There was actually a study done and published in the New England Journal of Medicine on duct tape on warts not quite 2 years ago. The conclusion was it does work - not like a miracle cure or anything.

Regarding the child - if the warts are spreading that fast and over such an area, the doctor should be called and given the information that the warts which previously covered 20% of his arm and 10% of his chest now cover 40 % of arm and 69% of chest and back. The percentages used are an example.

Best of luck