Death might be a little too far for me, unfortunately. But the next update will take the health issues further and feature hospitalisation!
Don't worry, hospitalisation won't appear unless you have the 'Health Issues' game setting from the Jobs & Home update enabled.
It'll only occur if you ignore a character's health, to add more incentive to go to the doctor and take medication/use the CPAP machine. The hospitalisation itself is a break from gameplay to recover and doesn't result in any permanent effects.