Mr Johnson confirmed that a man taken to hospital after arriving at Heathrow on a flight from Mexico City yesterday did not have the H1N1 strain of the virus, which has been blamed for more than 80 deaths in the central American country.
The Health Secretary told BBC1's The Politics Show he had no doubt that there would be more cases of travellers coming into the UK with flu-like symptoms and promised that they will be examined "very, very quickly" by the NHS. But he said that, so far, no cases of swine flu have been confirmed in Britain or the rest of Europe.
The NHS has a stockpile of more than £500 million worth of the Tamiflu anti-viral drug which has proved effective on patients in Mexico, and scientists are working on developing a vaccine against the new strain, said Mr Johnson.
The World Health Organisation rated Britain as one of the two countries best prepared for an outbreak, alongside France.
The global health watchdog today warned countries around the world to look out for unusual flu cases as the outbreak spread to the USA and probably New Zealand, where a group of 10 students is ill after a school trip to Mexico. Israel's Health Ministry has reported one suspected case and France is investigating four possible cases.
A male air steward was taken to hospital yesterday after complaining of flu-like symptoms when he arrived at Heathrow on board a British Airways flight from Mexico City, but laboratory tests at a north London hospital today showed that he does not have swine flu.
WHO director-general Margaret Chan said the outbreak, caused when the H1N1 strain associated with pigs crossed over to the human population, constituted a "public health emergency of international concern".
Nations will be expected to step up reporting and surveillance of the contagious respiratory disease, which she said had "pandemic potential".
Health authorities were monitoring the situation as tests were carried out to assess the potential of the virus to spread.
Mr Johnson told the BBC: "The WHO says that the UK and France are the two best-prepared countries in the world and we have about half a billion pounds worth of the anti-viral which has worked in Mexico.
"The good news from Mexico is that, given anti-virals like Tamiflu early enough, the people who get this recover.
"We have got a whole range of measures in place. The thing that has to happen now is the WHO must give an analysis that shows whether there is a propensity for this to spread.
"We have a stockpile of the drugs and we have an agreement with a manufacturer so as soon as the scientists can find a vaccine, they can produce it."
Mr Johnson ruled out a travel ban to try to prevent the virus being brought to the UK from Mexico by plane.
"That is usually futile and doesn't work at all, and it causes so many other problems that it really is not part of the planning," he said.
The advice for anyone experiencing symptoms including fever and fatigue after travelling abroad was to stay at home and contact the NHS Direct telephone helpline, he said.
"If you have flu-like symptoms, don't go to your GP but stay at home and call NHS Direct... The whole point about these kinds of epidemics is you don't want to spread them and you spread them by going out and mixing with other people."
The UK Health Protection Agency said it was keeping a close eye on the situation.
A spokesman said: "The HPA is monitoring this situation closely and is working with the UK Government to review the current incident and any threat it poses to UK public health."
There was currently a "very low level" of flu activity in the UK, according to the HPA, which added that H1N1 was treatable with antiviral drugs such as Tamiflu and Relenza.
"The HPA and the NHS have systems in place, which will alert public health authorities of any unusual strain circulating in the UK," it said.
Britons are not currently being advised to avoid travelling to affected areas of Mexico and the US, although anyone visiting those destinations or who has recently returned should consult a doctor if they experience flu-like symptoms.
To prevent the spread of illness, the HPA advised people to cover their nose and mouth when coughing or sneezing, to throw away dirty tissues promptly and carefully, and to wash hands and surfaces which are regularly touched.
Scotland's health secretary later revealed that two people have been admitted to a Scottish hospital after returning from Mexico with flu-like symptoms.
Nicola Sturgeon said the pair - who arrived in Scotland on April 21 - had not been in an area affected by recent outbreaks of swine influenza.
She said: "The patients have displayed mild flu-like symptoms and their current condition is not causing concern."
Britons planning to travel to Mexico were warned about the swine flu outbreak there.
The latest advice from the Foreign and Commonwealth Office stated: "Visitors to Mexico should be aware of an outbreak of influenza.
"The Federal Health Ministry issued a nationwide alert on April 22 after learning that a number of people had died from what appears to be a new form of influenza.
"Cases have been reported in Mexico City, together with the states of Oaxaca, San Luis Potosi, Mexicali and Baja California.
"Travellers should consult a doctor immediately if they show signs of flu-like symptoms.
"The Mexican Secretariat of Health has advised people to avoid large crowds, shaking hands, kissing people as a greeting, or using the subway.
"Maintaining a distance of at least six feet from other persons and frequent hand washing may decrease the risk of exposure."
Nicky Paull, president of the British Veterinary Association, said scientists would work together to study the virus.
She told Sky News: "Our role will continue in the background and the ongoing surveillance that we are doing in Europe and particularly in the UK will continue, but certainly from what we are hearing here today it very much now appears to be a human problem.
"We have amazing facilities in the UK, for instance at Weybridge, the laboratories there will be working on this.
"I am sure that the veterinary profession, the scientists if you like, will be working on this alongside the human ones because of the expertise that we will already have in identifying influenza in animals.
"So they will be looking at the fine-tuning and looking exactly at the genetic make-up of these viruses."
Mrs Paull added that eating pork products did not pose a risk, especially if people chose British meat.
"There is always concern because I know during avian influenza there was concern about handling meat," she said. "I am almost certain that there is nothing to be concerned about but where people are concerned, if they go for the quality standard mark they know that's British and we know for sure that in Britain we do not have that problem at this moment in time."
The Government's chief veterinary officer Nigel Gibbens added his reassurances, saying: "There is continual surveillance of pigs in this country and there is currently no evidence of this variant of the disease.
"Swine influenza cannot be transmitted by eating pork products.
"Eating properly handled and cooked pork and pork products is perfectly safe.
"Existing EU rules which prevent imports of all live pigs and pig meat from Mexico into the EU will continue to be upheld."
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