Julia Skala left her London home around 1pm on 8 January and has not been heard from or seen since.
In a direct appeal to Julia on Wednesday, her family said: “Julia, you are loved and missed terribly.
“No matter what the reason was for you to leave, you aren’t in any kind of trouble and we just want to know you are safe. Please, just let someone know you are ok.”
The teenager was reported missing by her mother on the morning of 9 January.
Although Julia lives in Mitcham, she also has links to Lambeth, Camden and Islington and it is possible she has travelled to these places via public transport.
Detective chief superintendent Clair Kelland, in charge of policing for Merton, said: “Since Julia was reported missing, our officers have been working around the clock to try and find her.
“We have gathered and examined hours of CCTV and doorbell footage, spoken to many people who know her and carried out enquiries with those who live in the area to establish whether they have any information that might help our investigation.
“Julia has never been away from home for this long and obviously her family are extremely worried.
“While there is no evidence to suggest she has come to harm, we are growing increasingly concerned about her welfare. She did not take her mobile phone with her and there has been no activity on social media or known contact with her friends.
“We also do not believe she had a credit or travel card, or any cash when she left the house.”
Anyone with information should call police immediately on 999 quoting the reference 24MIS000879. Share non-urgent information by calling 101. You can also contact the Missing Person’s charity anonymously on 116 000.