I have an employee who shows a pattern of calling in sick directly before or after his normal days off. I feel very strongly that he is being dishonest, but he knows exactly how to 'work the system'.

There may be several reasons for this employee working the system.

They could range from just lacking motivation to go to work, a mental health issue undiscovered, lack of engagement with their role, feeling lonely, poor understanding about the imperative of loyalty to one’s employer and maybe other priorities that are deemed more important than going to work.

I agree, from your perception, he is ‘working the system’ and you feel that he is taking advantage of you.

Nevertheless, as his employer, my advice would be for you to arrange a time to discover what is the background to his behaviour, noting your observations about the time taken off from attending to his responsibilities at work.

This should initially be a conversation of discovery. Furthermore, it will be important for you to have the facts that demonstrate the time taken off, patterns of time taken, responsibilities at work that have been neglected or forgone and impact that he has left with his colleagues.

More formally, you might want to include in this encounter or a subsequent one, a performance review against his previously agreed objectives. Clearly, is he delivering on his responsibilities or not! 

In short, you need to understand why he is taking time, agree to changes in behaviour and set a new course of expectations. If this is not possible then you may need to look to exit him.