Fooled by COVID Probabilities in Melbourne

How small variations in the distribution of super-spreader COVID infections make a huge difference in the start of community transmission

Here in Melbourne (Victoria) we can see the case movements and transmission of COVID in very granular detail because we have very low cases numbers and very good contact tracing.

This morning I was having a conversation with a friend of mine who was saying “they are saying the variant we are now experiencing is far more infectious, How can that be if we have not had that many cases (a few a day) before the lockdown started and people were moving around unchecked”. They were questioning the need for hard lockdowns.

Generally we are not very good at understanding exponential growth and how it affected by variations in the starting situation. So I have done a very quick and simple model looking at some differences. This is for story telling purposes, not for disease control or modelling and there are a myriad of other possibilities but I think that it good enough for understanding what can happen.

I have made the following assumptions for unchecked transmission:

1. On average 1 person infects 2.5 people so 10 people infect 25 people.
2. But 1 in ten people is a super-spreader and infects 16 other people. The other 9 people infect 1 person each (and that might vary from 1-3 but for simplicity I am just assuming they are all infecting one person). We have plenty of evidence that many people infect very few or no people.
3. Average time to transmission after infection is 5 days so have used this as a fixed number to simplify things.
4. This produces the following amazing table:

The path of each scenario is detailed at the end of the article but the key difference is that in the very low and low scenarios the initial case is not a super-spreader while the initial case in the very high scenario is a super-spreader.

I was going to produce a graph of this but there is such a huge difference the low and very low cases and the very high case are so stark that that the two lower cases would not be visible on a graph.

Again, this is a very simplistic model and is only for illustrative processes but it clearly shows that we can have situations of very low transmission when we are at very low case numbers but once a super-spreader is involved we can get rapidly climbing cases.

There is some discussion that the current variant is more infectious than the numbers I have used here and has a shorter time from infection to transmission. In the press conference on June 2nd on the current outbreak Chief Health Officer Brett Sutton stated that it might be that on average each person infects 5 people. The actual result on my model of such a situation would depend on whether

1. Everyone’s transmission goes up;
2. Just the 1 in 10 super-spreaders infect far more people or 
3. The overall percentage of super-spreaders increases. 

Again there are a myriad of possibilities but clearly there would be both faster spread and a lower percentage of low transmission scenarios. But there would still be low-transmission scenarios until super-spreaders got involved in the chain of transmission.

In summary it is clearly possible to get a highly infectious variant that does not show too many cases until it explodes exponentially. So low initial case numbers are not reflective of variant that is not that infectious (although of course it might be).

Assumptions for each scenario

Very Low

• The first case is not a super-spreader and so infects one other person 5 days later.
• The second case is not a super-spreader and so infects one other person 5 days later
• The third case is not a super-spreader and infects one other person 5 days later. The original case drops off the list after 14 days).
• The fourth case is not a super-spreader and infects one other person 5 days later. The second case drops off the list after 14 days).
• and so on

note: given that some people will not infect anyone at all this chain could be broken at any stage and the cumulative cases would go to zero.

Low

This is the same as the first very low scenario except on day 30 a super-spreader person is infected and therefore on day 35 infections start to spread much faster.

Very High

Initial infection is a super-spreader

• 5 days later they infect 16 others (Day 5)
• 2 of the 16 cases on day 5 are super-spreaders so five days later they infect 32 people and the other people infect 14 others. Therefore 46 new cases and 63 in total (Day 10).
• 5 of those new cases are super-spreaders so on day 15 there are 121 new cases (80 from the 5 super-spreaders and 41 others). The original cases drops off the list after 14 days (Day 15).
• 12 of those new cases are super-spreaders so 301 new cases (12 x 16 from the super-spreaders +109 single transmission cases). The 16 cases from day 5 disappear (Day 20).
• There are 30 super spreaders in the last lot of cases so on day 25 there are 751 new cases (480 cases from super-spreaders and 271 single transmissions). 46 cases from day 10 drop off the list (Day 25).
• On day 30 there are 75 super spreader infection events so there are 1876 new cases (1200 super-spreader transmission cases + 676 single). 121 cases from day 15 drop off the list (Day 30).
• On day 35 there are 187 super spreaders spreading infection so there are 2992 super-spreader caused infections + 1689 single transmissions for a total of 4681 new cases. 301 cases go off the list from day 20 (Day 35).

Notes on probabilities

Just because on average in our assumptions there is one super-spreader out of every 10 people that does not mean that every group of 10 people that are infected contains a super-spreader. Some groups of 50 would have no super-spreaders and some groups of 10 would have 5 super-spreaders for instance. The likelihood of such groups is just low.

p.s. I am not a human epidemiologist but I am a veterinarian with experience in prevention of and responding to large scale outbreaks in animal populations. You can see a presentation I Did for the National Australia Bank on the Pandemic last year by going to:

NAB Mortgage Brokers Presentation May 2020