Kim, who is a singer with boy band JYJ, told his Instagram followers on Wednesday that he was hospitalized after catching the virus, writing: "I'm infected with Covid-19 virus."
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In a now-deleted post, the pop star said he had ignored warnings by the government.
April Fools' Day is celebrated on April 1 every year and is traditionally a day when people play pranks on each other.
Many of the star's 1.9 million followers expressed outrage at his remarks, with one social media user writing: "A prank about coronavirus is not funny but childish and senseless."
The singer later apologized for his post, admitting that his judgment "was not right."
Still, Kim's actions have caused anger -- a petition calling for the singer to be punished was posted to the Office of the President's website, and had been signed by more than 13,000 people as of 5:20 p.m. local time [4:20am ET] on Thursday.
In a statement posted to Instagram, the star claimed he was trying to raise awareness about the virus, which has killed at least 169 people in South Korea, and sickened almost 10,000.
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"First, I sincerely apologize to those who were damaged and to those whose administrative work was hindered due to a post I wrote on social media and due to Covid-19," Kim wrote.
The singer added that he "wanted to convey the dangers of this moment to people who don't listen while there are big and small warnings on various information media and the internet."
When asked by a reporter on Thursday whether Kim would be punished for the post, Yoon Tae-ho, director-general for public health policy at the Ministry of Health and Welfare, said: "Currently, cases of providing false information to an epidemiological investigator or medical personnel during epidemiological investigation or treatment can be punished according to the Infectious Disease Prevention Law."
But he said Kim's case does not fall under these conditions.
"Whether there are other parts for punishment, (we) will have to see, but it's difficult to punish under the Infectious Disease Prevention Law," Yoon added during the press briefing.
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