# COVID-19 NZ: Using a Fibonacci sequence to estimate total case numbers V

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:
https://aaamazingphoenix.wordpress.com/2020/05/23/covid-19-nz-can-one-person-infect-2-7-to-2-8-others/https://aaamazingphoenix.wordpress.com/2020/05/23/covid-19-nz-can-one-person-infect-2-7-to-2-8-others/

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:
https://aaamazingphoenix.wordpress.com/tag/fibonacci/

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)

 F 27.44/24 1.14333333 Date Day# Actual Cases Factor 1.4 Fibonacci Fibonacci*F 16/03/20 18 8 10 10 17/03/20 19 12 14 14 18/03/20 20 20 19.6 19.6 19/03/20 21 28 27.44 24 27.44 20/03/20 22 39 38.416 33.6 38.416 21/03/20 23 52 53.7824 47.04 53.7824 22/03/20 24 66 75.29536 65.856 75.29536 23/03/20 25 102 105.413504 92.1984 105.413504 24/03/20 26 155 147.578906 129.07776 147.578906 25/03/20 27 205 206.610468 180.708864 206.610468 26/03/20 28 283 289.254655 252.99241 289.254655 27/03/20 29 368 404.956517 354.189373 404.956517 28/03/20 30 451 566.939124 495.865123 566.939124 29/03/20 31 514 793.714773 694.211172 793.714773 30/03/20 32 589 1111.20068 971.895641 1111.20068 31/03/20 33 647 1555.68096 1360.6539 1555.68096 01/04/20 34 708 2177.95334 1904.91546 2177.95334 02/04/20 35 797 3049.13467 2666.88164 3049.13467 03/04/20 36 868 4268.78854 3733.63429 4268.78854 04/04/20 37 950 5976.30396 5227.08801 5976.30396 05/04/20 38 1039 8366.82554 7317.92322 8366.82554

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:

My other COVID-19 posts can be found here:
https://aaamazingphoenix.wordpress.com/tag/coronavirus/

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