Do not sit upon the fence
Keep in your bubble’s defence
There may be trouble
Cases may double
Says Fibonacci sequence
22 April 2020
We use Fibonacci sequences to answer the above questions.
The usual Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, …
Where each number (after the first two numbers) is the sum of the previous two numbers.
Supposedly the above sequence may model the number of rabbits that there may be in each successive generation.
Help … They are breeding like … Rabbits!
On average it is assumed that six days (range 2 to 10 days or more) will pass before a person infected with COVID-19 will show symptoms and spread the disease. Some people appear to spread infection before symptoms appear.
We only will miss one day then add together the previous two numbers to estimate the cumulative number of new cases. Otherwise the curve rises too slowly.
We continue this until the date mid-point (Day 33-34) then add on to the midpoint number the previous number of cases (not cumulative) in reverse order.
The Cases column in the table below is the actual cumulative number of cases in New Zealand.
Re=2 is the column we are interested in.
Note on 12 March and 13 March we have changed the number of cases from 5 to 6 to generate a better sequence.
So for 14 March our estimate is 6 + 5 = 11
(remember we miss out the figure for 13 March).
For 15 March our estimate is 6 + 6 = 12.
For 26 March (Day 28) the number of cases is 283 and our estimate is 282.
The midpoint for the actual case numbers is between 31 March and 1 April (Day 33 and 34, say Day 33.5 maybe midnight?).
For our sequence we use the midpoint 1 April.
A four week Level 4 Lockdown started in New Zealand at 11.59 pm on Wednesday 25 March.
Until 27 March our estimate is very good.
For now ignore the last column in the table.
We see the table goes up to the date midpoint.
Our estimate for the actual total number of cases is 647 + 708 = 1355 cases.
Our estimate for the actual total number of cases using the sequence is 2 x 1525 = 3050.
This turns out to be very close to the results in the graph below (excluding any long tail).
So lets look at some graphs.
Until 27 March (two days after Lockdown, the estimate is quite close to the actual number of cases.
We see from the first line graph that the estimated total number of cases is double the actual number.
So what can we see from our estimate sequence?
Each day’s number appears twice in our calculations. It is used on two consecutive days to calculate estimates.
This means that since each day’s number is used twice, each person infects two others.
i.e. For the red line Re = 2. Re is the average number of people infected by a person (with isolation). Without isolation Ro (called R-zero) is used instead.
Up until the Day 29 (mid-point Day 33-34; 31 March, 1 April), the red curve is a good estimate for the actual number of cases.
Where the blue line falls below the red line Re may be less than 2 except where the gap is narrowing (coming closer again to the red line) where Re is more than 2.
Where the blue line falls below the red line before 23 March, the red line may be sometimes close to one day ahead of the blue line. Look at the table.
In fact early on Re is close to 3 (see below).
Look at the last column in the table. This sequence adds three previous days’ numbers together (once again missing out the nearest number).
8 + 8 +6 = 22 (18 March)
12 + 8 + 8 = 28 (19 March).
Since three numbers are added together, each number is used three times on successive days and where this (green) line appears close to the blue line, Re = 3.
For the sequence estimates, we have made the date mid-point match closely to the actual case estimate.
In reality, the date midpoint for the sequence estimates would be later.
Even with this change the red line has twice as many cases as the blue line.
Around the midpoint, Re ~ 2 (the red line and the blue line are close).
When you look at the graphs, most of the time up to Day 29, Re is at least 2.
If we did not go to Lockdown Level 4 the number of cases may have at least doubled (followed the red line).
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