Depends on what you mean by "civilization." You could make the case that almost all of modern history has been one contagious civilization. However, if you mean modern nation-state, generally speaking they collapse due to replacement population scenarios. Not so much because the native population is not breeding, but because they have expansionist policies that kill off vast swaths of their most viable breeders. The Romans found this out the hard way. It was a good run, but they eventually depleted their military nucleus with expansionist wars, i.e. their men were all on the frontiers. Eventually, Rome was forced to hire mercenaries to defend their holdings and the Gauls ended up sacking Rome.