Steve: I think it could burn itself out very quickly.

Highly likely. Today's BBC radio news says there are reported to have been 1 million cases last week. If that is correct, and it continues doubling every 2 days (unlikely as it would run out of prospective infectees) then a fortnight would mean that the UK population had been affected and were in recovery phase.

Number of available contacts available to be infected must be dropping rapidly, which means that doubling every 2 days cannot be sustainable.