Three of the new cases have been found in South Gloucestershire and are said to be close or household contacts with the two cases previously discovered in the area.
The other case was found in Bradford after the person tested positive late last month having travelled back from Brazil via Paris on February 14.
A PHE statement said: “Contact tracing teams have followed up close contacts of the individual and advised them to isolate and get a test.”
The new cases bring the number of cases of the variant, named P1, detected in the UK to ten – seven in England and three in Scotland.
Bradford Council’s Director of Public Health, Sarah Muckle, said: “We have been working closely with Public Health England and NHS Test and Trace since an isolated case of the P1 COVID-19 Variant of Concern that originated in Manaus, Brazil, related to international travel, was detected in the Bradford District.
“A very small number of local contacts have been identified and all precautions have been taken to minimise the risk of transmission.
“If we are to prevent infections, particularly from the potentially more transmissible COVID-19 variants, everyone should continue to keep their travel – in the UK and abroad – to essential journeys only while national restrictions apply. Hands, face, space should still apply as part of our day-to-day routines.”