Since the identification of a novel coronavirus (COVID-19) in Wuhan, China, in December 2019, the virus kept spreading around the world. One of the most remarkable characteristics of COVID-19 is its high infectivity, resulting in a global pandemic. In this complex scenario, tasks such as protecting the people from the infection and the global economy are considered two major challenges. In order to improve our knowledge about COVID-19 and its behavior in different countries around the world, we exhaustively explore the real time-series of cumulative number of confirmed infected cases by COVID-19 in the last few months until March 27, 2020. In our analysis, we considered the Asia, Europe, North America, and South America. Our main findings clearly show the existence of a well established power-law growth and a strong correlation between power-law curves obtained for different countries. These two observations strongly suggest a universal behavior of such curves around the world. To improve our analysis, we use a model with six autonomous ordinary differential equations, based on the well-known SEIR (Susceptible–Exposed–Infectious–Recovered) epidemic model (considering quarantine procedures), to propose efficient strategies that allow the government to increase the flattening of the power-law curves. Additionally, we also show that soft measures of quarantine are inefficient to flatten the growth curves.

The paper is divided as follows. Section II presents the power-law growth of confirmed infected cases of COVID-19 and the DC between pairs of countries is determined. Section III discusses numerical results using the proposed model showing many strategies to flatten the power-law growth. In Sec. IV , we summarize our results.

In line with the above publications, the present work analyzes the time-series evolution of the COVID-19 for the following countries: Brazil, China, France, Germany, Italy, Japan, Republic of Korea, Spain, and the United States of America (USA). In all cases, we observe a power-law increase for the positive detected individuals, where the exponentchanges for different countries. In addition to the power-law behavior, we also computed the Distance Correlation (DC)between pairs of countries. The DC is able to detect nonlinear correlations between data.We show that power-law data are highly correlated between all analyzed countries. This strongly suggest that government strategies to flatten the power-law growth, valid for one country, can be successfully applied to other countries and continents. Furthermore, a model of Ordinary Differential Equations (ODEs) is proposed and some strategies to flatten the power-law curves are discussed using the numerical simulations.

In fact, recent analysis regarding the behavior of the COVID-19 in China demonstrated a power-lawgrowth of infected cases.The authors found exponents around, which do not vary very much for different provinces in China. This suggests that socioeconomic differences, local geography, differences in containment strategies, and heterogeneities essentially affect theof the exponentbut not the qualitative behavior. A model of coupled differential equations, which includes quarantine and isolation effects, was used by the authors to match real data. Power-law growths for China were also obtained in another study and a possible relation to fractal kinetics and graph theory is discussed.

In general, the average reproductive number, which gives the number of secondary infected individuals generated by a primary infected individual, is the key quantity which determines the dynamical evolution of the epidemic.Usually, for values, the number of new infected individuals decreases exponentially. For, this number increases exponentially.However, the nature is full of surprises, and there are plenty of cases for which the exponential behavior is substituted by power-lawand are related to branching processes with diverging reproductive number,scale free networks, and small worlds.It was already suggested in the literature that the COVID-19 growth might be a small world.This is in agreement with recent resultssuggesting that, for many countries around the world, the COVID-19 growth has the tendency to follow the power-law.

The astonishing increase of positive diagnosed cases due to COVID-19 has caught the attention of the whole world, including researchers of many areas and governments. It is urgentto find explanations for the already known data and models that may allow us to better understand the evolution of the virus. Such explanations and models can hopefully be used to implement social policies and procedures to decrease the number of infections and deaths. Time is essential to avoid economic and social catastrophes.

, anddisplay the corresponding DC calculated between the countries. Results show that DC between the curves is relatively high in the beginning. The lowest values for the DC are obtained between Brazil and Italy, in, and between Italy and USA, shown in; in both cases, DC is around 0.4. DC decays substantially when the power-law starts in one country but not in the other. The exception is between Japan and USA. After some days, when both countries reach the power-law behavior, the values of DC become very close to. Thus, they arecorrelated besides distinct exponents. Furthermore, the DC is not necessarily related to the exponent. One example can be mentioned. Even though USA has the largest exponent and Japan the lowest one (considering the error in Table I ), they are highly correlated. Besides that a even though there are not many data available for Brazil, it seems to become more and more correlated with Italy and Japan.

presents specific results for the DC calculated between some selected countries, namely, Brazil, Italy, Japan, and USA. Italy was chosen due to their relevance to Europe, regarding the typical data of the virus. USA was chosen for being the top affected country, and Brazil and Japan represent distinct continents and distinct epidemic containment measures. Thus, we compute the DC between four continents., andare the cumulative number of confirmed cases in each country, as in, but considering data since the first day the infection was reported. In these curves, we clearly see the initial plateaus due to the incubation time. After the plateaus, a qualitative change to the power-law growth (the same from) occurs. The time for which the qualitative change occurs is distinct for each country.

The power-law observed in all cases fromis certainly not a coincidence but a consequence of virus propagation in scale free systems. To quantify the relation between the power-law growth, we use the DC, which is a statistical measure of dependence between random vectors.Please do not confuse the word distance with the geographical distance between the analyzed countries. The most relevant characteristics of DC is that it will be zero if and only if the data are independent and equal to one for maximal correlation between data. Details about the numerical procedure to obtain the DC are given in Appendix A

The most desired behavior is that if the exponentbecomes smaller it leads to the flattening of the curves. But, this is apparently not that easy. Besides USA and Germany, which have a distinct inclination in the beginning of their power-laws, and China and Republic of Korea, which are stabilizing the epidemic spread, for all other countries the growth remains strictly on the fitted curve andessentially does not change in time. In Sec. III , we discuss some possibilities to flatten the power-laws.

The regimes with power-law growth are the most relevant to be discussed since they provide essential information of what is expected for the future and possible attitudes needed to flatten the curves. The exponentchanges for distinct countries and the complete fitting parameters are given in Table I . Results in Table I are presented in decreasing order of the exponent; the USA [] has by far the largest exponent and, therefore, became already the country with epidemic records. Even though Germany [] reported a small number of deaths, it has the second large exponent, followed by Spain [], France [], and Italy [], in this order. China [], Brazil [], Japan [], and Republic of Korea [], in this order, are the last in the list. In the case of China and Republic of Korea, the power-laws are more clear due to the number of available data. For these two countries, a flatten is observed after the power-law. The jump observed after 30 days in China data is due to a change in the counting procedure of infected cases (see the situation report on the February 17, 2020, in Ref.). Republic of Korea, on the other hand, focused on identifying infected patients immediately and isolating them to interrupt transmission.It is interesting to note that for Japan, another country that adopted similar measures, we obtained a similar value for

displays data of the cumulative number of confirmed positive infected cases by COVID-19 of nine countries as a function of the days. The analyzed countries are (in alphabetic order): Brazil, China, France, Germany, Italy, Japan, Republic of Korea, Spain, and the USA. Data were collected from the situation reports published daily by the World Health Organization (WHO).We notice that the values in the vertical axis inchange for different countries. Initial data regarding the incubation time were discarded since they do not contribute to the essential results discussed here. Black-continuous curves are the corresponding fitting curves, whereis the time given in days and, andare parameters. Insets in all plots show the data in the log–log scale. Straight lines in the log–log plot represent the power-law growth. The fact that the growth increases as a power-law is good news since it increases slower than the exponential one. However, that is not good enough.

III. PREDICTIONS AND STRATEGIES Section: Choose Top of page ABSTRACT I. INTRODUCTION II. REAL DATA ANALYSIS III. PREDICTIONS AND STRA... << IV. CONCLUSIONS REFERENCES CITING ARTICLES

19,20 Global dynamics of a SEIR model with varying total population size ,” Math. Biosci. 160, 191– 213 (1999). 19 M. Y. Li, J. R. Graef, L. Wang, and J. Karsai, “,” Math. Biosci., 191–(1999). https://doi.org/10.1016/S0025-5564(99)00030-9 Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: A modelling study ,” Lancet 395, 689– 697 (2020). 20 J. T. Wu, K. Leung, and G. M. Leung, “,” Lancet, 689–(2020). https://doi.org/10.1016/S0140-6736(20)30260-9 8,21–23 8 B. F. Maier and D. Brockmann, “Effective containment explains sub-exponential growth in confirmed cases of recent COVID-19 outbreak in mainland China,” arXiv:2002.07572v1 (2020). 21 K. M. Khalil, M. Abdel-Aziz, T. T. Nazmy, and A. B. M. Salem, “An agent-based modeling for pandemic influenza in Egypt,” in Handbook on Decision Making (Springer, 2012), pp. 205–218. Modeling influenza epidemics and pandemics: Insights into the future of swine flu (H1N1) ,” BMC Med. 7, 30 (2009). 22 B. J. Coburn, B. G. Wagner, and S. Blower, “,” BMC Med., 30 (2009). https://doi.org/10.1186/1741-7015-7-30 et al., “ Transmission dynamics and control of severe acute respiratory syndrome ,” Science 300, 1966– 1970 (2003). 23 M. Lipsitch, “,” Science, 1966–(2003). https://doi.org/10.1126/science.1086616 a priori from other studies and (ii) those related to adjusting the model to the real data (for more details, see The model proposed in this work for the numerical prediction and strategies is presented in detail in Appendix B . It is a variation of the well known Susceptible–Exposed–Infectious–Recovered (SEIR) epidemic modelto propose efficient strategies that allow the government to increase the flattening of the power-law curves. Our SEIR model takes into account the isolation of infected individuals.In this case, quarantine means the identification and isolation of infected individuals. The parameters are divided in two categories: (i) those related to the characteristic of the virus spreading, definedfrom other studies and (ii) those related to adjusting the model to the real data (for more details, see Appendix B ). These parameters can change according to social actions and government strategies.

Numerical results of this section take into account possible interferences or strategies from the government of each country, what means that some parameters must be changed after the last day of the real data. For each distinct strategy, we use distinct colors, which are then plotted.

Fig. 3 For a detailed explanation of variables and parameters, see Appendix B . The colors used infor the distinct scenarios are the following (for continuous curves):

Red curves: the tendency which follows from the behavior of the last points of real data (last values for θ e f f and κ s e f f ). This is what happens if we do not change the current scenario on March 28th.

Blue curves: reduction of social interactions by using smaller values of θ . Dark blue for θ = 1.2 , medium blue for θ = 1.0 , and light blue for θ = 0.8 .

Green curves: reduction of social interaction together with tests to identify and isolate asymptomatic and mild symptomatic cases. Here we use θ = 1.0 and κ a = 0.01 .

Magenta curves: reduction of social interaction together with tests to identify and isolate asymptomatic and mild symptomatic cases. Here, we use θ = 1.0 and κ a = 0.05 .

Orange curves: identification and isolation of asymptomatic and mild symptomatic cases with rate κ a = 0.20 . In this strategy, we do not increase the social distance and use the last value for θ obtained in the adjustment.

For the dashed curves, the configurations are the same inside each color. However, in these curves, the asymptomatic and mild symptomatic identified cases are not accounted for. We notice that without the realization of tests in the population, the asymptomatic individuals would not be computed.

Figs. 3(a) 3(d) κ s = 1 , which means that we assume that all symptomatic individuals are properly isolated. Figures 3(c) 3(d) 120 days. The vertical axis is the cumulative number of positive infected individuals in the population. In the horizontal axis, we have the days since the first computed case in these countries. Black circles are the real data starting from the power-law-like behavior discussed in Sec. θ and κ s that better adjust the simulation results with the data. In the cases shown in Fig. 3 θ and κ s , namely, the values θ e f f and κ s e f f given in the figures. As a consequence, the red curves are in full agreement with the data in this time interval. When the available data end, the simulation continues and the red curves can be used to predict the asymptotic number of confirmed cases since they represent the scenario following the tendency demonstrated by the data. In the case of Italy, we obtain 5.6 × 10 6 and for France, 1.0 × 10 7 . See the tendencies in Figs. 3(c) 3(d) θ e f f and κ s e f f obtained for these countries. Besides θ e f f being larger for France, we obtain κ s e f f = 0 , which can be interpreted as the nonexistence of quarantine measures or the inefficient isolation of symptomatic individuals. We are aware that such asymptotic behavior can be hardly trusted with numerical simulation of models. However, our intention in displaying such asymptotic behavior is to show that the proposed model converges to reasonable values. We start discussing the cases of Italy and France, shown in, respectively. In these cases,, which means that we assume that all symptomatic individuals are properly isolated.andshow the evolution of scenarios fordays. The vertical axis is the cumulative number of positive infected individuals in the population. In the horizontal axis, we have the days since the first computed case in these countries. Black circles are the real data starting from the power-law-like behavior discussed in Sec. II . During the times for which real data are available, the model chooses the values of the parametersandthat better adjust the simulation results with the data. In the cases shown in, we needed three values ofand, namely, the valuesandgiven in the figures. As a consequence, the red curves are in full agreement with the data in this time interval. When the available data end, the simulation continues and the red curves can be used to predict the asymptotic number of confirmed cases since they represent the scenario following the tendency demonstrated by the data. In the case of Italy, we obtainand for France,. See the tendencies inand. The considerable difference between these projections is explained by the last values ofandobtained for these countries. Besidesbeing larger for France, we obtain, which can be interpreted as the nonexistence of quarantine measures or the inefficient isolation of symptomatic individuals. We are aware that such asymptotic behavior can be hardly trusted with numerical simulation of models. However, our intention in displaying such asymptotic behavior is to show that the proposed model converges to reasonable values.

Now, we discuss results for some emblematic scenarios for the model when specific strategies are applied to Italy and France on day March 28th. For both countries, we assume κ s = 1 for all strategies, which means that all symptomatic individuals per day are put into quarantine. We can see that the strategy represented by the orange curves is not sufficient to significantly reduce the total number of confirmed infected individuals for France, since the last value θ e f f = 1.90 indicates a large level of social interaction in this country. On the other hand, for Italy, a considerable reduction is observed, specially for the orange-dashed curve, which indicates only the number of symptomatic cases. Strategies related to the blue curves mitigate the growth of the number of confirmed cases and, with exception of the dark blue case for Italy, lead to smaller asymptotic values when compared to the red and orange scenarios. The light blue curves, related to large social distance ( θ = 0.8 ), are the most efficient scenarios to induce an accentuated reduction of the growth and a fast convergence to the maximal number of confirmed cases. Furthermore, green curves tend to approach the medium blue curves, which means that, for θ = 1.0 , there is no significant difference between isolating 1 % ( κ a = 0.01 ) of the asymptomatic individuals per day or doing nothing. However, increasing the daily ratio of detection and isolation of asymptomatic individuals to κ a = 0.05 , a noticeable reduction of the asymptotic value of infected individuals is observed (see magenta-dashed curves). Nevertheless, none of these strategies are better than increasing the social distance, scenario represented by the light blue curves.

Figs. 4(a) 4(d) Figures 4(c) 4(d) Fig. 4 Next, we discuss cases for Brazil and USA using other strategies. Results are shown in, respectively.andfurnish predictions for the number of infected individuals. Due to the distinct scenarios, we had to change the colors a bit (see also color labels in):

Red curves: have the same meaning as before.

Blue curves: are still related to the reduction of social interaction so that θ can take the values θ = 2.0 , 1.5 , and 1.0 , going from dark blue to light blue.

Green curves: reduction of social interaction together with tests to identify and isolate asymptomatic and mild symptomatic cases. Here, we use θ = 2.0 and κ a = 0.20 .

Magenta curves: reduction of social interaction together with tests to identify and isolate asymptomatic and mild symptomatic cases. Here, we use θ = 1.5 and κ a = 0.15 .

Orange curves: reduction of social interaction together with tests to identify and isolate asymptomatic and mild symptomatic cases. Here, we use θ = 1.0 and κ a = 0.10 .

For the dashed curves, the parameters are the same as those from the continuous curves above but represent the total number of confirmed symptomatic individuals. We notice that asymptomatic individuals, or those with very light symptoms, would not be identified without realization of tests and are not computed in the number of confirmed cases.

Fig. 3 θ and κ s that better adjust the simulation results with the real data. In the case of Brazil and USA, we obtain two values of θ e f f , as shown in Figs. 4(a) 4(b) θ = θ e f f and κ s = κ s e f f ), the red curves increase very much for both countries. Very high asymptotic values of infected individuals are reached, 3.4 × 10 7 cases for Brazil and 5.9 × 10 7 cases for USA. As in, during the times for which real data are available, the model chooses the values of the parametersandthat better adjust the simulation results with the real data. In the case of Brazil and USA, we obtain two values of, as shown inandwith the corresponding numerical values. Red curves fit nicely the data as long as they are available. For the USA case, there were some difficulties in adjusting the parameters since the data show some irregularities. In the case for which the strategy does not change (and), the red curves increase very much for both countries. Very high asymptotic values of infected individuals are reached,cases for Brazil andcases for USA.