Pauline Cafferkey, who is being treated at London's Royal Free Hospital, contracted the disease while working as a volunteer at a British-built Ebola treatment centre in Sierra Leone.
She felt unwell as she returned to Britain and was diagnosed in Glasgow on December 29, before being transferred to the Royal Free, which has the only isolation ward in Britain equipped for Ebola patients.
"We would like to thank all our friends, family and the members of the public who have contacted us with support following Pauline's diagnosis with Ebola. We have been very touched by the kind words," her family said.
"Pauline's condition could remain the same for some time."
Health Secretary Jeremy Hunt said Monday that the 39-year-old's condition was critical but had stabilised.
Cafferkey has been treated with blood plasma from an Ebola survivor containing virus-fighting antibodies as well as an experimental anti-viral drug.
Centred on Sierra Leone, Liberia and Guinea in west Africa, the world's worst-ever Ebola outbreak has infected more than 20,700 people, of whom 8,235 have died.
As of January 4, a total of 838 healthcare workers were known to have contracted the virus, and 495 of them had died, the World Health Organisation said.
