Here is how it works at my carrier. If you call in sick you lose the whole trip, unless it transits your domicle. If it transits your domicle and you become well, you can pick up the trip later on. If you lose a trip that doesn't transit your domicle, and you become well, you can make yourself available for reserve and not lose subsequent sick leave.
We get 71.5 hours of sick leave per year...that will cover 11 to 14 days of flying, depending on your trip. Unused sick leave is bought back by the company once per year at your hourly rate.
If you call in sick in domicle, the company can easily put a reserve on your trip on short notice. If you call in sick during the middle of a trip in say, Seattle, the company may be able to find someone to fill in for you on short notice. They can call around to off duty crewmembers in that area and, if you answer your phone, make you cover the trip if you are "available".
If you call in sick a short time before reporting for duty in the middle of nowhere...chances are that airplane isn't going anywhere.
In my 16 years of 121 airline flying I've had two interesting sick experiences.
One was when I was flying Convairs and should have called in sick. As it turned out, I puked all over as soon as we landed in LAS and, let's just say, was not at my best on the approach. Live in learn....
The second deal was when I ate something bad in ONT and had to call in sick a couple of hours before showing for a trip. The plane didn't go. They figured out some other way to move the boxes and I never heard another word. Since there was no one around to take my trip and I had purged the illness by the next morning, scheduling elected to let me continue the rest of the trip that night. It was best for them and best for me.
Boys and girls. Don't fly sick and don't fly fatigued...and be sure to show well rested and able to go for 16 hours if you've had legal crew rest.