The 46-year-old was last seen at Cross Lane Hospital in Scarborough on the evening of Wednesday May 8.
Peter's wife, Natalie, said: "If anyone sees Peter please take a discreet photo, call the police and try not to let him out of your sight."
Peter, from Whitby, is described as white, about 6ft 1in tall, of muscular/athletic build, unshaven with greying/brown hair which is slightly thinning.
He was last seen wearing a blue waist length jacket with a blue hoodie underneath, loose grey cotton jogging bottoms, and blue Nike trainers.
It is believed that he could also have been wearing a blue baseball cap and may have changed into blue jeans.
However, police believe he may be sleeping rough so his appearance may be more dishevelled than in pictures.
Since Peter went missing there has a huge community response in the bid to find him and get him back to his family.
A Facebook group - Pete Brown Search & Updates - has been set up to share new information and search groups in the area.
He has also been listed on the Missing People website.
There have been several sightings of a man possibly matching Peter's description in the area but unfortunately they haven't been confirmed as Peter.
Extensive police searches are continuing to locate Peter. Anyone with information about Peter's whereabouts is asked to contact police on 101 and quote reference number 12190083137.