Depends on what diseases the cattle had and whether they were transmissible to humans or not.
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No, cattle and cows are often kept together with no problems. If there where any risk of horses catching a disease from cattle than they would not enter horses in gaming classes with cattle (calf roping, team penning etc).
Jersey bull calves are slaughtered for veal. Jersey cows, when they're too old to be productive any more or are culled for productivity reasons, are slaughtered for beef, which is often made into ground beef for hamburger and such. Ultimately, though, Jerseys are not used for meat, but for dairy or milk production.
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Spitting out cud is one of the first signs of disease in cattle. It is often indicative of abscesses or bad teeth. It is also associated with diseases such as tick fever and anaplasmosis.
Often blocky.
Feeder cattle are young cattle (steers and heifers) that are carrying more weight or condition than stocker cattle and are ready to be put on a "hot" ration (or high-energy ration like grain) for finishing prior to slaughter. Quite often these cattle have started out as stockers or backgrounders and have been raised and fed on a high-forage-based ration to the point where they've grown enough and put on enough weight (in both fat and muscle; this is known as "condition") to be considered as feeder cattle. Feeder cattle are often between the 10 to 18 months, depending on the length of backgrounding/stockering phase they went through.
Tapeworms can often spread to humans when people eat the infected meat of cattle, and swine. Other diseases are transferred by insect vectors, often blood-feeding insects that carry the cause of the disease.
In Italy, they were often used to guard cattle from wolves and brown bears.
Texas longhorns can be slaughtered at any age and weight above 200 to 300 lbs. Cows and bulls can be slaughtered at any age, young or old, and heifers and steers are often slaughtered between 800 and 1200 lbs, be they around the age of 8 months to 14 months of age.
Depending on the vaccinations involved, most cattle should be done once or twice a year.
Diseases are often caused by viruses and bacteria or genetic factors.