Are there any lessons as Australia comes out of the Second Wave

Now that the second wave is coming under control are there any lessons from past events? Here are some possible lessons - and below them, the reasons for the suggestions.

• A very small number of new cases a day, even if the number is low and stable, is not a good situation. When a small, or very small, number of new cases is reached the likely only mathematically reasonable number of cases to aim for is zero.
• It is likely that Zero is the only mathematically reasonable number of cases to aim for. It will not remain zero because of 'leak' from other populations but the aim should still be to get back to zero.
• With each new cluster go hard and go soon.
•  R is the observed infection ratio and R8 is defined here as the mean observed infection ratio over 8 days. R8 appears to be a useful measure of the infection ratio.
• A rising R8 near 1 is not safe. 
• Don't ease up on restrictions when the mean R8 is anywhere near 1 and rising.
•  Keep R8, a mean observed infection ratio, low. Nowhere near 1.
• R8 rising above 1 heralds a new cluster (or wave) and should be addressed hard and soon.
• Beware of compliance fatigue. Compliance with restrictions appears to be lower with each successive wave and between waves (or clusters).

And here are the reasons for the above suggested lessons from the Australian new cases till now:​

• A very small number of new cases a day, even if the number is low and stable, is not a good situation. R is the observed number of new cases now, compared to the number of new cases in the previous time period.  If the number of new cases each day is unchanged day to day then R is 1 no matter how few the new cases are. And if R is 1 it is at the critical point at which an epidemic or wave or new cluster will start. If the number of new cases have been brought to a low number and then kept stable, the R will be 1, and an epidemic or new wave or cluster is  likely to start.
• It is likely that Zero is the only mathematically reasonable number of cases to aim for.
• It will not remain zero because of 'leak' from other populations but the aim should still be to get back to zero. Leak from other populations is probably inevitable. So effective border control, quarantine and contact tracing is probably needed to keep aiming at zero new cases. 
• With each new cluster go hard and go soon.
•  R is the observed infection ratio and R8 is defined here as the mean observed infection ratio over 8 days. R8 appears to be a useful measure of the infection ratio. R8 is used in the above and below charts.
• A rising R8 near 1 is not safe. It heralds a new epidemic, wave or cluster.
• Don't ease up on restrictions when the mean R8 is anywhere near 1 and rising. The blue arrow below marks 1 May 2020 when restrictions were eased. The R8 was near 1 and rising, which heralds a new cluster at least. The R8 had been rising after the perceived 'end' of the epidemic, perhaps because of an anticipation of the end of lock-down or because of compliance fatigue. Also the purple arrows on the chart below shows the start of the second wave with R8 rising above 1 by 13 June 2020 even though the number of new cases was low and had been stable.
•  Keep R8, a mean observed infection ratio, low. Nowhere near 1.
• R8 rising above 1 heralds a new cluster (or wave) and should be addressed hard and soon. Suggested by the purple arrows on the chart below where that did not occur.
• Beware of compliance fatigue. Compliance with restrictions appears to be lower with each successive wave and between waves (or clusters). The rate of fall of R8 was greater in the first wave than it was in the second wave suggesting compliance fatigue (the slope of the blue line during the first wave in the above chart compared to the lower slope of the purple line in the second wave. Also R8 did not fall on two occasions during the second wave suggesting non-compliance and compliance fatigue.