We look again in this post at using a Fibonacci sequence to estimate cases numbers in New Zealand.
In the previous post we concluded that an infected person could infect other people for two cycles. See:
This suggested that a Fibonacci sequence could work to estimate case numbers.
In the post we also saw that a factor of 1.4 (staring with 10) could be used to estimate case numbers.
We have already looked a using Fibonacci sequences in other posts. See:
In these posts we saw that to allow for an incubation period, it proved useful to miss a day before adding on the previous two day’s results to generate the estimated number of cases for the next day. i.e.
F[D] = C[D-2] + C[D-3]
(be patient this is just our first step)
where F[D] is our Fibonacci estimate and C[D] is the estimated number of case numbers on Day D (from the previous column).
We obtain the following table (see second to last column):
(the previous values come from the previous column)
|Date||Day#||Actual Cases||Factor 1.4||Fibonacci||Fibonacci*F|
We see that the estimates are not very good and become a thousand less in the last row.
We compare the first estimate (24) with the calculated estimate (27.44).
We look at the ratio of these numbers, F=27.44/24 which is equal to 1.14333333.
When we multiply (scale) the numbers in our Fibonacci sequence (column 5) by F = 1.14333333, the estimated case numbers match exactly (see columns 4 and 6; keep the three starting vales).
We obtain the formula (when using factor 1.4):
C[D] = 1.14333333 * ( C[D-2] + C[D-3] ) for D > 20
We tried other factors instead of 1.4 (see 4th column) and see that this always is true for the new factor F calculated in the same way.
Since the factor 1.4 matched the actual number of cases well, so does the scaled Fibonacci sequence since the same numbers are generated.
Below are the graphs again from the previous post:
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