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| Logistic curve |
Using the data provided every day by the Spanish Ministry of Health, about
the number of new cases affected by the pandemic each day, and the number of
deaths, we have carried out the following analysis:
- We have manipulated slightly the
real data to take into account the weekend effect, pointed
out by the minister, which consists in the fact that the figures for new
cases for March 28, 29 and 30 are not faithful to the reality, because
fewer cases are declared on the weekend (we are used to waiting until Monday
to do it), and then, on Monday, all the missing cases get accumulated. As by then the curve had flattened, what we’ve done is replacing the data about new cases
for those three days by the average of the three, thus redistributing the Monday
excess between Saturday and Sunday. The following weekend, April 4, 5 and 6, we
made the same correction. The attached
figure represents the number of new cases for each day (red curve) and the
number of deaths (green curve). The numbers in the abscissa of the figure
represent days counted from the beginning of the data. Day 1 corresponds
to March 3. Day 36 is April 7, the last day about which there were data when this post was published.
- A curious detail: if you look
at the figure above, you’ll see that the maximum number of new cases was
reached on March 25. On the evening of that day the Bishop of Fatima consecrated
Portugal and Spain to the Virgin of Fatima. The following day began the
decrease in the number of new cases in Spain. Draw your own conclusions.
- If we look at the mortality
curve (green in the previous figure) it will be seen that it is delayed
about a week compared to the red curve of new cases, which is logical, as deaths
usually take place several days after the detection of the disease. Thus,
the maximum, which was reached around a week later, was followed by a
decline and a small rise, just as in the red curve. We are just there right
now, so it can be expected that in the coming days there will be further
decreases in mortality.
- It can be expected that the
final accumulated curve of new cases will approach the logistic curve,
which, as I have indicated in several previous posts in this blog, faithfully
represents the evolution of many biological, economic, and social
processes. Growth is exponential at first; then it changes to a linear
(constant) growth stage; finally there is a negative exponential approximation
towards a maximum value.
The equation for the
simplest form of the logistic curve is this (see its image at the beginning of
this post):
P(t) = 1/(1+e-t)
This curve takes
values between 0 and 1. A slightly more complex version, which includes parameters
that can be adjusted to fit the observed data, is this:
P(t) = Max/(1+n.e-t/τ)
This curve takes
values between 0 and Max. Max, n and τ are parameters that can be adjusted appropriately.
By adjusting the
parameters, so that the logistic curve is as close as possible to the real
curve of accumulated data provided by the Ministry, by minimizing the sum of
the squares of the errors, we have got the following values:
Max=204,000, n=0.29, τ=1.47
- The attached figure shows the
approximation of the accumulated real data (green curve) by the logistic curve
(red curve). It will be seen that the approximation is quite good. This allows
predictions to be made regarding the future. Notice that by the 67th day (May
8) the number of new cases will be small, less than 100 a day for all of
Spain, as the curve of accumulated cases flattens towards the final part
of the logistics curve.
- The midpoint (the inflection
point) of the curve would correspond to the point where the curve goes
through one half of the maximum value (i.e. 102,000 reported cases). This
took place on April 1. So we are already in the second part of the
logistics curve, and things should improve from now on.
- A fact that has not been given much
publicity, but which is found in the figures provided by the Ministry, is
that the number of male deaths is 62% of the total, while women are only 38%.
In other words, this pandemic is mainly aimed against men.
It is being said that the figures offered by the Ministry are not
reliable, as there are many asymptomatic cases, and even undeclared symptomatic
cases, while the number of deaths seems to be much higher, according to data
from funeral homes, because many deceased are not classified as infected
because the tests have not been applied. But if the error in the official
figures affects specific values, but not trends (for example, if the published data
are a fixed percentage of the true ones), the considerations made in this
article could still be valid.
Many mistakes have been made, together with reckless
decisions that have probably worsened the evolution of the pandemic in Spain.
If more appropriate measures had been taken, the logistic curve followed by the
disease could have been different, with a smaller number of affected and
deceased. Even now things could get worse if mistakes continue to be made. The
current logistics curve could link to another, giving rise to a second phase of exponential increase of the pandemic.
Manuel Alfonseca
Julio A. Gonzalo
Thanks for this interesting article, gentlemen.
ReplyDeletePlease can you give a few more details on the "mistakes" and "reckless decisions" to help the rest of us to avoid them?
Mistakes: Buying thousands of tests and having to return them because they gave 30% correct results. Forbidding the regional governements to buy material, because the government of the country wanted to keep all the strings in their hands, and then doing nothing.
DeleteThere are many more, but this is a good sample.
Reckless decisions: Allowing the feminist demonstrations on March 8, in spite of warnings against it, even by the WHO, because of ideological reasons. After the mass meetings, the rate of contagion escalated. Five days later, they had to order us confined.
Another mistake, taken from a press headline dated today (4/10): "The Ministry of Health waited a week before deciding the urgent protection of old people residences." Thousands of old people in residences have died.
DeleteThe Spanish Ministry of Health appears to have lost control of the pandemic data in Spain. See what they have done in the last few days:
ReplyDeletea) According to data published on 4/16, the cumulative number of deaths in Spain was 19,130 that day. The data are here: https://www.mscbs.gob.es/en/profesionales/saludPublica/ccayes/alertasActual/nCov-China/documentos/Actualizacion_77_COVID-19.pdf
b) According to the data published on 04/17, the accumulated number of deceased was that day equal to 19,478. The difference with the previous day was, therefore, 348. However, the number of deaths in the previous day provided by the Ministry is 585. The data is here: https://www.mscbs.gob.es/en/profesionales /saludPublica/ccayes/alertasActual/nCov-China/documentos/Actualizacion_78_COVID-19.pdf
c) To explain this discrepancy, a footnote indicates that the Generalitat of Catalonia has sharply increased the number of its deceased, so that the Ministry will introduce corrections progressively in the next days.
How can one follow the data about the pandemic and make predictions, if the meaning of the data is changed every few days? This is the second time they've done it; a few days ago, they started to count asymptomatic people who had been detected through tests, but who hadn't been considered before, as tests were not available. This led to an upwards jump of some 2,000 new cases detected per day, which is not due to a flare-up of the pandemic, but rather to the fact that data collection rules have been changed.
In order to draw consequences from statistical data and make valid predictions, rules should not be changed every few days. If this is done, the data have no predictive value.