George H.W. Bush has been in and out of the hospital in Houston since early November. (Tom Pennington/Getty Images)

Former president George H.W. Bush remains in intensive care at a Houston hospital on Friday, where he has been since developing a fever since Sunday. 

A spokesman for the family says that Bush's condition is improving and that he is alert and in good spirits -- even singing. Doctors continue to be cautiously optimistic. 

The 88-year-old has been in and out of Methodist Hospital since early November, battling a severe cough. Doctors expected he would be able to go home for Christmas. But while undergoing physical therapy to rebuild his strength, Bush developed a rising fever that forced him into intensive care. 

"President Bush remains in the intensive care unit at Methodist Hospital, where he continues to improve. The President is alert and, as always, in good spirits -- and his exchanges with doctors and nurses now include singing," spokesman Jim McGrath said. "The Bushes ... like their doctors, are cautiously optimistic that the current course of treatment will be effective."

Bush's chief of staff, Jean Becker, told family and friends in an e-mail obtained by CBS News that Bush  "would ask me to tell you to please put the harps back in the closet." The former president "has every intention of staying put," she added. 

Bush served in the presidency from 1989 to 1993. The oldest living former president, he marked his 85th birthday in 2009 by skydiving and was hoping to do the same for his 90th. However, he has developed a form of Parkinson's disease that limits his mobility.