Insights into the dynamics of SARS-CoV-2 pandemic via Shannon-Fisher causality plane

This paper performs a systematic investigation into the temporal evolution of severe acute respiratory disease coronavirus 2 (SARS-CoV-2) pandemic considering 15 diverse countries. Based on the foundations of Information Theory, we apply the Shannon-Fisher causality plane (SFCP), to map the dynamics behavior inherent to SARS-CoV-2 and their respective locations along the (SFCP). Our results show that this dynamics varies widely along the SFCP from the lower-right region, characterized by high entropy and low degree of reliability in relation to the information extracted from the analyzed data set to the top-right region, characterized by the less entropic and high degree of reliability in relation to the information extracted from the analyzed data set. It reveals that we have three different groups of countries in controlling the SARS-CoV-2 pandemic. A country that was proactive in implementing measures such as social distancing, quarantine, orders to stay at home, testing symptomatic and asymptomatic loads and hygienic measures to limit the impacts of SARS-Cov-2 (China) and that today is clearly in the decay phase with the number of cases tending to zero and is no longer in a pandemic situation (efﬁcient). Moderately proactive countries, ie, implemented measures only when the spread of SARS-Cov-2 was already reaching the country (France, Germany, United Kingdom, Spain, Sweden, Italy, Ireland, USA, Austria, and Canada) (moderately efﬁcient) and the reactive countries, which took a long time to implement the measures and/or the infection came later and as a result are not managing to reduce the number of daily cases of SARS-CoV-2 (Russia, Iran, Brazil, and India) (inefﬁcient) and are the new epicenters of the SARS-COV-2 pandemic. Besides, we applied the Bandt Pompe permutation entropy ( H ) and the Fisher Information ( F ) to obtain the rank of the most efﬁcient countries to the ﬁght against the SARS-CoV-2. To the best of our knowledge, no researches have been ranking the most proactive countries in the ﬁght against the SARS-CoV-2 dissemination. We truly believe that the empirical results showed in this research draws new perspectives that can collaborate in the formulation of more efﬁcient healthy public policies to combat SARS-CoV-2 spread.


Introduction
A variety of preventive strategies have been put in place by different governments across the world. These strategies include testing symptomatic and asymptomatic carries, social distancing, quarantine, stay-at-home orders, and hygienic measures 1,7 . These strategies had different effects in diverse countries according to the time they were applied and other factors.
Thus, the chief goal of this research is to investigate the temporal evolution for SARS-CoV-2 pandemic consider the daily incidence number of COVID-19 cases for 15 countries based on the Shannon-Fisher causality plane (SFCP) 8,9 . Thus, we applied the Bandt & Pompe permutation entropy 10 and Fisher Information measure 11,12 which allowed us to map and rank the countries most efficient in controlling the SARS-CoV-2 pandemic. The SFCP is a powerful tool of Information Theory to investigate the global and local characteristics of the Bandt Pompe's probability density function (PDF) related to the dynamics of SARS-CoV-2 dissemination.
The highlights of this research for the literature are: First, it analysis the complex dynamics of SARS-CoV-2 pandemic through the estimation of the Bandt & Pompe permutation entropy combined with Fisher Information measure. Second, it builds up the SFCP for SARS-CoV-2, which allowed us to map the virus dynamic behaviour and their respective locations along the SFCP for each country. Third, it shows the ranking of the countries most efficiently in controlling the SARS-CoV-2 pandemic taking into account the complexity hierarchy.

Results
The insights promoted by our fitting procedure are based on data comprising of the number of daily incidences related of COVID-19 cases in 15 countries (Methods section for more details). For each time series, we investigate the phenomenology inherent to the temporal evolution of SARS-CoV-2 pandemic. These types of time series are widely known to be non-stationary. Despite that, the time series of the daily number of COVID-19 cases can be used to characterize the fractional Brownian motion, also a non-stationary process [13][14][15] .
Given this, taking into account the permutation entropy as a way to discriminate time series, the previous empirical results showed that better results 16,17 are provided by original series, in this case is the daily incidence of SARS-CoV-2 infection. For each country, the time series of temporal evolution of SARS-CoV-2 pandemic are shown in Fig. 1.
The temporal evolution of the number of daily incidences related to COVID-19 depicts three distinct phases: an exponential increase phase, a plateau phase, and a decrease phase. In view of this, it seems that China, France, and Germany are in the decrease phase, while Brazil and India are in the exponential increase. In turn, Iran deserves attention once it peaked at April followed by a decay phase and then a new increase since May. While, the US presented a critical exponential increase phase, but it seems systematically decrease.
The SARS-CoV-2 pandemic generated strong negative effects into the investigated countries in this research, causing abrupt changes in the worldwide status quo of the people lives 7 . For each country, we perform the Boxplot 18 to verify the anomalous values (outliers). Fig. 2 presents the Boxplot.
The analysis of Fig. 2 reveals the number of outliers for each country. It is an indication that extreme events in the time series of the daily incidence of COVID-19 are more recurrent for Austria, Brazil, China, France, Germany, India, Ireland, and Spain. Extreme events can be understood as statistically improbable occurrences. Even with a high probability of not occurring, these extreme events are observed in financial, social, and natural systems, bringing hard consequences 19 .
In this way, we also applied the univariate quantile-quantile (Q-Q) plot 20 , which is a powerful tool used to examine the distributional similarities and differences between two independent samples [20][21][22] . Fig. 3 depicts the Q-Q plot.
The Q-Q plot is a non-parametric approach more robust than the classical histogram to study the statistical properties inherent to measures of central tendency, dispersion and skewness. For all countries investigated in this analysis, there is a standard statistical behavior in relation to the daily observations of COVID-19 cases. Especially at the ends of the Q-Q plot, the daily incidences follow a strong non-linear pattern, revealing that the data are not distributed by a Gaussian (X ∼ N (0, 1)).
Based on the Information Theory quantifiers, Permutation entropy (H), Statistical complexity (C), and Fisher Information (F), all measures evaluated by Bandt & Pompe method, it is possible to define three causality information planes 23 . Here, we use the causality Shannon-Fisher plane (HxF) which performs an analysis based on the global and local characteristics of the Bandt Pompe's probability density function (PDF). We emphasize that there are some peculiars characteristics between the causality Shannon-Complexity plane (HxC) and Shannon-Fisher causality plane (HXF). The focus of (HxC) comprises the global characteristics of the associated time series Bandt Pompe's probability density function (PDF). Thus, the range variation of (HxC) includes [0, 1] x [C min ,C max ] 24 .
The Shannon-Fisher causality plane considers the global and local characteristics of the Bandt Pompe's PDF. The range is [0, 1] x [0, 1]]; no limit curves have been shown to exist so far 25 . This approach has already been used successfully to distinguish noise from chaos 26 , to investigate the characterization of motor, imagery movements in electroencephalogram 27 , to study the physiology of cerebral cortex 23 , to analyze the complex dynamics relative of observed and simulated ecosystem gross primary productivity 25 and in the info-quantifiers to the logistic map 9  For each country, the temporal evolution of the number of daily incidence related to COVID-19 day from January 22, 2020 until May 28 2020 with 128 observations. These time series present peculiar characteristics such as non-linear dynamics, noisy and chaotic characteristics.

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China is the only country that is located within the ideal position zone to a stochastic process related to a random walk. It reveals that China is no longer in a pandemic situation and presents no community contagion. France is the country that is closest to entering this ideal position zone followed by Germany and the UK and will soon no longer be in a pandemic situation and is without presenting community contagion.
The other countries are far from this ideal position zone to a stochastic process related to a random walk. However, with the exception of Sweden, Brazil, India, Iran, and Russia the other countries are already in the decay phase. Sweden has a behavioral anomaly in terms of the spread of SARS-CoV-2. Apparently, it is the only one on the plateau phase. Brazil, India, Iran, and Russia are in the exponential growth phase related to the daily number of COVID-19 cases.
The SFCP method measures the magnitude of the total impact of the SARS-CoV-2 pandemic for the entire period of analysis. Thus, a temporal analysis of the number of COVID-19 cases is not carried out, but a general picture of the pandemic situation for the countries investigated. In this sense, what is being compared is the general picture of 15 countries, which allows us to verify which ones were most impacted considering the number of daily incidences SARS-CoV-2 infection.
The temporal evolution of the number of daily COVID-19 cases tends to have their starting positions close to the lower-right region at the Shannon-Fisher causality plane. The countries located in this region or in its surroundings are characterized by high entropy and low degree of reliability in relation to the information extracted from the analyzed data set, so their behavior is closer to a random walk. These countries have adopted more efficient measures in controlling the SARS-CoV-2 pandemic. These measures are directly associated with the fulfillment of the determinations of the World Health Organization (WHO) and with a greater political and social engagement we can highlights social distancing, quarantine, stay-at-home orders, testing symptomatic and asymptomatic loads and hygienic measures to control the SARS-CoV-2 pandemic.
While the countries located in the SFCP intermediate region or in its surroundings are characterized by a less entropic degree and a high degree of reliability in relation to the information extracted from the analyzed data set. These countries took a long time to adopt the measures recommended by the WHO and the political and social engagement was delayed as a result of that they still have a moderate risk of spreading by SARS-CoV-2.
The countries located to the top-right region at SFCP in its surroundings are characterized by lower entropy and higher degree of reliability in relation to the information extracted from the analyzed data set. These countries took even longer to adopt the measures recommended by the WHO and have a low level of political and social engagement in complying with these measures, thus presenting a high risk in relation to contagion by SARS-CoV-2. In addition, the situation related to this health crisis is enhanced by the collapse of the health system, poverty and a few years of educational studies.
The permutation entropy H and Fisher information F are used to quantify the degree of efficiency in combating the impacts of the spread of SARS-CoV-2 considering the time series of daily incidences of COVID-19 cases. For time series of the daily incidence of SARS-CoV-2, the time evolution is characterized by a temporal pattern derives in deviations from the ideal position to a stochastic process inherent a random walk. The higher distance to this random ideal position reflects a higher level of efficiency in the controlling the SARS-CoV-2 dissemination. Table 1 presents a rank of successful countries in controlling the SARS-CoV-2 pandemic and these impacts based on the complexity hierarchy. For each country, the higher distance to point (1.0) reflects a higher level of efficiency in controlling the SARS-CoV-2 pandemic. In this sense, the results presents that China, France, Germany and U.K. are near to the lower boundary of the Shannon-Fisher causality plane. Because of this, they can be considered more efficient countries 16,[28][29][30] . Moreover, the Euclidian distance presents a relevant discrepancy in the distance between China and France. It reveals that China is less complex per entropy value than France.
Otherwise, the lower distance from point (1.0) reflects less level of efficiency in the combat against to the SARS-CoV-2. Our results show that India, Brazil, Russia and Iran are lying significantly farther from the right corner leads behavioral dynamic related to SARS-CoV-2 far from a random walk. Thus, these countries are less efficient 16 and show long-term correlated. Moreover, these countries are characterized by lower entropy and a lower degree of randomness. Given this, these countries are the most inefficient in the combat against to the SARS-CoV-2.

Discussion
Based on the empirical results presents in Fig. 1, Fig. 2, Fig. 3, Fig. 4 and Table 1 we perform individual analyzes to understand the complex phenomenology inherent of the number of daily incidence of COVID-19 cases. Our results show that we can cluster the 15 countries investigated into 3 different groups related to the SARS-CoV-2 pandemic, the efficient countries, the moderately efficient countries and the inefficient countries.
China is located in a region of the highest entropy and lowest Fisher information. These results may be a consequence of a quick and proactive stance in controlling the SARS-CoV-2 spread since had emerged the epidemic outbreak disease 31 , although it had a high peak in the daily incidence of COVID-19 cases in the beginning of the pandemic. China has been preparing to contain future pandemics by applying lessons learnt from SARS ever since 2003 32 . Within a matter of weeks, China implemented all the tools ranging from case detection with immediate isolation, and contact tracing with quarantining and medical observation of all contacts 33 .
Although Europe does not have the same preparation than China, Germany, France and U.K. are the countries closest to reaching this stage that today is China, a country that has a lower daily incidence of SARS-CoV-2 infection rate tending to zero. These countries clearly have a drastic reduction in COVID-19 cases in the last days studied. However, they did not invested in the strategy of mass testing, contact tracing, and physical distancing in the early stage of SARS-CoV-2 spread which led them to failures in inicial stage containment of COVID-19 pandemic 34, 35 But, we believe that public health policies implemented together with some factors such as population adherence to social isolation measure, genetic and age composition 36 and high human development index (HDI) may have helped in decreasing the impact of SARS-CoV-2 in these countries.
Spain, Italy, Ireland, United States, Austria and Canada, already presenting a curve of cases confirmed by the decaying SARS-CoV-2, but still indicating a moderate risk of having new outbreaks if sanitary measures are not followed and there is early relaxation. Sweden has adopted more relaxed strategy to control the SARS-CoV-2 pandemic with no massive testing on suspected individuals and no strict lockdown in her most affected regions and, probably, because of this, we can observe a difference in a pattern of COVID-19 cases.
In the top-right region investigated by the Shannon-Fisher causality plane are Russia, Iran, Brazil and India. These countries are presenting high daily cases of SARS-CoV-2, their curves are growing and, therefore, their position on the Shannon-Fisher causality plane, present greater distance to the point of efficiency. Brazil and Iran had confirmed cases later than the others countries already mentioned, which contributed to being in the pandemic control stage presented. Among the factors that may be influencing Russia in its low efficiency and the response to COVID-19, the lack of clear political leadership has been pointed out, just like in Brazil 37 .
Brazil and India are two very populous countries that are undergoing economic development. Facing an outbreak of COVID-19 with these characteristics is quite challenging. Specifically in the case of Brazil, where we are familiar with the situation, several factors such as a permissive culture, political polarization, 38 high social inequality, poor sanitation in many regions, public transportation often crowded in large cities, among other characteristics may be contributing enormously to the low efficiency in controlling the pandemic.
In conclusion, our results reveal different behavioral dynamics of the SARS-CoV-2 epidemic in the diverse countries evaluated. It is important to have instruments in order to analyze the country's efficiency in controlling the pandemic so that it is possible to know the strategies that were used by the most successful countries. Thus, a more effective response can be taken in the face of a new SARS-CoV-2 outbreak, or by countries that are still experiencing the first outbreak. However, it is worth noting that each country has its own intrinsic economic, social, behavioral, genetic, and age characteristics, among others, that will influence the final result, bringing about a more or less effective control of SARS-CoV-2 outbreaks.

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Methods This section has been segregated into 2 distinct subsections in which we present the theoretical framework of the methods used in this research. The Bandt & Pompe method (BPM) and the Shannon-Fisher Causality plane (SFCP). Fig. 5 depicts the flowchart related to the theoretical framework used in this research.

Data
The main database used in our analysis is the time series of daily incidence related to SARVS-COV-2 taking into account 19 countries. For each country, the periods cover 122 day from January 22, 2020 until May 28 2020 with 128 observations. The data were obtained via public service on the internet at COVID-19 Dashboard by the Center for Systems Science and Engineering (CSSE) at Johns Hopkins University,https://github.com/CSSEGISandData/COVID-19. According to the terms of use, the data is public and can be used for public health, educational, and academic research purpose. A list of countries considered in this research with country name and the respectively geographical coordinates and Human Development Index (IDH) is presented in Table 2. Specifically, the HDI is an index that considers in its calculation 3 variables, life expectancy at birth, expected years of schooling for children and mean years of schooling for adults, and per capita income.
In this research, we taking into account countries that present very high human development (Austria, Canada, France, Germany, Ireland, Italy, Russia, Spain, Sweden, U.S. and U.K.), high human development (Brazil, China and Iran) and medium human development (India).
For each country, regardless of the value related to its HDI, it is notable that SARVS-CoV-2 has dramatically altered social, economic structures and the health system. Enforcing the isolation of people on a global scale.

Permatutation entropy
The Permutation Entropy (PE) approach was proposed by Bandt & Pompe 10 as the Shannon's entropy 39 to quantify the probability distribution of ordinal patterns which are constructed through the symbolization technique that considers time causality by comparing neighboring values of the time series under analysis 40,41 .
The permutation entropy is a measure for information content in time series 42 which is apply to evaluate uncertainty, disorder, state-space volume, and lack of information 43 inherent in a process as displayed by discrete measurements of a parameter of the system 44 .
Thus, more predictable signals (which show a tendency to repeat only a few ordinal patterns) have lower permutation entropy than less predictable signals (which tend to exhibit all possible ordinal patterns).
This method taking into account a symbolic sequence of ordinal patterns of segments (words) of a given size (embedding dimension) denoted by d 16,40 . Then, for each ordinal pattern the respective instances are counted in order to construct an 6/15 ordinal pattern probability distribution 17 . More specifically, these ordinal patterns related to the quantifying of Bandt & Pompe probability distribution are calculated by observing the local ordering of consecutive values within each word.
In view of this, the Bandt & Pompe method (BPM) can be described as follows. For a given time series x t ,t = 1, ..., T initially T − (d − 1) overlapping segments X t = (x t , x t+1 , ..., x t+d−1 ) of length d (incorporation dimension) are generated and within each segment the values are classified in ascending order, which provides the set of indices r 0 , r 1 , ..., r d−1 such that x t+r 0 ≤ x t+r 1 ≤ ... ≤ x t+r d−1 . The corresponding sequences denoted by π = r 0 , r 1 , ..., r d−1 can take on any of d! possible permutations of the set {0, 1, ..., d − 1} and are symbolic representatives of the original segments. The permutation entropy of order d ≥ 2 is now defined as a Shannon entropy of probability distribution P(π), given by where {π} represents the summation over all the d! possible permutations of order d, and p(π) denotes the relative frequency of occurrences of the permutation π. It follows 0 ≤ H(d) ≤ log d! t follows that 0 ≤ H(d) ≤ log d! where the lower limit is reached for a strictly increasing or decreasing series (only one permutation appears), and the upper limit for a completely random series where all d! Possible permutations appear with the same probability. The ideal d strongly is related to the phenomenology of each event studied, but in order to promoted a goodness of fit statistics as a rule of thumb it is typically recommended 40 to choose maximum d such that T > 5d!. In addition, the application of BPM is recommended not only to time series of low dimensional dynamical systems, but also to any type of time series such as regular, chaotic, noisy or reality based) 29 . Given this characteristic, the BPM has been successfully applied in distinct research areas such finance 28,29,41 , geophysics 45 , engineering [46][47][48] , physiology 49-51 , hydrology 52, 53 and climatology 54 Fisher information Fisher (1922) proposed a statistical measure of indeterminacy now called Fisher Information 55 . There are some possibilities for interpreting this statistical measure (i) as a measure of the ability to estimate a parameter, (ii) as the amount of information that can be extracted from a set of measures (the "quality" of the measurements) and (iii) as the disorder state measure of a system or phenomenon 11,12,56 being its most important property is the so-called the Cramer-Rao Bound (CRB) for nonlinear parameter estimation 9, 57-60 . The Cramer-Rao inequality 26 is obeyed by all time series of quarterly GDP'S.
Specifically, Fisher information is a robust measure given by which evaluate the gradient content of the distribution f (continuous PDF), thus being quite sensitive even to tiny localized perturbations.
In view of this, it is essential to note that the gradient operator significantly influences the contribution of tiny local f -variations to the value of Fisher's information, so that the quantifier is called "local". Note that Shannon's entropy decreases with the skewed distribution, while Fisher's information increases in this case. Local sensitivity is useful in scenarios whose description requires an appeal to a notion of "order" 8,9,26 .
The concomitant problem regarding loss of information due to discretization has been thoroughly studied, for more details see ( 61-63 and their references) and, in particular, it implies the loss of Fisher's shift invariance, which has no importance for our current purposes. For the calculation of the Fisher information measure (discrete PDF), we follow the proposal of Dehesa and collaborators 64 based on the amplitude of the probability f (x) = ψ(x) 2 so, its discrete normalized version (0 ≤ F ≤ 1) is defined by

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In this paper the normalization constant F 0 reads F 0 = 1, if P i * for i * = N and P i = 0∀i = i * , 1/2 otherwise.
Based on the level of ordering of the investigated system, we can considering that it describe by a probability density function (PDF) given by P o = {p k ∼ = 1; p i ∼ = 0 ∀ i = k; i = 1, ..., N} (with N, the number of states of he system) in consequence we have a Shannon entropy S [P o ] ∼ = 0 and a normalized Fisher's information measure F [P o ] ∼ = F max = 1. Otherwise, if the investigated system is represented by a very disordered state, we can understand this particular state is described by a PDF given by the uniform distribution P e = p i = 1 N ∀ i = 1, ..., N .