The chart below shows Australia's (pretty much Victoria's) locally acquired new cases and the mean daily R averaged over the previous 8 days. The mean R has been in the epidemic range recently and is still at a worrying 0.9. Contact tracing and compliance with restrictions is still necessary. Carefully ease some restrictions but still aim for zero new local cases.
Victoria is easing restrictions just as the R goes down to 0.9 which is risky. An R of 1 or above signals epidemic spread, and the R has been above 1 for 9 out of the last 11 days.
The chart below allows yellow, brown and pink straight lines to be fitted to the curve made by actual new cases in Australia (green). They are drawn by putting a value for R and the start date of the coloured graph into the appropriately coloured box on the left. That way the best fit of the coloured curves to the green Australian actual cases suggests that the value of R entered is the apparent R of the actual Australian cases. So the yellow graph suggests the apparent R was 0.75 during the late part of the second wave. The brown graph suggests a relapse during which the R rose to 1.2. And the pink graph suggests that the current Australian apparent R is about 0.85.
Another way to assess what the recent value of R in Australia has been doing is to try to fit a known curve (grey) to the curve of the Australian new cases curve (green). If a good fit can be made then the values used to draw the grey curve may represent the actual values in Australia. The best fit appears to be with the values input into the grey box to the left of the chart. The values producing the best or closest fit were an R starting at 1.15 on the 3/10/2020 and decreasing by 0.02 every day until 21/10/2020.
Why set R to decrease by an small amount each day?
Because it appears that R behaves in that way. It is counter-intuitive. You would expect R to decrease immediately, or at least quickly, to a new value as soon as restrictions are put in place. Everybody would comply with restrictions and the R would fall rapidly. But in fact, as can be seen in the top chart above, during the first and second wave the R fell consistently but slowly. The fact is that when restrictions are put in place the R falls much more slowly than the usual mathematical model of infection spread would suggest. Some have called this the wash-in period. But no part of the SIR model accounts for a wash-in period.
It might be that individuals in the population have different risk factors for disease spread, like big families or habitually visiting bars, or alternatively by living alone and staying home. Different individuals may differ in their compliance. The initial observed high values of R may represent the virus spreading more easily through a network of high risk individuals (individuals with a high personal Ri). And the observed R may decrease once individuals in the high risk networks are progressively infected and the virus then can only spread through lower risk individuals (lower personal Ri). Perhaps there is another explanation.
It may intuitively seem that fining or punishing individuals or companies for non-compliance would increase compliance. But, counter-intuitively, this strategy often fails. Individuals are more likely to better hide their noncompliance if they face a fine or some other punishment.
So is the best strategy to create a bit of a fuss but then not do anything much in the way of punishment? Or is the best strategy to offer rewards. Or rewards for whistleblowers?
Maybe the best strategy would be to punish dishonesty when asked about compliance and hiding noncompliance once asked about compliance by an authority.
That way individuals and companies would be encouraged to improve compliance, not because noncompliance is punishable, but because, once questioned, hiding or being dishonest about noncompliance will be punishable. It would also mean that an individual questioned about their movements, contacts or compliance would benefit from being truthful, which would be a great help to authorities and contact tracers.
With a mean R of 1.3 this appears to be no time to ease restrictions. During the first wave the mean R fell rapidly from well over 2.5 to about 0.6 (green graph below). Unfortunately restrictions were eased on 1 May just as the mean R was rising and approaching 1 (the critical epidemic point). A series of clusters followed with the mean R slowly rising over time and compliance fatigue set in. Then the second wave hit. The decrease in mean R was sluggish and finally failed. The second wave was therefore higher and lasted longer than the first wave (blue graph below). Just when the second wave was nearly over compliance decreased and mean R has risen to 1.3 and rising. Signalling a new cluster at least and possibly a third wave.
On the chart below straight lines of different colours are draw to 'best fit' the curving green graph of the new daily Australia cases. The straight lines can only fit a part of the curved green graph, but the straight lines give a good indication of where the green graph is going at that point in time. The straight lines are drawn by putting values into the coloured box of the same colour on the left.
The yellow line shows that Australian new cases were decreasing during the second wave with an indicated R of 0.75. And the brown straight lines shows that the Australian new cases are now increasing with an indicated R of 1.2. Which agrees with the mean R over 8 days (R8) of 1.3.
It is easy to see this cluster or wave in hindsight, but it was probably predictable by the 4 October by the fact that the mean R was rising and approaching 1. That was no time to be contemplating an easing of restrictions.
The number of new cases in Australia is falling, but falling less than required. The current mean R is 0.87 which is perilously close to the critical epidemic value of 1. And at the moment R is rising. These numbers probably signal a risk of another cluster or possibly the start of a third wave.
What a pity that the low number of new cases seems to always make people and the government relax just at the time they should be holding firm and driving new cases towards zero and keeping the mean R low.
Instead of falling the mean R has stayed the same or risen since 27th August 2020. The mean R has not dropped for 5 weeks (blue horizontal arrows on the chart below).
It only requires a few members of the population to be non-compliant with restrictions for there to be a potential pathway connecting them up. If any one of them gets the virus then all of them get the virus and so a cluster starts and spreads.
In the past a new cluster or new wave started each time the mean R was rising and approaching 1 (the blue vertical arrows on the chart below). When the mean R becomes 1 it means there are one or more pathways or network connecting non-compliant individuals together in such a way as to make it possible for each infected person to infect one further person. If anyone in the pathway or network is even less compliant then they can catch and spread the virus more easily and the mean R will rise. At that point there is a new cluster or new wave starting.
Maybe NZ and QLD are right that this is not quite the time to ease restrictions. It is probably the time to lessen non-compliance in those few who contribute to the rising mean R. Mean R probably does not lie.
As a side issue: What is the meaning of R or mean R?
At its base, R, or a mean of R over a few days, represents the number of individuals infected by one infected individual. It seems common for an individual to be considered infected until they can be removed to the recovered or dead groups. In practice, though, an individual is only infectious until they are removed from the population by by the authorities by isolation or hospitalisation. After that they should not often be able to infect anyone. Thus R should represent the number of individuals infected by one Infecting individual. Not 'infectious' individual. That is by one individual who is potentially infecting others. Once the individual is isolated they may still be infectious but they are not infecting others and therefore do not contribute to the value of R.
But at a less granular level, R may be a property of each individual (say Ri) in the population and may measure that individual's risk of catching the virus and of infecting others. Thus some individuals would have a low Ri and others would have a high Ri. Some might have an Ri of 0.2 and be of low risk of catching or transmitting the virus, while others may have an Ri of 2 and have a high risk of catching and transmitting the virus. Ri would probably have a 'normal' distribution. Thus there would be high Ri individuals in the population even if the mean R was relatively low. So there would be individuals with an Ri of 2, for example, in a population whose mean R was 0.5
An individual's Ri would be increased by living in a big family, sharing a dwelling with many others, for example a care facility or dormitory, working in an environment where isolation is difficult or in which the virus thrives, for example the cold room in a meat works, visiting pubs and clubs and shopping centers where many individuals meet, visiting many contacts, traveling on public transport, being careless with social distancing and hand-washing, being non-compliant with restrictions and being unable to understand.
An individual's Ri would be decreased by living alone or with a few, staying home, keeping contacts to a minimum, not travelling on public transport, being careful with social distancing and hand-washing, being compliant with restrictions and by understanding the issues at play.
Ri measures an individual's risk of catching or infecting others with the virus.
A person with a high Ri is in contact with more individuals than a person with a low Ri and therefore with more high Ri individuals. Thus there is likely to be a pathway or network joining high Ri, or high risk, individuals together. As long as no one on the pathway or network catches the virus, nothing happens. Once one person on the pathway or network catches the virus there is already the potential path through which it infects the others on the pathway or network. And 'nothing happening' turns to a cluster in a short time.
The concept of Ri may explain the fact that R does not fall quickly when restrictions are introduced as would be expected. Instead R follows a flat reverse S shape, but mainly follows ,a linear decrease over the course of the cluster or wave.
The concept of Ri would suggest that many individuals would rapidly reduce their risk by reducing their Ri by following restrictions and infection control measures. But others would be less compliant or risk adverse and would not reduce their risk, Ri, to the same degree. Hypothetically the virus would continue to spread through these high risk networks. The observed R would be high because only those catching the virus or infecting other individuals contribute to the observed R in the population. R is the number of new infections on a day divided by the number of infecting individuals who have not yet been isolated. The rest of the population does not contribute to R in any way. As the cluster or wave continues two thing may happen. First, individuals may reduce their Ri further. Second, higher risk, high Ri, individuals are removed from the population by being infected leaving individuals who are high Ri, but not as high as the removed individuals. So the virus would still spread, but not as vigorously as before. The observed R would decrease. And the observed R would decrease slowly to a low value towards the end of the cluster or wave.
Unfortunately, human behaviour being what it is, there is a tendency to ease up on compliance once the perceived danger is over. That is, before the virus is eradicated in the community. At that stage single community cases at intervals of less than 6 days signals the presence of networks of individuals with Ri of 1 or more, resulting in an observed R of 1. That is at the critical R for the start of a cluster or wave. (curve matching of Australian cases suggests that individuals are isolated after 6 days and play no further part in infecting others. One case ever 6 days suggests that one case is still infecting one other individual before being isolated and therefore the R is 1).
A low number of community cases is not a good number of cases to be easing restriction on.
An apparent R rising and approaching 1 is not a good time to be easing restrictions.
Aim for zero community cases. An observed R of 0.6 or less is much safer than an R of 0.8 and higher.
No value of R is completely safe.
R is a summary value representing only those just infected and those who did the infecting. It is mathematically unconnected to the rest of the population. It is an assumption that the observed R represents a characteristic of the whole population.
The concept of Ri is that Ri is a statistic of each individual in the population and is a summary measure of that individual's risk of catching and transmitting the virus. Each individual has a number of contacts. Potential pathways or networks for viral spread exists between individuals with high Ri.
Early in a cluster, networks of high Ri individuals are more likely to spread the virus.
Later, networks of lower Ri individuals will become more prominent.
The concept of Ri suggests that the assumption that the observed R represents a characteristic of the whole population is not correct.
Herding humans is as difficult as herding cats.
There are good economic and psychological reasons for easing restrictions whenever possible. The problem is in deciding if and when it is possible. After the first wave Australia lifted restrictions on 1 May 2020, just when the R was near 1 and climbing. In retrospect (or predictably, since an R over 1 means an epidemic is under way) it may have not been the best time to ease restrictions as much as they did. The blue arrow in the chart below.
Now Victoria is lifting restrictions while the R is above 0.8 and not definitely falling. It may be a time for caution. A low number of cases each day is not a good sign. It means the R is at or near 1 and a third wave is possible. To be a good sign the daily cases must be low AND falling. That means the new cases a day has to aim to drop to zero (for community transmission). Cautious lifting of restrictions so that R is kept at below 1 in the whole population is probably a reasonable approach.
Astonishing. The non-compliance with restrictions and sound medical advice. But predictable from the the mean infection ratio over 8 days (R8) in Australia during the second wave. The reduction in R has been slow compared to the reduction in R during the first wave. See the chart below. And the R has actually climbed at times during the second wave, demonstrating a complete disregard for the restrictions and other people's health by a small number of the population.
Compliance Fatigue or increasing distrust of authority may well explain the recurrent recent period where the average R (infection rate) for Australia has actually gone up instead of down (orange curve below). When the number of new cases is low that probably represents the behaviour or views of a small number in the population. But it represents a risk because those very individuals may be networked to other similar individuals, making an easy path of 'least resistance' for viral spread.
Perhaps (human) behavioural science could suggest ways in which compliance and trust could be improved and fatigue reduced.
Human Behaviour probably also accounts for the fact that the R8 (apparent average infection rate averaged over 8 days) fell faster in the first wave than the second wave, and fell cleanly in the first wave but often unfortunately rose again in the second wave. (The orange curve above, and the green curve below.)
If everyone was 'on board' and compliant as soon as restriction were put in place, the R or R8 should fall immediately and rapidly. Instead R or R8 falls over a number of days, probably indicating that more individuals become adequately compliant over time. Maybe there is a learning curve or a necessary preparation time or a peer effect (a greater pressure to do what others are doing).
The earlier prediction that Australia would be down to single digit new cases a day by mid-September was based on the inputs shown in the grey box to the right of the chart below. The inputs were based on the shape of the curve of new Australian cases a day (green curve below) during the early part of the second wave. The R was expected to start at 1.55 on 28 June 2020 and decrease by 0.015 a day until 30 August (the purple line or curve below). The earlier prediction was fairly accurate except that the R did not fall steadily as expected, but often rebounded again and is nor rising towards 1 (a definite risk of another cluster or wave; the purple arrow versus the blue arrow).
Various coloured curves (below) were fitted to the actual Australian new cases a day (green curve) at different periods during the second wave. Early on an R of 1.5 in the mathematical infection model matched the actual cases (brown curve). Later the pink curve, plotted from inputs in the pink box to the left of the chart, with an input for R of 1.25 matched the actual cases. As the second wave began to come under control an R of 0.87 generated a blue curve which best fitted the actual cases and finally, now, it appears that an R of 0.75 (the yellow curve) best fits the current Australian cases. The earlier prediction seems increasingly unlikely, but a least the cases will continue to fall as long as the R does not continue to rise. The way forward may lie in social behavioural science.
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.
And here are the reasons for the above suggested lessons from the Australian new cases till now:
It appears that Victoria (and Australia) are still on track to get down to single digit New Cases per day by mid-September as the second chart below suggested on 26 July 2020. Admittedly, it looked shaky a few times. Some residents were non-compliant with restrictions during the two periods shown with light blue arrows below. And the rate of reduction in the infection ratio (shown by R8, the 8 day mean of the daily R calculated from the actual Australian cases) was slower in this second wave (the purple trend-line) than it was during the first wave (the blue trend-line).
The grey curve shows the number of New Cases a day predicted by the epidemic model using the inputs in the grey box to the left of the chart. The curve is seeded from the actual Australian cases for the 8 days prior to the start date of the curve (set at 28/6/2020) and the start date, the initial R, the rate of fall of R, the i , and the stop date for the fall of R (the infection ratio). The best match to the curve of the actual Australian cases was as shown. R is best set set to fall by 0.015 each day from 1.55 between the start date and the stop date (30/8/2020) after which it remains unchanged. R therefore falls from 1.55 to 0.62 during this period. The actual Australian new cases per day is shown in green, and they appear to be following and well matched to the grey curve.
The chart below shows the actual Australian new cases a day on 26 July 2020 in green. Various values had been input into the grey box on the left of the chart in order generate the grey curve to best match the actual Australian new cases up to that time. R increases and i (the number of days a case remains infectious before the cases is removed from the population by isolation) decreases the slope of the grey curve at any point. Thus, only one value of R/i will best match the slope of the curve of actual Australian cases. The value of 6 for i best matched the first wave of Australian cases. R had fallen steadily during the first wave, and so was expected to fall steadily during this second wave. It should be noted that i is not the number of days a case is actually or medically infectious. It is the number of days that a cases can infect another in practice because it is the number of days after infection until the case is removed from infecting the population by being isolated or medically managed.