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Improve pre-milking hygiene. Reduce the amount of time goats are milked. Identify and segregate or cull affected goats.

Article:
Improving the hygiene of pre-milking and milking conditions can significantly decrease the rate of mastitis transmission. This includes better sanitation and cleanliness in the goats' housing and in the milking area.  Goats should not be overcrowded. Every goat should have sufficient room in the barn as well as in the yard. The paths between your milking area and the goats' housing or fields should be kept clean. The paths should be free-draining and should be kept clear of feces and slurry. Do a dry wipe and a thorough washing of the udders and teats with clean, potable water. Make sure you also wash your hands before and after milking. Use teat dips and sprays to disinfect the mammary glands before milking, and keep any milking equipment you use clean and sanitary. Some preliminary studies suggest that there may be a link between mastitis outbreaks and the amount of time that goats are attached to milking units. Though this may not conclusively prevent cases of mastitis, it is worth further consideration and it may warrant minimizing the time spent hooked up to a milking unit. If any of your goats have mastitis, they should not be kept with the rest of the herd. Most sanitation and mastitis-prevention regimens recommend either isolating the affected goats from the herd or culling them to prevent recurring outbreaks.