A metric for detecting the effects of de-confinement : Italy

Figure 1. Showing rate of change of new daily cases, the break away from initial spread of infection, and an extrapolated empirical model of the post confinement period.

Introduction

Most presentations of COVID-19 case statistics show either cumulative totals of fatalities or reported confirmed cases, or the daily increments: daily fatalities or new daily cases.

Examining higher differentials will more clearly expose rapid changes than cumulative totals. In this analysis, the rate of change in the daily number of cases is examined. That is to say, the daily change in the daily new case numbers, measured in cases / day / day.

This analysis detects the break from this initial spread of the infection and compares this to the timing of the introduction of confinement in Italy. It further characterises an apparent weekly cycle and the underlying trend in the post confinement period and extrapolates this into the immediate future to provide a metric against which effects of de-confinement can be assessed. This should provide a sensitive means of monitoring the expected increase in case numbers as restrictions are eased and help inform the difficult task of balancing the continued destruction of the economy with the implied long term damage to all aspects of society, with the urgent need to postpone new infections and spread them out to more manageable levels over a longer period.

Method

Here a simple logistics curve model for the evolution of a pandemic ( without specific intervention measures ) is fitted to the Italian data. The motivation is to provide a direct indication of the early decline in reported cases expected to occur as a result of confinement measures introduced by central government announced on 8th March 2020 to come into force the following day. The model chosen is very naive and the fitting to early rise does not well constrain later behaviour of the curve. For this reason no value should be placed on the latter part of the curve other than its general form. In particular the height, peak, timing of peak and timing of zero crossing should not be accorded any significance at all. The sole purpose of the comparison is to reveal the break from the generally rising early expansion of infection. A number of alternatives such as Gompertz function, SIR or SIER models could equally have been used in this context without any significant change to the detection of a breakpoint.

The second part of this analysis aims to characterise an apparent weekly cycle and the underlying longer term trend then, by speculative extrapolation of the near future trend and the addition of the mean weekly cycle, attempts to provide an estimation of the continuation of the data had the restrictions remained in place. This provides a metric against which the actual data can be compared to provide the earliest indication of resultant changes in hospitalisations.

Since the data sample is too short to usefully apply a low-pass filter to remove the weekly cycle, a light 2nd order binomial filter was applied to reduce the considerable day to day variation. The minimum and maximum points were then extracted from the rate of change of daily reported new cases. Successive midpoints of max and min data were used to decimate the daily data to a set of twice weekly points and thus remove the weekly cycle. The resulting points were joined with a C-spline and this curve was used to detrend the daily data. From this the average weekly cycle was derived.

Once the effect of confinement settled, the last four midpoints up to and including the date of de-restriction were regressed to a straight line whose slope was found to be indistinguishable from zero. This fixed trend is then extrapolated forward from the last point ( day 106 ) and the mean weekly cycle added back in, providing an empirical, data-based expectation of how the number of daily new cases would have developed under continued confinement.


Figure 2. Showing repetitions of the mean weekly cycle superimposed on the detrended case data.

Results

The break between data and model is clear and occurs at about 10 days after the national confinement order. This fits closely with the timing expected due to established development of the disease. The incubation period before symptoms has been found to be between 2 and 12 days with a median value of 5 days. One may speculate 2 to 3 three days of worsening illness before those who eventually require emergency hospitalisation reach this stage. There may be a further delay of 1 or 2 days for a PCR result in the severely overloaded hospitals. There was some reduction of growth 2 or 3 days earlier but this was within normal variability of the data. Some spreading of magnitude of the change would be expected due to the variation of incubation period. This would be consistent with the earlier dip, though the evidence is inconclusive.

Standard epidemiological models predict a reduction in the rate of infections and were the main motivation for application of draconian confinement regulations. This analysis shows a rather sharp and well defined deviation can be found in the published clinical data with a timing consistent with the expected reduction in infections and thus number of reported cases.

Another notable feature is the appearance of a surprisingly regular weekly oscillation of significant magnitude in the data after confinement takes effect which was much less apparent previously. The trough of this cycle falls on Monday and Tuesday of the week. This cycle was isolated, as described. Figure 2 shows a repetition of the mean weekly cycle superimposed on the detrended case data.

The midpoints after day 95 ( Saturday 4 th April ) form a near straight line with a slope indistinguishable from zero. As a result, the last mid-point was taken as a fixed base to apply successive weekly mean cycles, both hind-cast and extrapolated.

Figure 1. shows the empirical model is a good match with recent weeks of case data and extrapolates it 10 days hence. Assuming that the initially detected lag between regulations and the impact on hospital admissions is still pertinent, the main effects of the relaxation of confinement rules would be expected to come into effect around day 115 ( Friday 24th April 2020 ) at the peak of the next weekly cycle, with initial signs 2 to 3 earlier.


Figure 3. Showing the deviation of current case date in Italy from the extrapolation of the model fitted during restriction period.

Conclusion

The analysis successfully characterised a rather stable weekly cycle and the underlying trend in the post-confinement period in Italy. Comparison of the extrapolation of this model to the rate of change of new detected daily cases should provide a sensitive and fast responding metric to assess the initial effects of relaxation of confinement regulations. This will help inform the difficult balance between the immediate need to manage case loads by postponing infection rates and the massively destructive counter effects of continued confinement.

This will provide useful advanced knowledge to other countries who are a few weeks behind Italy in the evolution of a COVID-19 epidemic in their own populations.

UPDATE: Reprocessing the ECDC data released on 27 April 2020 shows a deviation from the model showing an upturn in the data points on the days which would usually be the lowest in the weekly cycle and a significant drop at would have been the peak of the weekly cycle under confinement. This shows a clear break from the established, stable pattern, but overall there is no net increase of decrease evident so far in the limited relaxation of movement restrctions.

   
Appendix


Datasource:

ECDC : multiple download formats are proposed:
https://www.ecdc.europa.eu/en/publications-data
current daily update in cvs format is usually available at:
https://opendata.ecdc.europa.eu/covid19/casedistribution/csv

Data processing

Data processing was done using plotting software http://gnuplot.info and command line text processor ‘awk’. Both are multi-platform, freely available, open source software. In this instance processing was done on Linux OS but it should be readily run on other platforms.

There is also use of some simple awk scripts for data processing and organisation, the source for which is also provided. The code has been structured to allow rapid adaptation to different countries by changing a single variable. (This will need changing in three places). How suited other countries data are to this processing is not investigated, though Spain does seem to display a similar repetitive structure.

Supplementary Information

The code for reproducing the analysis and the above graphs can be downloaded from the following link:
https://climategrog.wordpress.com/covid_italy/