They did. Chickens and turkeys, anyway. https://www.vox.com/future-perfect/23053296/bird-flu-chickens-turkeys-cull-depopulation-ventilation-shutdown Horses and donkeys are not considered a major food source (yet). Nor are sheep or goats. As for pigs, another swine flu (H1N1) on the way?
A pig influenza, AKA African Swine flu, has decimated pigs in Asia, including in the Philippines.
Bird flu has been a problem for poultry for years
And right now Indonesia has an outbreak of foot and mouth disease.
All these things are ongoing problems as old time farmers can tell you. The problem is modern farming with thousands of animals being raised together so they can spread disease to each other.