Dr Meirion Evans, a public health expert at Cardiff University, says there is a backlog of older children who have not had the MMR vaccine, and has urged people to make sure their children are immunised.
He told our Correspondent Richard Morgan that tens of thousands of children are potentially at risk if the outbreak spreads beyond the Swansea area.
Fresh measles cases have been reported in Powys and Cardiff. A public health expert warned tens of thousands of children could be at risk.
Public Health Wales has information and advice on measles - a highly-infectious viral illness which can have life-threatening complications.