Police are trawling through 117 reported sightings of the missing student nurse Owami Davies amid suspicions that she could be sleeping rough.
The last confirmed sighting of the 24-year-old from Essex was in Croydon, south London, on 7 July, and she has now been missing for more than six weeks.
Five people have been arrested in connection with her disappearance, but a senior police officer working on the investigation said on Monday that there is “no evidence that she’s come to harm”.
Detective Chief Inspector Nigel Penney told reporters: “We’re still hopeful that we’ll find her alive and well.”
Asked whether she could be sleeping rough, he said: “We’re open to all possibilities and that is one current working hypothesis, that she is in that current situation with her life.”
Ms Davies left her home in Grays, Essex, on 4 July and was reported missing to Essex Police two days later.
She was found asleep in a doorway in Clarendon Road in Croydon on 6 July while waiting for a friend, but told Metropolitan Police officers that she did not need help and left.
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The officers who spoke to her did not yet know that she was a missing person.
The Independent Office for Police Conduct, the police watchdog, is considering whether to investigate the police contact with Ms Davies that day.
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She is believed to have no money left on her Oyster card, used to pay on public transport, and no access to her phone or bank cards.
Around 50 officers are currently working on the investigation, and they are getting advice from the National Crime Agency.