Some companies operate a system where each employee is allowed a fixed number of paid sick days per year, say, five.
If the employee doesn't take them off as a result of sickness, they are allowed add them to their holidays at the end of the year.
Maybe the introduction of something like this would help. First of all, it removes the expectation of unlimited paid sick days and secondly provides an incentive not to use them to recover from hangovers, etc, etc.