Stop counting sick days
Do you or your employees have a fixed number of sick days allowed per year?
Do you really want to limit the number of sick days?
No, you don't, even if you think you do.
You don't want sick employees at work
First of all, a sick employee is not very productive. He won't do as much work as he usually does on normal days and the quality of the work won't be as good. Furthermore, the employee will stay sick for longer since his body won't have all the energy it needs to fight the sickness which leads to more unproductive work.
However, the worst threat is that he's going to contaminate his colleagues who will get sick and then take a day or more off work to rest. One day at home for an employee could save many days, so you should encourage your employees to stay home when they're sick.
Employees can count too
Depending on the sick days scheme in use, the employer may be paying for the allowed sick days both used and unused. Most employees will claim their sick days pay if they didn't use all of them. If you don't count sick days and let them use them as needed, you wouldn't need to pay the healthy ones more.
What about the fakers
Yes, this system opens the doors to fakers and I understand that it's not realistic for a big majority of companies, especially the bigger ones. Fakers could abuse the system and take sick days when they are not sick, taking more than you would have allowed. Fakers are either employees that you don't want or a symptom that working for you is boring. It's your responsibility to provide a stimulating working environment and to weed out the employees that don't care enough for their job to come into work.