Officers are becoming increasingly concerned for the welfare of Andrew Lambert, 58, from Camborne, Cornwall, who is known to have visited the south east of the country recently.
Andrew (pictured) went missing from Treliske Hospital in Truro, Cornwall, on Tuesday 5 May at around 2.30pm and has been travelling around the country since then.
He suffers from a condition which causes amnesia so he may be in a confused state, and may not even respond to his name.
Officers believe Andrew has visited Broadmarsh Centre in Nottingham; Wisbech in Cambridgeshire; Lowestoft in Suffolk; Woolwich, Docklands Light Railway (DLR) and Stratford stations in London; Eastbourne in East Sussex; and West Malling in Kent since he went missing.
Andrew is known to have visited Oxford on Monday 18 May at around 3pm which is the most recent information officers have of his whereabouts.
Andrew is described as 6ft tall, of medium build, with grey hair and a beard. He wears glasses, walks with a limp, uses a stick and usually wears shirts and jeans. He was last seen carrying a sleeping bag.
As well as having family in Norfolk and Kent, Andrew also has links to Hertfordshire.
Chief Inspector Mark Bolt said: “Andrew has disappeared before and has been found in locations across the UK such as South Wales and a forest in Somerset.
“We believe that he is using the bus as his main method of transport and we urgently need to speak to him to check on his welfare.
“Officers are focussing their efforts in the south east of the country at present and are appealing for anyone who knows of his whereabouts to get in touch.”
Anyone who has seen Andrew since he went missing or knows of his whereabouts is asked to call Devon and Cornwall Police on 101 immediately, quoting reference 252 of 7 May.