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Two more migrant boats have been intercepted in the 10th and 11th such incidents since Christmas.
The Home Office says four men were picked up in a dinghy in the early hours today and another eight in a small boat six hours later.
Both incidents happened off Dover, and now Home Secretary Sajid Javid has called the rising number attempts by migrants to cross the English Channel a "major incident".
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He has asked for an urgent call with French interior minister Christophe Castaner to discuss the issue.
Dover MP Charlie Elphicke said: "For too long the Home Office has not been taking this seriously enough and the crisis has continued to escalate to unprecedented levels.
"So it's welcome that the Home Secretary has rightly declared the situation a major incident, taken personal control and appointed a Gold Commander.
"Now we need a clear strategy to defeat the traffickers.
"Let's start by bringing back our cutters to the English Channel."
A Home Office spokesman said: "At around 3am, HM Coastguard received reports of a RHIB (rigid-hulled inflatable boat) containing four people off the coast of Dover.
"Border Force was deployed and four adult males, one Syrian and three Iranian, were brought to Dover.
"At around 9am the Maritime Intelligence Bureau were notified of a small boat with eight people on board.
"Border Force was deployed and eight Iranian adult males were brought to Dover."
All 12 men were given medical assessments and transferred to immigration officials for interview.
A spokesman for the Maritime and Coastguard Agency said: "We have been assisting Border Force with an incident off the Kent coast today
"HM Coastguard is committed to safeguarding life around the seas and coastal areas of this country.
"Attempting to cross the Channel in this way is extremely dangerous and they [organised crime groups] are putting their lives at risk" - Immigration Minister Caroline Nokes
"We are only concerned with preservation of life, rescuing those in trouble and bringing them safely back to shore, where they will be handed over to the relevant partner emergency services or authorities."
Immigration Minister Caroline Nokes said: “The number of incidents over recent days is deeply concerning. Some of this is clearly facilitated by organised crime groups while other attempts appear to be opportunistic.
“We are in close and continual contact with the French authorities and law enforcement partners including through the new UK-France Co-ordination and Information Centre which opened in Calais at the end of last month to tackle criminality at the border.
“Attempting to cross the Channel in this way is extremely dangerous and they are putting their lives at risk.
"People should also be in no doubt that Border Force and the Police Aux Frontieres are working together 24 hours a day to prevent attempts, protect life and prosecute offenders.
“At the same time the UK’s NCA-led Organised Immigration Taskforce is working to stop people smuggling at source.”
Responding to Home Secretary Sajid Javid's comments about the rising number of refugees and migrants attempting to cross the Channel, Steve Valdez-Symonds, Amnesty UK's Refugee and Migrant Rights Programme Director, said: "It's extremely worrying that women, men and children have been compelled to undertake such very dangerous sea crossings in an attempt to find a place of safety and security.
"The Home Secretary needs to understand that we’re in this situation because of the persistent refusal of our Government and others to share responsibility in providing asylum and safe passage to people fleeing desperate situations.
"This has left people with no other option than to take extreme risks to find safety - often at the hands of ruthless or careless smuggling gangs."
These latest two incidents mean that a total 79 people who have tried to cross the Channel to reach the UK.
There were five incidents on Christmas Day involving 40 people and three more people were picked in the early hours of Boxing Day, mid-Channel.
Three cases yesterday involved a total 24 people.
In all there have been 33 recorded incidents of 270 known or suspected migrants, trying to reach Britain since November 8, nearly all in small craft.