Archive for March, 2020

Covid-19 Luxembourg: narrowing of the uncertainty range (3/x)

March 27, 2020

In the previous blog (link) I wrote about the fact that with few data points, the Gompertz fit asymptotic value (= the maximum of infected people to be expected) has an extremely large uncertainty. The interval may extend to impossible negative and also impossible high positive values.

As an example I will show how the situation changes if we extend the full data series (starting the 29th February) from the 22nd March to the 24th and finally the 26th March.

If we stop at the 22nd, the lower bound is about -22430 and the higher 43532, and the parameter a of the Gompertz fit (i.e. the asymptote) is not statistically significant.

When we extend the period to March 24, and than again up to March 26, everything changes: the computed asymptotes become statistically significant, and the uncertainty interval narrows spectacularly:

The blue circles represent the asymptotic values, and the red boxes the range of the uncertainty interval. I will continue this investigation when data up to the 28th March are available.

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PS: here is the situation with the 28-March data included:

Luxbg_uncertainty_range_29Mar20

 

(to be continued)

COVID-19: does the number of deaths increase exponentially? (2/x)

March 19, 2020

In this 2nd and very short comment on the COVID-19 situation in Italy (data for 22 days now available) I will show that the usual remark on the exponential development of the death cases is not correct. The next figure shows as points the number of observed deaths (black dots) up to 18-Mar-20, and also what 3 different fits to the Gompertz function give when made at days 14, 19 and 22 (the points given by Death_GPZxy)):

Italy_Death_18Mar20_Obs_GPZ_Expontial

Firstly, the 3 GPZ fits numbers are nearly identical to the observations (the 4 markers mostly coincide). The blue curve is an exponential fit for the complete series of 22 days: clearly from day 19 on the real numbers lie below the exponential curve. This difference will probably increase in the future.

Conclusion:

In the ongoing COVID-19 in Italy, the number of deaths does not increase exponentially, but follows very closely the Gompertz function y=a*exp(-b*exp(-c*x)), and so will be gradually lower to what an exponential increase would suggest.

 

(to be continued)

 

COVID-19: calculations and curves (1 of x)

March 18, 2020

 

1. Introduction

With the spread of the new and nasty corona virus modelling and calculations about pandemics and spread of infection spring up about everywhere. Now the interest and the mathematics of handling such a situation are not new, and are a standard subject for all students of differential equations. As an example, let me give an example from a publicity brochure for the Hitachi 200x hybrid analog computer from the 60’s:

Hitachi_200x_brochure_title

(~1967, click here to download the full brochure).

Here is the example problem:

Hitachi_analog_computer_infection_spread

A system of 3 non-linear differential equations is enough to model the situation, and the plotter driven by the analog computer outputs this graph:

Hitachi_200x_infection_graph

A very long time ago, Benjamin Gompertz (1771 – 1865) published a sigmoid-type function that often fits well to the first or all parts of the infection (above for instance the curve Z). The Gompertz function has only 3 parameters and is:

y(t) = a*exp(-b*exp(-c*t)).

When t tends to infinity, exp(-c*t) -> 0  and exp(-b*0) = 1; so when t -> infinity, a becomes an asymptote ( in the curve Z all persons become immune in the long run).

Willis Eschenbach has an excellent article in the Wattsupwiththat blog titled “The Math of Epidemics“, where he applies the Gompertz function to the Covid-19 cases in South Korea, and the fit is excellent:

Eschenbach_GPZ_Korea

His article put me on the rails to do the same with the situation in Italy (start of the epidemic is 25 Feb 2020), all relevant data are published live here.

2. Gompertz function and Covid-19 in Italy

This is the Gompertz-function applied to the death cases, as published today 18-March-2020:

Italy_18_Mar2020_deathuntiltoday

The fit is excellent, and all 3 parameters are statistically significant; the interval between the lower and upper confidence levels is [6741, 45889], and the fraction of (std.error of a)/a has become 35%. The relative errors and the width of the confidence interval will narrow with more data points available. When only few points is all you have, do not make any prediction! The next figure shows the plot from above, extended up to 100 days, and also the same exercise when only 14 and 15 data points were available:

Italy_17Mar2020_withconfintervals

When only 15 or 14 data points are available, the confidence interval becomes ridiculous; it is only for the latest plot with 21 points that all three parameters become significant at the 95% level. So beware to massage your asymptote for any serious meaning, if you have too few data!

 

3. The Gompertz curve and Covid-19 in Luxembourg.

Tiny Luxembourg (~675000 inhabitants) only started the epidemic in Feb. 29th. Until today we have only 6 data point for the total infected (up to 203 today) and 2 cases of death. Here what the Gompertz fit looks like for the total infected:

Luxbg_18Mar2020_only6points_withconinterval

But look at the enormity of the confidence interval! Even if the Gompertz calculation gives near identical results as the observations, all parameters are not significant and one should never see the asymptote in the next graph as an intelligent predictor for the maximum number of infections!

Luxbg_18Mar2020

 

(to be continued)