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28 September 2015 Last updated at 12:13 BST
Many have travelled from countries such as Eritrea, Syria and Sudan and have sought asylum in the UK.
BBC Look East found that in Bedfordshire 389 people were arrested in the first eight months of this year - a rise of 82% compared with the same period last year.
Bedfordshire Police said the increase was "unprecedented".
In Northamptonshire the latest figures show a 79% increase between August and July 2013-14 and the same period in 2014-15.
In Cambridgeshire arrests are up 73%.
The programme followed the route many of the migrants are taking, in lorries from Calais, across the Channel, and then up the M1.
A rise of up to 82% in the number of people being arrested after arriving in the East of England is reported by police.