said.
"It is not yet clear that it will become a pandemic - the number of cases in China has been slowing over the last couple of weeks and we are relatively confident that information is correct."
In other developments worldwide:
China reported 508 new infections on Monday as the death toll rose by 71 to 2,663In South Korea, 10 people have died after contracting the virus, with the number of people infected reaching 977
Japan has now confined more than 850 people with the virus, mostly from the quarantined Diamond Princess cruise linerThree more deaths have been recorded in Iran, state media says, bringing the death toll there to 15
Mr Hancock said if other countries where the number of cases are growing "get their response right", there is still the possibility that the outbreak will remain small in the UK.
He said: "We still expect more cases but at the moment there are 13 cases in the UK."
The latest cases in the UK were four passengers who returned from the Diamond Princess cruise ship, which had been quarantined in Japan.
They were among 30 Britons and two Irish citizens who arrived at Arrowe Park Hospital in Wirral for two weeks of isolation on Saturday