Dan,
Here is a link to various studies about the use of carpets in homes
www.hostdry.com › pdf › IAQ_research
As you will find out after reading these articles carpets can serve as big filter systems, perhaps aiding in homes healthy environments - if taking appropiate steps to maintain them adequately. However if your concerns are related to an existing health problem it would be wise to have the individual suffering of such problem tested for allergies -if that is the case- as opposed to trying finding the source first. If mold is the concern, walls and ceilings are actually a better host to them than carpets and perhaps testing these areas would be wise as well before looking into carpet issues. When looking at sources of respiratory problems in a residential environment, while carpets do represent a possible source, the risk of other elements in the household and other particles in the air and physical structure of the building such as wood, insulation, and the spaces that these confine should be looked at in first place