Anti-viral drugs have been issued to sufferers
Ten new cases of swine flu have been confirmed in England, taking the total number in the UK to 65, the Health Protection Agency has said.
Another seven adults and three children have been diagnosed with the virus.
The news comes as a UK analysis concludes the World Health Organization was right to raise the alert over a potential global flu pandemic.
It says the latest outbreak is likely to be comparable to the pandemics of the 20th century.
The study, led by Professor Neil Ferguson, of Imperial College, London, is published in the leading journal Science.
It finds that - as suspected - the virus is more infectious than normal.
Seasonal flu normally infects one in ten of the population. So far swine flu has infected a third of the people that have come into contact with it in Mexico.
However, Professor Ferguson admitted it was difficult to quantify the impact on human health at this stage.
His study suggests swine flu could kill between four in every 1,000 infected people, and 14 in every 1,000.
Professor Ferguson said: "The World Health Organization was correct in its judgement that this is a virus that should not be ignored, but these figures suggest at this stage it is not going to be catastrophic."
Of the new cases in England, four are in eastern England and three are in the north-west. Three more are in London, including a pupil at Hampton School in south-west London.
The school has been closed for a week from Monday as a precaution. Four other schools in England have reopened.
Hampton School, which is an independent boys' secondary, shut its doors after a Year 7 pupil fell ill after travelling overseas.
The school said the boy was recovering well at home, and GCSE and A-level exams, the bulk of which commence this week, would continue as planned.
It is offering anti-viral drugs to all children in Year 7, any staff who had close contact with the pupil and any other children who shared school coach journeys with him.
These include some girls from the nearby Lady Eleanor Holles School in Hampton.
A Department of Health spokesman said on Monday that all infections in the UK so far had been "mild". Thanks to early diagnosis and treatment with anti-virals, the spread of the virus is being limited and symptoms reduced.
"But it is right to prepare for the possibility of a global pandemic. The UK's arrangements are continuing to ensure that we are well-placed to deal with this new infection."
The two cases in London are both connected with Alleyn's School in Dulwich, which was closed on 4 May after five pupils were confirmed with the virus.
NHS London said the two latest cases were a 12-year-old pupil and a parent.
Despite the development, the school reopened on Monday and the Health Protection Agency stressed that because of the time taken to diagnose swine flu, all 10 newly-confirmed patients may already have recovered and be symptom-free.
NHS East of England said the four cases in its area included a man from North Weald, Essex, who had close contact with an already confirmed case and a child from Canvey Island, also in Essex, who recently visited Mexico.
Another case involved a man from Lowestoft, Suffolk, who recently visited Florida and the fourth was a woman from the Huntingdonshire district in Cambridgeshire.
Investigations are continuing into the origin of her infection.
In the north west, two of those infected are adults who have returned from abroad and the third adult is a case of close contact.
Three other schools also reopened on Monday. They are the Dolphin School in Battersea, Paignton Community and Sports College in Devon and Downend School in Bristol.
Alleyn's has reopened despite another pupil being confirmed with the virus
Antony Faccinello, senior deputy head teacher at Alleyn's, said all staff and pupils who had not developed infections and were symptom-free could now return.
He said the school was in close contact with exam boards and had already made arrangements to postpone some oral and practical exams.
But he stressed that the impact of the closure on students sitting exams was expected to be minimal: "Some people take the view that most courses last two years and this is a week at the end.
"I think a lot of it is about reassuring people."
Meanwhile, holiday companies Thomson and First Choice said they were cancelling all flights to the Mexican resorts of Cancun and Cozumel up to and including 18 May.
They also said that their last holidaymakers still in Mexico would be returning home on Monday.