No special measures are normally needed as far as hygiene and the sharing of the breasts. Normal baths and showers will suffice. There is no reason to clean the nipple area between children and, as always during lactation, use of soap on the nipple is not recommended except under special circumstances.
In general, even if one child becomes sick, there is no reason to take any special precautions. Lots of hand washing is more important for limiting the spread of infection, and the anti-infection properties of breastmilk will also help to prevent the spread of illness.
An exception to this is thrush – if anyone involved in the nursing relationship gets thrush, it is important that everyone be treated to prevent reinfection. One source suggests that assigning each child a separate breast might also help to prevent the child without thrush from becoming infected.
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By the time symptoms occur in a viral illness, the two nurslings will have been in close contact for days (both in the home and at the breast), and thus the well child will have already been exposed to the illness. In addition, your breasts will have already started to produce antibodies specifically for that illness. These antibodies will help the sick child get well quicker and will also help protect your other child from illness.
There is some speculation that tandem nursing actually hastens the process of antibody production in breastmilk. The breast starts making antibodies soon after exposure to the illness via your child’s breastfeeding. If an older child who is not nursing gets sick, the breast will not start making antibodies until mom picks up the virus independently or from baby’s mouth (if baby gets sick) — tandem nursing cuts out the middleman in this process. Washing the breasts between nurslings might even reduce the breasts’ exposure to the pathogens and therefore reduce the breasts’ efficiency at manufacturing antibodies.