Travellers at airports wear masks in a bid to combat the spread of the virus
Doctors in Scotland are carrying out tests on two people who recently visited Mexico, where over 100 people have died from suspected swine flu.
Twenty cases of the virus have so far been confirmed in the USA.
Officials there have declared a public health emergency, despite the fact that no one has yet died from the disease outside of Mexico.
Cases have now been confirmed in Canada and Spain, with suspected cases reported in France and New Zealand.
Anybody who has returned from Mexico and is displaying any symptoms should, of course, seek advice
Scottish health secretary Nicola Sturgeon
Dr Nick Phinn from the UK Health Protection Agency says medical staff have been told to look out for signs of the virus.
"We've sent out advice to the NHS, which is really saying, if someone has been to Mexico and they've got these symptoms, then this is what we're recommending in terms of investigation and management," he said.
"And we hope that that will help us to detect at a very early stage any possible cases of influenza coming into the country."
Chris Clarke, who's just come back from Mexico, was told not to leave his house in Northamptonshire after he developed flu-like symptoms.
However, him and his family were told they do not have swine flu on Monday morning.
"I was reasonably relaxed and felt statistically the chance of having picked up anything in Mexico was relatively low," he told BBC News.
"But of course I have two young children and it's concerning to think you might have brought something untoward back in to the household."
1918: The Spanish flu pandemic remains the most devastating outbreak of modern times - infecting up to 40% of the world's population and killing more than 50m people, with young adults particularly badly affected
1957: Asian flu killed two million people. Caused by a human form of the virus, H2N2, combining with a mutated strain found in wild ducks. The elderly were particularly vulnerable
1968: An outbreak first detected in Hong Kong, and caused by a strain known as H3N2, killed up to one million people globally, with those over 65 most likely to die
Londoner Jeremy Bailey is on holiday in Mexico City and revealed what's going on in the central American capital.
"The last couple of days you've seen the military handing out masks to various people on the Metros and in busy parts of the city," he said.
"They've closed down the museums and sporting activities and a lot of restaurants have closed down."
Doctors in the US have confirmed that eight children from the same school have swine flu.
There are also cases in California, Texas, Kansas and Ohio.
Officials say the public health emergency sounds more serious than it is - and their advice is to wash your hands at stay at home if your get sick.
But government disease specialists are warning it's likely swine flu will claim lives in the US.
Ten pupils in New Zealand are believed to have the virus, after returning from a language trip to Mexico.
David Hodge, the head of Rangitoto College in Auckland, says they hope to know within the next 48 hours if children from his school have swine flu.
"Samples have now gone to Melbourne, Australia, to see if it's a normal A strain or whether it's this dreaded swine flu that is causing the major concern," he explained.
US officials are urging people to be sensible and take precautions
There is no hard evidence that swine flu has reached the UK yet, though.
Scottish Health Secretary, Nicola Sturgeon, stressed there had so far been no confirmed cases in Scotland.
"We're acting on a precautionary basis," she said. "The patients in question are not particularly ill, their symptoms are very mild, their clinical condition is showing absolutely no cause for concern, but given the international situation, it's appropriate that we carry out these tests."
Sturgeon explained that Scottish health officials were well prepared for a possible flu outbreak.
"We have, for example, significant stockpiles of the anti-virals that are proving effective in Mexico and the US.
"Anybody who has returned from Mexico and is displaying any symptoms should, of course, seek advice."
Justin McCracken, the Chief Executive of the Health Protection Agency, says they're monitoring the situation closely, and explained he believes they're ready to deal with any outbreak.
"All the preparations have been going on for years," he said, "and the considerable expenditure in buying these drugs will put us in a very good position if the virus is found here, and we definitely think we do have enough of the drugs."
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