Former South African President Nelson Mandela, at home after a long hospital stay, is alert and playing with his grandchildren, his granddaughter Zaziwe Manaway told CNN exclusively Friday.

Mandela, who was born in 1918, is aware of social media rumors that he's close to death, Manaway said.

"That is absolutely not true. My grandfather is well," she said. "It can be very, very hurtful for us to hear these messages out there in the social media that our grandfather is going to go home to die. It is insensitive."

Mandela is revered in his country, CNN's Robyn Curnow writes, because he reminds South Africans of how far they have come. The former president embodies the South Africa that was promised in the election of 1994, and many South Africans worry that their country no longer reflects the democratic ideals Mandela spent his life advocating.

Manaway said she wants to appeal to people spreading rumors to stop, and to be more sensitive to the family and to Mandela, a global icon of peace and South Africa's defeat of apartheid.

"My grandfather still wakes up in the morning (and) reads the newspaper," she said. "So he is also aware of what is being said around him."

Mandela was treated for an acute respiratory infection in 2011. He was hospitalized for a lung infection on December 8, and on December 15, he underwent surgery for removal of gallstones.

Because Mandela is in his 90s, it's understandable that "once in a while, he needs medical care and medical attention," another granddaughter, Swati Dlamini, told CNN. "And we're very grateful he's surrounded by the best medical team. He's very well taken care of, and he's very comfortable, and he's very happy."

On Wednesday, a spokesman for South African President Jacob Zuma told media that Mandela had been discharged from the hospital and would continue receiving treatment at his home in Houghton.

He's received well wishes from around the globe, his granddaughters said.

"We'd just like to thank the whole world for sending us messages and keeping us in their prayers," Manaway said.

"We know that people worry and we know that people are concerned," Dlamini said. "But, you know, we'd just like people to know that he's doing very well and he's in good spirits and he's very cheerful."

Mandela has not made a public appearance since the 2010 World Cup hosted in his country. In 2011, South Africans got a rare glimpse of him when he voted in local municipal elections at his home in Johannesburg.

There's been secrecy surrounding his health.

"He has every right to his privacy," Dlamini said. "As the family, we call on people and urge people to give us the privacy to deal with whatever we're going through as a family in private."

"I think people need to remember that my grandfather played a huge role -- and not only him, many other South Africans played a huge role -- to get us where we are now," she added. "My grandfather said this when he was resigning from public life -- it is now (up) to South Africans to take this country forward, that a legacy like his should be carried by as many people as possible."

Dlamini said she is telling her children about their great-grandfather's life.

He was the president from 1994 to 1999, making him the first president chosen in a democratic election and the country's first black president. In his younger years, he fought against apartheid and was sentenced to life in prison for his activism. He spent 27 years behind bars and was released in 1990. Mandela was awarded the Nobel Peace Prize in 1993.

"He's always dedicated his life to the struggle," Dlamini said. "For our children to be able to spend time with him now ... they can sit on his lap and hear stories from him. It's great; something we didn't have growing up."

On Mandela's birthday in July this year, former U.S. President Bill Clinton reflected on the conversations the two have had. Mandela kept his wife and daughter in mind, Clinton said.

"He didn't call me a single time, not once, when he didn't ask about Hillary (Clinton) and Chelsea," Clinton said of their conversations during the time both were in office. "If it wasn't too late, he'd ask me to go get Chelsea, bring her to the phone, ask about her homework."

Clinton said the anti-apartheid icon never lost touch with his humanity.

"I saw in him something that I try not to lose in myself, which is no matter how much responsibility you have," Clinton said, "he remembered you were a person first."

CNN's Kim Norgaard contributed to this report.