Morrowbay asylum would normally be an high-security prisonlike building for the especially troublesome patients with no social ties to the outside world. It is located on a small island in the middle of a large, deep and ice-cold lake. The water of this lake comes directly from the icy peaks of the nearby mountains.

a mental hospital

Before the apocalypse started about 350 people + 70 something staff members. However, once the apocalypse had undeniably started all the staff members ran away and left the patients behind.
Those who managed to escape their cells in search of whatever and didn't try to swim across the lake upon escape right away are the only survivors and have formed the group of main characters.

We take care of your unwanted troubles.

Do as the people in charge say and you'll get food and medicine

France, Pyrenees

a small island in the middle of the lake named Etang Majue

The island was bought in 1976 by William Quintence who had supposedly intended to make it into a private vacation location for his family. However, he disappeared shortly after and his business associate, Tyler Beaumant, claimed the island as payment for a debt.
Beaumant proceeded to build a high-security prison but changed the purpose halfway through as he had reasoned an asylum would be more lucrative

1983: two patients manage to take over the building by locking out the staff, declaring they are now the kings of pumpernickel-land.
1996: ten inmates kidnap a staff member in the early morning and escape to Morrowbay where they take the bus to the large city. They return the next morning, drunk out of their minds. The staff member reported it was a very enjoyable surprise birthday party.
2004 up to now: a "new" inmate manages to convince every other patient in the facility that he is the rightful emperor of the world, the ghost of William Quintence, the devil/angel/god, the other patients are but figments of his imagination. the list is still growing.
2012 up to now: one of the inmates frequently starts fires wherever she goes. no-one knows how she hasn't been confirmed to be near or in possession of flammable materials at any of the times.
2013: a mayor explosion goes off on the south wing of the asylum. Though there is no major damage to the structure or physical damage to inhabitants, an unknown gas has spread through the hallways that puts people to sleep within seconds.

1981

The lake is called Etang Majue (France, Pyrenees) and there is a small village to on the edge called Morrowbay. Founded by some English people trying to make their own fortune in France.