Wouldn't the adult cases be through household transmission from schools plus school staff?
Yes, unfortunately, it’s not a great graph (I am not as good with graphs as boys3 so I have to copy them from random but trusted sources). However what it does show (in my personal opinion) is that the patterns in transmissions in older groups changed from September to October. In September the trend was predictable - infections in kids caused gentle but slow increase in parents age group (40-49), exactly as expected.
However, the pattern changed on October 3. From that date on, the growth across all older groups (including 20-24 year olds) picked up speed simultaneously. It is unlikely that many families across the whole country held multigenerational parties on Sep 30 with schoolchildren infecting parents, grandparents and older 20 year old siblings. Most likely, there was another factor at play. In retrospect, the change looks like a one-off behavioural shift (increase in socialisation across all age groups). We saw similar trends previously and many of them were weather related.