It’s so odd the way it does come in waves. I don’t really understand how this happens.
I think this probably could be modelled fairly easily mathematically-like the waves you get on a congested motorway.
I'd guess simply:
Start with 1 person has it. Say they meet 10, pass it on to 4, all those 4 meet 10 and pass it onto 4, and all those pass it to 4 more.
So it's currently increasing rapidly:
1
4
16
64
256...
But then at some point things change. Maybe it's lockdown, taking the contacts from on average 10 down to 2, so even assuming both the contacts get it, instead of 256, only 128 will have it at that point.
Plus people tend to see the same people. So, for example, over a week I regularly see 7 people each week through work or my household. I probably see about another 5 on top of that, but they would be quick meets, possibly outside etc, so less likely to catch it.
The chances are I caught it from one of the regular people I see. So one of the 7. The work group (including me) is 4, the home group (including me) is 5. But if they've infected me, there's not an insignificant chance they will have also infected the rest of the group. So I could only give it to the other group-roughly half the number, so you might find that 128 only give it to 128 others-stabilising the cases.
But then add to as numbers get bigger people change their behaviour. They don't go where they expect there to be many people, they remember not to hug, wash their hands, wear their face mask.
As numbers increase, on average, people are more careful. They're also more likely to think they may have caught it and isolate, they're less likely to go round and see old Aunty Mavis. So that slows numbers down.
But then as numbers decrease again the opposite happens. They get more careless about remembering to keep the distance. They forget to sanitise their hands on the way in, and they might as well go and see Aunty Mavis because they don't know anyone who has it, so they're sure they're fine.
It's never going to be able to totally model it, because human behaviour isn't predictable, but I think the waves do make sense as people adapt to the situation.