Who comes up with this shit. If someone is sick they shouldn't come to work at all, ever. You should encourage people to stay home so they don't get others sick. If they can manage, have them do a little work from home.

Moving from a small (10 person) company to a building with 1000 people in it I went from almost never being sick to getting sick 3+ times every winter.

Strongly encourage your employees to stay home when they're sick as long as they're getting their jobs done. If they're out frequently or for a long period of time (4+ days) require a doctor's note. That's your sick policy. Done.