Detective Superintendent Lewis Basford said the baby is “at the heart of the investigation” and that concerns are rising “day by day” for the child’s wellbeing.
He said the public should be aware that the couple have made an effort to remain inconspicuous, and that their “change of disguise” as well as movement and “shielding” of the baby has made it hard to locate them.
“We know they’ve worn masks, we know they’ve changed coats at various times just within the two to three days since the Met Police have taken the inquiry and followed them,” he said.
He added that the couple have “now removed the stroller” and are “more aware and paranoid that actually the police and members of the public will be more cognisant of a baby”.
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In a message to the couple, Det Supt Basford said: “Constance and Mark, your baby has spent the first month of its short life exposed to the elements when it should be safe and warm and, most importantly, seen by medics.
“After a month you must be running low on cash. Please pick up the phone and let us know, at the very least, that you are okay. We are ready to come to you and see that you and the baby get medical attention.
“We know that the baby was still alive on 8 January and finding the baby remains our top priority.”
He also appealed particularly to those who may have spent time in the countryside near Newhaven and elsewhere in Sussex, where the couple are believed to be camping.
He urged that any detail, no matter how small, was important, including if an individual gave the couple directions or saw them buying food, drink or nappies.
Ms Marten, who is from a wealthy family, has lived an isolated life with Mr Gordon, a convicted sex offender, since they met in 2016.
It is thought most of the couple’s belongings were destroyed in the blaze that affected their vehicle.
Since then, Ms Marten and Mr Gordon have taken a taxi to Liverpool, and from there to Harwich in Essex. They have also been seen in east London, before travelling to north London and then on to East Sussex.
Pictures released on 8 January showed the pair walking near Brick Lane, east London, at around 11.45pm where they dumped a pushchair.
When spotted in East Sussex, they were carrying bags containing a two-man tent, sleeping bags and pillows that they had bought from Argos.
The last known sighting of the pair was at 6.15am on 8 January, where they walked along Cantercrow Hill, East Sussex, into the fields beyond.
They were carrying a number of bags and their blue tent, although given the time that’s passed they could have moved some considerable distance.