Plantar warts (verrucae plantaris) are usually caused by type 1 human papillomavirus (HPV).
The route of infection is thought to be through an abrasion or cut
The bottom line is that it's not clear how infectious they are and under what conditions. The one study that is always referenced made a conclusion that public shower use heightened risk. However, that study was rated as "fairly low on the level of evidence scale" by the Healthcare Journal "Bandolier". This Thrive-Online article states "warts are an infectious disease of low infectivity caused by a virus". The same article also says that for prevention "avoid direct skin contact with another wart".
An extensive search of the medical literature revealed only one study indicating a possibly greater risk of plantar warts (verrucae plantaris) due to communal shower use. It is interesting that Bandolier, a journal of evidence-based healthcare, also found only the same study when they performed a search. Those conducting the study collected no data on whether the shower users wore sandals or not. In this article Bandolier noted that "In terms of levels of evidence, this study comes fairly low on the scale."
This (really detailed) report states "Evidence that socks prevent the transmission of infection is circumstantial and little is known about the infectivity of subclinical warts". It also says that "Cutaneous warts are harmless and transient" and that "Treatment is unnecessary".
In general, (here's our 2 cents) if you are really worried, have your children wear
completely enclosed shoes whenever they are using public shower facilities. Other than
that, don't worry. No study has been done on this, but we suspect children that
habitually go barefoot are less likely to get the virus that causes Plantar Warts since
the skin on their soles has been toughened from this activity. They also get the
additional benefit of avoiding the moist and hot environment found inside shoes.
The symptom details for Plantar Warts (also known as Verrucas) and so on can be found here.
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