key: cord-0336761-ulg8ixon
authors: Pesenti, R.; Pesenti, K. W.
title: Covid spirals: a phase diagram representation of COVID-19 effective reproduction number Rt
date: 2021-09-06
journal: nan
DOI: 10.1101/2021.09.02.21262861
sha: 09c03cbec27974757719f930b1653d9d15b6de52
doc_id: 336761
cord_uid: ulg8ixon

In this paper, we propose a phase diagram representation of COVID-19 effective reproduction number R_t. Specifically, we express R_t as a function of the estimated infected individuals. This function plots a particular clockwise spiral that allows to easily compare the evolution of the number of new infected individuals at different dates and, possibly, provide some hints on the future progression of the infection.

https://github.com/pcm-dpc/ In this work, we use the Civil Protection data to estimate both R t and I t values. Specifically, we compute 24 these two values using the methodology proposed in [2] for both the regions and the provinces of Italy. Finally, 25 we plot the values assumed by R t and I t over time in the space I × R t to obtain a spiral graph. 26 Our estimates and the associated graphs are available at http://virgo.unive.it/pesenti/tekwp/dashboard.php. 27 Here, a word of caution must be spent. The data recorded by the health authorities may sometimes present 28 some errors that must be corrected, otherwise a negative number of new infected individuals may be obtained on 29 some days. Our way of correcting these errors may differs from the one by other authors. Then, our estimated 30 values for R t may present small discrepancies, e.g., from the values estimated by the Italian National Institute 31 for Nuclear Physics at https://covid19.infn.it/.

In this section, we describe the relations that occur between the graph of the function R t (I t ) and the number is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted September 6, 2021. ; https://doi.org/10.1101/2021.09.02.21262861 doi: medRxiv preprint Figure 5 and its zoom in (Fig. 6 ) for the months of July and August, 2020, allow the reader to compare the information that can be deduced from the graph function R t (I t ) with the one deriving from the following index: This index is practically identical to the RIC-index introduced in [1] . In [1] , the authors suggest that abrupt 54 increases of the value of the RIC-index may be precursors of new pandemic waves. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted September 6, 2021. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted September 6, 2021. ; Figure 9 : Ratio between R t and the number of days it took to double/halve the 7-day moving average of the number of new infected individuals observed in the region of Sicily between August 25th, 2020 and August 24th, 2021.

7 . CC-BY-NC-ND 4.0 International license It is made available under a perpetuity.

is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted September 6, 2021. ; https://doi.org/10.1101/2021.09.02.21262861 doi: medRxiv preprint Figure 10 , i.e., the zoom in of Fig. 7 , allows us to compare the last pandemic wave that started at the very 60 beginning of July, 2021, with the pandemic wave that started in September-October, 2020. Specifically, both 61 waves are preceded by a parallel steep increase of R t , followed by a similar slow decline of this index. We remark 62 that the first measures to limit the October 2020 wave in Sicily wave were adopted on October 24th, 2020. On 63 that day, high-schools were closed, the number of passengers allowed to board trains and buses were reduced 64 by 50% and a night curfew was imposed. Differently, no measure is imposed to the date of August 25th, 2021, 65 to limit the effects of the last wave. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

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If we accept a certain level of arbitrariness, we can distinguish four phases, plus possibly an initial one (hereinafter 68 referred as to Phase 0), in the loops of the graph of function R t (I t ). is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

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The copyright holder for this this version posted September 6, 2021. ; https://doi.org/10.1101/2021.09.02.21262861 doi: medRxiv preprint of function R t (I t ). The periods of occurrence of these oscillations in the two graphs may coincide (see, e.g., 78 Fig. 14) or Phases 0 may appear slightly earlier (see, e.g., Fig. 15 ). is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

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The copyright holder for this this version posted September 6, 2021. ; https://doi.org/10. 1101 /2021 Needless to say that the occurrence of a Phase 0 does not necessarily imply a new pandemic wave if immediate 80 measures to isolate the infected individuals involved in the outbreaks are implemented. Apparently, one can 81 deduce from Fig. 12 and Fig. 16 that authorities were not able to contain the relatively small outbreaks in 82 September and October, 2020, whereas so far they succeeded in July and August, 2021. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

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The copyright holder for this this version posted September 6, 2021. ; https://doi.org/10.1101/2021.09.02.21262861 doi: medRxiv preprint Figure 19 : Graph of the values of the R t (I t ) function observed in the province of Taranto between August 25th, 2020 and August 24th, 2021. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted September 6, 2021. ; https://doi.org/10.1101/2021.09.02.21262861 doi: medRxiv preprint 3.3.2 Phase 1 90 Phase 1 is characterized by an horizontal trend or a slight decline of the R t value in presence of an increasing 91 number of infected individuals (see Fig. 21 and Fig. 22) , with a possible stabilization of R t to a value greater 92 than 1.

A Phase 1 corresponds to the growth of a pandemic wave in the new infected individuals graph. The longer 94 (along the abscissa axis) is a Phase 1, the higher is a pandemic wave. The higher is the average value of R t in 95 a Phase 1, the steeper is a pandemic wave.

Both Fig. 21 and Fig. 22 show a relative rapid decrease of the R t value during Phases 1 if the R t has reached 97 a high value. As an example, the province of Milan adopted the first containment measures for the October 2020 98 wave on October 22nd, when an initial decrease of the R t value had already occurred (see Fig. 21 ). The region 99 of Sicily adopted almost no special measures to contrast the July-August 2021 wave, which indeed is greater 100 than in other Italian regions, but the R t value decreased all the same. Possibly, the population spontaneously 101 adopts a precautionary behavior when a high intensity of risk is perceived. Possibly, a good communication 102 strategy by the authorities may foster this behavior. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

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Phase 2 is characterized by a steep, some times almost vertical, drop of the R t value that goes below 1 in 105 presence of a relative stability in the number of infected individuals (see Fig. 23-25) , 106 These phases may be associated to: Consequently, these phases may or may not occur at the same time in the different Italian regions and provinces.

As an example, we can observe a Phase 2 around November 22nd, 2020, in the graphs of all three Fig. 23 -25. 112 Differently, only the province of Rome (the largest Italian province) and Milan present a Phase 2 in August, 113 2021, (see Fig. 23 and Fig. 24 ) possibly, because the percentage of vaccinated is less in Sicily (see Fig. 25 ) than 114 in the provinces of Rome and Milan, or because Sicily is a holiday destination in August.

A Phase 2 corresponds to the period that goes from the slowdown of a pandemic wave growth to its initial 116 decline in the new infected individuals graph. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

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The copyright holder for this this version posted September 6, 2021. ; https://doi.org/10.1101 https://doi.org/10. /2021 3.3.5 Phase 4 126 Phase 4 is characterized by a steep, some times almost vertical, increase of the R t value that goes above 1 in 127 presence of a relative stability in the number of infected individuals (see Fig. 27-29) .

A Phase 4 corresponds to a new increase in contagions and, therefore, to the beginning of the rise of a new 129 pandemic wave in the graph of the new infected individuals.

In Fig. 27-29 , we can observe that there is not a clear cut that allow us to distinguish a Phase 4 from a 131 Phase 0, when the number of infected individuals is relatively low. It can be also observed that a Phase 4 is is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

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The copyright holder for this this version posted September 6, 2021. ; https://doi.org/10.1101/2021.09.02.21262861 doi: medRxiv preprint 4 Conclusions 137 In this work we presented possible uses of the representation of the effective reproduction number R t in the 138 I × R t space. 139 We showed that empirical data suggest that Phases 0, 2, and 4 have a brief duration and correspond to 

Metastable states in plateaus and multi-wave epidemic dynamics of Covid-19 spreading in Italy