key: cord-0956635-78xmjtmc
authors: Adachi, S.
title: Novel indicators for evaluating topological threats to populations from pandemics applied to COVID-19
date: 2020-05-29
journal: nan
DOI: 10.1101/2020.05.29.20116491
sha: 3b49d2600a156159cc01dbcc8c76b58bb8b40e7a
doc_id: 956635
cord_uid: 78xmjtmc

BACKGROUND To deal with pandemics, evaluating the temporal status of an outbreak is important. However, prevailing standards in this respect are mostly empirical and arbitrary. As an alternative, we focus on a novel approach which configures indicators that evaluate topological threats to populations due to the COVID-19 pandemic. METHODS We extended the current PzDom model to calculate a threshold of the model for accelerated growth, an indicator of growth extent Re(v), covariance Re(s), a topological number E(l), and expected sums of possibly increasing numbers of infected people. We term this the exPzDom model. RESULTS The indicators in the exPzDom model adhere well to the empirical dynamics of SARS-CoV-2 infected people and align appropriately with actual policies instituted by the Japanese government. CONCLUSIONS The described indicators could be leveraged pursuant of objective evaluation based on mathematics. Further testing of the reliability and robustness of exPzDom model in other pandemic contexts is warranted.

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To understand and respond appropriately to pandemics, it is important to evaluate the temporal status of a disease outbreak. However, standards in this case are often empirical and arbitrary. It is difficult to apply SIR and related models in some cases, particularly where a given pandemic is not significantly influenced by herd immunity amongst the population [e.g., Koroveinikov and Maini (2004) 1 3 ]. For example, a small decrease in new infections following a sharp increase or flattened peak beforehand, as has been frequently observed in SARS-CoV-2 infection dynamics, is difficult to explain by a simple model. Therefore, an alternative approach that tackles the issue qualitatively with topological analysis might have important merits. Our aim here is to put forward a novel approach for indicator development that evaluates topological threats to populations from COVID-19 in Japan and the rest of the world.

Data were taken on new cases of SARS-CoV-2 infections from 3/12/2020 to 5/21/2020 (5/4/2020 was omitted because of possible data error) in 47 prefectures in Japan . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted May 29, 2020. . https://doi.org/10.1101/2020.05.29.20116491 doi: medRxiv preprint (https://github.com/kaz-ogiwara/covid19) and from 2/24/2020 to 5/21/2020 in 16 highly affected countries around the world (https://github.com/CSSEGISandData/COVID-19).

The PzDom model, proposed in Adachi (2019a) 4 Next, Re(v) is developed in Adachi (2017) 5 and Adachi (2019b) 6 . Re(v) = ln N k /ln(Im(s)) and it is an indicator of non-Archimedean valuation of N k . It represents . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 29, 2020. . https://doi.org/10.1101/2020.05.29.20116491 doi: medRxiv preprint how far an observed group has a tendency for potential growth. E(l) is proposed in Adachi (2019b) 6 and it represents a categorical value of the observed set of groups, converged to a certain value. That is, a number of external symmetries of observed groups. Note that the convergence value is a function of the applied dataset and this value would be different between empirical applications even where data overlap exists. is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

(v) = exp(∑N k *Re(v)/b), E(l) = ln N k /ln Im(v).

The copyright holder for this preprint this version posted May 29, 2020. . https://doi.org/10.1101/2020.05.29.20116491 doi: medRxiv preprint

We performed calculations of the indicators described above and obtained the following results.

There was a significant increase in the "threshold" in Japan until mid-April.

Approximately 2 weeks after the declaration of the state of emergency and partial lockdown by the Japanese government on April 7, 2020, the threshold value started to decrease and this well represents the decrease in new cases during this period. Tokyo was prominent in dynamic behavior (Fig. 1A) . In terms of the global situation, the poor situation of the U.S.A. is obvious, with diverging growth of COVID-19, followed by Brazil, Russia, and India (Fig. 1B) .

Re(v) in Japan can evaluate qualitative differences among the 13 prefectures subjected to special cautions defined by the Japanese government (from April 16, 2020

to May 14, 2020; probable infections with the disease were supposed to be 2 weeks before this, possibly during April 2020) and the remaining 34 prefectures. The former ranged mostly between 0.3 and 0.8 (disregarding Ibaraki and Gifu with lower values), while the latter ranged mostly between 0.1 and 0.4 during April (Fig. 3A) . For the world, again the U.S.A. is notable with diverging growth of COVID-19. Brazil, Russia, Peru, . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 29, 2020. . https://doi.org/10.1101/2020.05.29.20116491 doi: medRxiv preprint India, Mexico, and Saudi Arabia follow this trend to a lesser extent. The remaining countries appear to be showing signs of improvement (Fig. 2B ).

Re(s) is an indicator related to the covariance of the data, and 0 < Re(s) < 1 means the extent of acceleration in the population is decreasing, while 1 < Re(s) < 2 means increasing, within the category of Gaussian fluctuations. Re(s) = 1 is a neutral situation. Re(s) > 2 means an explosive increase/decrease. Further information is available in Adachi (2019a) 4 . For Re(s) in Japan, infections were sporadic until March 22, 2020; the growth acceleration mode then shifted (in this case, increased) until May 6, 2020. Subsequent to this, the mode shifted back to a rather repressive mode. These interpretations well represent the actual dynamics in the growth of infected people (Fig.   3A ). For the world, explosions ended in mid-March. Explosions in individual countries such as Spain, U.K., and Russia were observed during April. China seems to have a prominent output representing improvement as Re(s) > 3 (Fig. 3B) .

Moving on to E(l), in Japan, empirically E(l) ~ 10 until March 24, 2020 was a safer course; E(l) ~ 15 during April 2020 was problematic; and E(l) ~ 20 at the beginning of May 2020 represents danger, followed by a significant decrease to E(l) ~ . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 29, 2020. . https://doi.org/10.1101/2020.05.29.20116491 doi: medRxiv preprint 10 again afterwards (Fig. 4A) . For the world, the top 8 countries still seem to be in a dire situation, while the next 8 countries have settled down to a safer course, followed by possibly gradually reentering problematic situations once again (Fig. 4B) . Note that E(l) would converge to a particular value at each time point, regardless of overlapping sets of data. If individual datums are somewhat different, it would converge to a different value.

For "expected sums" in Japan, it seems that hundreds of people in each prefecture still have potentials for infection (Fig. 5A) . For the world, it appears that tens to hundreds of thousands of people in each country still have potentials for infection ( Fig. 5B) .

Through the newly developed exPzDom model, we can calculate threshold, Re(s), Re(v), E(l), and expected sums. As shown, these indicators provided a good fit to the actual development of policies by national governments and could therefore be usefully leveraged in decision-making contexts pursuant of objective evaluation based on . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted May 29, 2020. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted May 29, 2020. . https://doi.org/10.1101/2020.05.29.20116491 doi: medRxiv preprint

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Exploring group theory and topology for analyzing the structure of biological hierarchies

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The copyright holder for this preprint this version posted May 29, 2020. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)The copyright holder for this preprint this version posted May 29, 2020. . https://doi.org/10.1101/2020.05.29.20116491 doi: medRxiv preprint