Yoi lived a normal, happy life with his parents until his illness manifested; by the time of the Survival Game, he was hospitalized. His deteriorating condition weighed heavily on his father, inspiring him to pursue godhood in the belief that he could cure Yoi's illness. He also struck a bargain with Minene Uryu for his son's sake, promising her access to the Sakurami Police Department's data files in exchange for her protecting his family in the event of his own death.
One of the major changes between the first and second worlds involved when Keigo learned of Yoi's prognosis. In the first world, he only discovered his son's grim prospects near the end of the Survival Game; in the second world, Muru Muru went out of her way to inform him well in advance of when he was "supposed" to. This led to Keigo betraying his alliances much sooner than Deus Ex Machina had anticipated, tipping him off to Muru Muru's interest in manipulating the events of his succession.
When Minene goes to the third world created by Yuno Gasai's time leap, she anonymously calls Keigo to warn him about Yoi's heart condition, recommending that he seek treatment immediately. It is revealed in the credits of the anime's last episode that Yoi's condition is successfully treated.