In Russian animals that died of a disease can be translated as fallen animals e.g. fallen sheep.
Is there a similar term in English?
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The term for farm animals that have died of a disease is, in fact, fallen stock:-
Animals that have died on the farm. These need to be removed as on-farm burial is prohibited
So you would have fallen sheep, fallen cows, fallen chickens (for all I know). A guide as to what to do with them can be found here.