Anthony, 82, was last seen on January 18 at Higham station, heading towards Northfleet via train.
He was reported missing on January 21.
Police describe him as being white with grey hair and grey/blue eyes 5’10 and of medium build.
He left wearing grey trousers, brown shoes and a burgundy jumper, complete with a long beige coat and a walking stick.
CCTV footage released by police shows missing Anthony Brown heading for Northfleet train station on January 18.
Police revealed he boarded a 4.50pm train, arriving at Northfleet by 5.02pm.
Investigating officer Insp Lee Cheesman said: “We now know that Mr Brown got off the train at Northfleet station at around 5pm but we do not know where he went to next.
“While he may have travelled on another train and maybe elsewhere, I want to ask everyone in the Northfleet area to check their garden sheds and outbuildings, in case he may need assistance.
“We are carrying out a thorough search to locate Mr Brown.”
Anthony, who is from Reading, had been in the vicinity after his car broke down on the M25.
He has links to Gravesend, Woodley in Berkshire and Richmond, London. He may also have travelled to Surrey.
Anyone with information is asked to call 999 quoting reference 21-1299.