If you arent willing to make a barn, then be willing to make a lot of supplements.
Barns are ridiculously expensive though, needing like 900000000 boards.
The only time I have had any animals die is if I didn't have the medicine already made or the materials to make said medicine. I do agree that the chance of sickness is a little silly since if you feed an animal medicine they shouldn't get sick again for at least another day or so instead of them all being sick again 4-6 hours later.