The Coexistence of Infection Spread Patterns in the Global Dynamics of COVID-19 Dissemination

The novel coronavirus SARS-CoV-2, commonly referred to as COVID-19, triggered the global pandemic. Although the nature of the international spread of infection is an important issue, extracting diffusion networks from observations is challenging because of its inherent complexity. In this paper, we investigate the process of infection worldwide, including time delays, based on global infection case data collected from January 3, 2020 to December 31, 2022. We approach the data with a complex Hilbert principal component analysis, which can consider not only the concurrent relationships between elements, but also the leading and lagging relationships. Then, we examine the interactions among countries by considering six factors: geography, population, GDP, stringency of countermeasures, vaccination rates, and government type. The results show two primary trends occurring in 2020 and in 2021-2022 and they interchange with each other. Specifically, European, highly populated, and democratic countries, i.e., countries with high mobility rates, show leading trends in 2020. In contrast, African and nondemocratic countries show leading trends in 2021-2022, followed by countries with high vaccination rates and advanced countermeasures. The results reveal that, although factors that increase infection risk lead to certain trends at the beginning of the pandemic, these trends dynamically changes over time due to socioeconomic factors, especially the introduction of countermeasures. The findings suggest that international efforts to promote countermeasures in developing countries can contribute to pandemic containment.

3 Analyses of the second eigenvectors The second eigenvector for the 2020 data shows wide ranges for lead and lag, in contrast to the first eigenvector for the 2020 data (SI Figure 21).In terms of regions, the results are entirely different from those of the first eigenvector, with Africa leading and Europe lagging.This suggests that even during this period, Africa and Europe already show leading and lagging trends, respectively, as observed from 2021 to 2022.In terms of population, a lead-lag relationship can be observed, with countries with larger populations leading; however, population does not significantly affect the amplitude.The same results are found for the stringency index.Regarding the democracy index, the lead-lag relationships and amplitudes are completely opposite those observed with the first eigenvector.The leading trend is associated with a lower democracy index, and the lagging trend is associated with a higher democracy index.Moreover, a larger amplitude is associated with a lower democracy index, and a smaller amplitude is associated with a higher democracy index.
The second eigenvector for the 2021 data shows that the spread in the argument is the same as that for the first eigenvector (SI Figure 22), but there are some differences in the results.The first eigenvector does not show outstanding features in region or democracy index, but the second eigenvector captures the leading trend of European countries and a high democracy index.There are no distinctive results for the population or stringency indeces, similar to the results of the analyses for the first eigenvector.

SI Figure 14 :
Figure 5 in the main text.SI Figure 18: The second eigenvectors for the entire period data.The panels are coloured by region, population, stringency index, and democracy index.SI Figure 20: The second eigenvectors for the 2020 data.The panels are coloured by region, population, stringency index, and democracy index.SI Figure 22: The second eigenvectors for the 2021 data.The panels are coloured by region, population, stringency index, and democracy index.SI Figure 24: The second eigenvectors for the 2022 data.The panels are coloured by region, population, stringency index, and democracy index.

SI Figure 5 :
Scree plots of CHPCA and the RRS.Each panel covers different years, i.e., 2020, 2021, and 2020.The error bars for the RRS indicate the 1% significance level, which is 2.33 times the standard error.

SI Figure 6 :
Eigenvectors with the first eigenvalue coloured by region.The abscissa corresponds to the argument, and the ordinate corresponds to the absolute value (amplitude) of the eigenvector.Note that time progresses from right to left.Panels (a-d) show the entire period, 2020, 2021, and 2022, respectively.

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Figure 7: Eigenvectors, with the first eigenvalue coloured by region.The abscissa corresponds to the real axis, and the ordinate corresponds to the imaginary axis.Panels (a-d) represent the entire period, 2020, 2021, and 2022, respectively.Note that time progresses from right to left.

SI Figure 8 :
Eigenvectors with the first eigenvalue coloured according to population.The abscissa corresponds to the argument, and the ordinate corresponds to the absolute value (amplitude) of the eigenvector.Note that time progresses from right to left.Panels (a-d) represent the entire period, 2020, 2021, and 2022, respectively.The population data for each year are used for the analysis of the corresponding year, while the 2020 population data are used for the analysis of the entire period.In addition, the colour denotes the rank of the analysed countries by population.If a country has no population record, it is coloured grey.

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Figure9: Eigenvectors, with the first eigenvalue coloured according to the GDP per capita.The abscissa corresponds to the real axis, and the ordinate corresponds to the imaginary axis.Panels (a-d) represent the entire period, 2020, 2021, and 2022, respectively.The population and GDP data for each year are used for the corresponding year, while the 2020 population and GDP data are used for the analysis of the entire period.In addition, the colour indicates the rank of the analysed countries by GDP per capita.If a country has no population or GDP record, it is coloured grey.

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Figure 10: Barycentres for five groups of GDP per capita based on rank.The abscissa corresponds to the argument, and the ordinate corresponds to the absolute value (amplitude) of the eigenvector.Panels (a-d) represent the entire period, 2020, 2021, and 2022, respectively.Note that time progresses from right to left.

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Figure11: Eigenvectors, with the first eigenvalue coloured according to the stringency index.The abscissa corresponds to the argument, and the ordinate corresponds to the absolute value (amplitude) of the eigenvector.Note that time progresses from right to left.Panels (a-d) represent the entire period, 2020, 2021, and 2022, respectively.The mean of each year's stringency index data is used for the corresponding year, while the mean of the entire period is used for the analysis of the entire period.In addition, the colour indicates the rank of the countries by stringency index.If a country has no stringency index record, it is coloured grey.
SI Figure 12: Barycentres for five groups of each stringency index based on rank.The abscissa corresponds to the argument, and the ordinate corresponds to the absolute value (amplitude) of the eigenvector.Panels (a-d) represent the entire period, 2020, 2021, and 2022, respectively.Note that time progresses from right to left.
(a) Entire period (b) 2020 (c) 2021 (d) 2022 SI Figure 13: Eigenvectors, with the first eigenvalue coloured according to the containment and health index.The abscissa corresponds to the argument, and the ordinate corresponds to the absolute value (amplitude) of the eigenvector.Note that time progresses from right to left.Panels (a-d) represent the entire period, 2020, 2021, and 2022, respectively.The mean of each year's stringency index data is used for the corresponding year, while the mean of the entire period is used for the entire period analyses.In addition, the colour indicates the rank by the containment and health index among the analysed countries.If a country has no containment and health index record, it is coloured grey.

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Figure 14: Barycentres for five groups of the containment and health index group based on rank.The abscissa corresponds to the argument, and the ordinate corresponds to the absolute value (amplitude) of the eigenvector.Panels (a-d) represent the entire period, 2020, 2021, and 2022, respectively.Note that time progresses from right to left.

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Figure 15: Eigenvectors, with the first eigenvalue coloured according to the vaccination rate.The abscissa corresponds to the argument, and the ordinate corresponds to the absolute value (amplitude) of the eigenvector.Note that time progresses from right to left.Panels (a-d) represent the entire period, 2020, 2021, and 2022, respectively.Each year's vaccination rate is used for the corresponding year, while the average vaccination rate is used for the entire period.In addition, the colour indicates the rank of the countries by vaccination rate.If a country has no vaccination record in the entire period or in 2020, it is coloured grey.

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Figure 16: Barycentres for five groups of the vaccination rate based on rank.The abscissa corresponds to the real axis, and the ordinate corresponds to the imaginary axis.Panels (a-d) represent the entire period, 2020, 2021, and 2022, respectively.Note that time progresses from right to left.

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Figure17: Eigenvectors, with the first eigenvalue coloured according to the democracy index.The democracy index was obtained from the Economist Intelligence Unit[1].The abscissa corresponds to the argument, and the ordinate corresponds to the absolute value (amplitude) of the eigenvector.Note that time progresses from right to left.Panels (a-d) represent the entire period, 2020, 2021, and 2022, respectively.Each year's democracy index data are used for the analysis of the corresponding year, while the 2020 democracy index data are used for the analysis of the entire period.In addition, the colour represents the rank of the analysed countries by the democracy index.If a country has no democracy index record, it is coloured grey.
SI Figure18:The second eigenvectors for the entire period of data.The panels are coloured by (a) region, (b) population, (c) stringency index[2], and (d) democracy index[1].If a country has no population, stringency index, or democracy index data, it is coloured grey.The abscissa corresponds to the argument, and the ordinate corresponds to the absolute value (amplitude) of the eigenvector.Note that time progresses from right to left.
SI Figure19: Barycentres of the second eigenvectors for data of the entire period.The panels are coloured by (a) region, (b) population, (c) stringency index[2], and (d) democracy index[1].The abscissa corresponds to the argument, and the ordinate corresponds to the absolute value (amplitude) of the eigenvector.Note that time progresses from right to left.
SI Figure20:The second eigenvector for the 2020 data.The panels are coloured by (a) region, (b) population, (c) stringency index[2], and (d) democracy index[1].If a country has no population, stringency index, or democracy index data, it is coloured grey.The abscissa corresponds to the real axis, and the ordinate corresponds to the imaginary axis.Note that time progresses from right to left.
SI Figure21: Barycentres of the second eigenvectors for the 2020 data.The panels are coloured by (a) region, (b) population, (c) stringency index[2], and (d) democracy index[1].The abscissa corresponds to the argument, and the ordinate corresponds to the absolute value (amplitude) of the eigenvector.Note that time progresses from right to left.
SI Figure22:The second eigenvectors for the 2021 data.The panels are coloured by region, population, stringency index[2], and democracy index[1].If a country has no population, stringency index, or democracy index data, it is coloured grey.The abscissa corresponds to the real axis, and the ordinate corresponds to the imaginary axis.Note that time progresses from right to left.
SI Figure23: Barycentres of the second eigenvectors for the 2021 data.The panels are coloured by (a) region, (b) population, (c) stringency index[2], and (d) democracy index[1].The abscissa corresponds to the argument, and the ordinate corresponds to the absolute value (amplitude) of the eigenvector.Note that time progresses from right to left.
SI Figure24:The second eigenvectors for the 2022 data.The panels are coloured by region, population, stringency index[2], and democracy index[1].If a country has no population, stringency index, or democracy index data, it is coloured grey.The abscissa corresponds to the real axis, and the ordinate corresponds to the imaginary axis.Note that time progresses from right to left.
SI Figure25: Barycentres of the second eigenvectors for the 2022 data.The panels are coloured by (a) region, (b) population, (c) stringency index[2], and (d) democracy index[1].The abscissa corresponds to the argument, and the ordinate corresponds to the absolute value (amplitude) of the eigenvector.Note that time progresses from right to left.
SI Figure26: Barycentres of the third eigenvectors for the data of the entire period.The panels are coloured by (a) region, (b) population, (c) stringency index[2], and (d) democracy index[1].The abscissa corresponds to the argument, and the ordinate corresponds to the absolute value (amplitude) of the eigenvector.Note that time progresses from right to left.