Strong correlations between power-law growth of COVID-19 in four
continents and the inefficiency of soft quarantine strategies

The astonishing increase of positively diagnosed cases due to COVID-19 has called the attention of the whole World, including researchers of many areas and governments. It urges Puevo (2020) to find explanations for the already known data and models which may allow us to better understand the evolution of the viruses. Such explanations and models can hopefully be used to implement social policies and procedures to decrease the number of infections and deaths. Time urges to avoid economic and social catastrophes.

In general, the average reproductive number $R_{0}$, 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 Vazquez (2006). Usually, for values $R_{0}<1$, the number of new infected individuals decreases exponentially. For $1<R_{0}<\infty$ this number increases exponentially Vazquez (2006); Hethcote (2000). However, nature is full of surprises and there are plenty of cases for which the exponential behavior is substituted by power-law Viswanathan et al. (2011) and are related to branching processes with diverging reproductive number Vazquez (2006), scale free networks and small worlds Watts (2004). It was already suggested in the literature that the COVID-19 growth might be a small world Ray (2020). This is in agreement with recent results Singer (2020) suggesting that for many countries around the World the COVID-19 growth has the tendency to follow the power-law.

In fact, recent analysis regarding the behavior of the COVID-19 in China demonstrated a power-law $t^{\mu}$ growth of infected cases Maier and Brockmann (2020a). Authors found exponents around $\mu=2.1\pm 0.3$, which do not vary very much for different provinces in China. This suggests that socio-economical differences, local geography, differences in containment strategies, and heterogeneities essentially affect the value of the exponent $\mu$, but not the qualitative behavior. A model of coupled differential equations, which includes quarantine and isolations effects, was used by the authors to match real data. Power-law growths for China were obtained also in another study and a possible relation to fractal kinetics and graph theory is discussed Ziff and Ziff (2020).

In line to the above last week 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 United States of America (USA). In all cases we observe a power-law increase for the positive detected individuals, where the exponent $\mu$ changes for different countries. In addition to the power-law behavior we also computed the Distance Correlation (DC) Székely et al. (2007) between pairs of countries. The DC is able to detect nonlinear correlations between data Mendes and Beims (2018); Mendes et al. (2019). 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.

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 Section IV we summarize our results.

The model proposed in this work for the numerical prediction and strategies is presented in details in Appendix B. It is a variation of the well known Susceptible-Exposed-Infectious-Recovered (SEIR) epidemic model Li et al. (1999); Wu et al. (2020) to propose efficient strategies which allow the government to increase the flattening of the power-law curves. Our SEIR model takes into account the isolation of infected individuals Khalil et al. (2012); Coburn et al. (2009); Lipsitch et al. (2003); Maier and Brockmann (2020b). In this case, the 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, defined a priori from other studies and (ii) those related to adjusting the model to the real data (for more details please 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.  

The colors of the subtitles represent the application of distinct strategies which leads to distinct scenarios. For a detailed explanation of variables and parameters, please see the Appendix B. The colors used in Fig. 3 for the distinct scenarios are the following (for continuous curves):

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.

We start discussing the cases of Italy and France, shown in Figs. 3(a), 3(c) and Figs. 3(b), 3(d), respectively. In these cases $\kappa_{s}=1$, which means that we assume that all symptomatic individuals are properly isolated. Figures 3(c) and 3(d) show the evolution of scenarios for $120$. 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 parameters $\theta$ and $\kappa_{s}$ that better adjust the simulation results with the data. In the cases shown in Fig. 3 we needed three values of $\theta$ and $\kappa_{s}$, namely the values $\theta^{eff}$ and $\kappa^{eff}_{s}$ 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 $5.6\times 10^{6}$ and for France $1.0\times 10^{7}$. See the tendencies in Figs. 3(c) and 3(d). The considerable difference between these projections is explained by the last values of $\theta^{eff}$ and $\kappa_{s}^{eff}$ obtained for these countries. Besides $\theta^{eff}$ being larger for France, we obtain $\kappa_{s}^{eff}=0$, 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.

Now we discuss results for some emblematic scenarios for the model when specific strategies are applied to Italy and France on day March 28${}^{\mbox{\footnotesize th}}$. For both countries we assume $\kappa_{s}=1$ for all strategies, which means that all symptomatic individuals per day are putted into quarantine. We can see that the strategy represented by the orange curves is not sufficient to reduce significantly the total number of confirmed infected individuals for France, since the last value $\theta^{eff}=1.90$ 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 curves and orange scenarios. The light curves, related to large social distance ($\theta=0.8$), 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 $\theta=1.0$, there is no significant difference between isolating $1\%$ ($\kappa_{a}=0.01$) of the asymptomatic individuals per day or doing nothing. However, increasing the daily ratio of detection and isolation of asymptomatic individuals to $\kappa_{a}=0.05$, 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.

At next we discuss cases for Brazil and USA using other strategies. Results are shown in Figs. 4(a), 4(c) and Figs. 4(b), 4(d), respectively. Figures 4(c) and 4(d) furnish predictions for the number of infected individuals. Due to the distinct scenarios, we had to change a bit the color subtitles (Please see also color labels in Fig. 4):
Red curves: have the same meaning as before.
Blue curves: are still related to the reduction of social interaction so that $\theta$ can take the values $\theta=2.0,1.5$ 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 $\theta=2.0$ and $\kappa_{a}=0.20$.
Magenta curves: reduction of social interaction together with tests to identify and isolate asymptomatic and mild symptomatic cases. Here we use $\theta=1.5$ and $\kappa_{a}=0.15$.
Orange curves: reduction of social interaction together with tests to identify and isolate asymptomatic and mild symptomatic cases. Here we use $\theta=1.0$ and $\kappa_{a}=0.10$.
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.

As in Fig. 3, during the times for which real data are available, the model chooses the values of the parameters $\theta$ and $\kappa_{s}$ that better adjust the simulation results with the real data. In the case of Brazil and USA we obtain two values of $\theta^{eff}$, as shown in Fig. 4(a) and (b) with the corresponding numerical values. Red curves nicely fit the data as long they are available. For the USA case there were some difficult to adjust the parameters since the data show some irregularities. In the case for which the strategy does not change ($\theta=\theta^{eff}$ and $\kappa_{s}=\kappa_{s}^{eff}$), the red curves increase very much for both countries. Very high asymptotic values of infected individuals are reached, $3.4\times 10^{7}$ cases for Brazil and $5.9\times 10^{7}$ cases for USA.

Now we discuss results when new government strategies are applied to Brazil and USA on March 28${}^{\mbox{\footnotesize th}}$. All blue curves (dark to light) tend to mitigate the growth of the asymptotic number of cases for both countries. However, values of $\theta=2.0$ and $1.5$ still lead to large asymptotic values. The case with $\theta=1.0$ is the most relevant strategy to flatten the curve efficiently. From the strategies which combine social distance with quarantine for the asymptomatic and mild symptomatic cases, stands out the inefficiency of the scenario with $\theta=2.0$ and $\kappa_{a}=0.20$ to flatten the green curve. On the other hand, strategies with larger social distance (magenta and orange curves) lead to more promising scenarios. The better prediction emerges when using $\theta=1.0$ and $\kappa_{a}=0.10$ (orange curves). For comparison, if we consider only the symptomatic cases, we obtain the asymptotic values $3.0\times 10^{4}$ (light blue curve) and $1.6\times 10^{4}$ (orange-dashed curve) for Brazil and $8.7\times 10^{5}$ (light blue curve) and $4.6\times 10^{5}$ (orange-dashed curve) for USA. As in the case of France and Italy, also for Brazil and USA we observe that the social distance is the key element to reduce the growth rate of the maximum of symptomatic cases.

The power-law growth of the cumulative number of confirmed infected individuals by the COVID-19 until March 27${}^{\mbox{\footnotesize th}}$, 2020, is shown to be the best description scenario for the countries: Brazil, China, France, Germany, Italy, Japan, Spain, Republic of Korea, and USA. Distinct power-law exponents for the countries are found and summarized in Table 1. The power-law behavior suggests that the underlying propagation dynamics of the virus around the countries follows scale free networks, fractal kinetics and small world features Watts (2004); Ziff and Ziff (2020); Ray (2020). While power-laws with distinct exponents may look similar visually, it is necessary to quantify this similarity. For this we compute the Distance Correlation Székely et al. (2007); Mendes and Beims (2018); Mendes et al. (2019) between all countries mentioned above (not shown). However, using representative countries of four continents, namely Brazil, Italy, Japan, and USA, results for the DC are presented in Fig. 2. They show that the power-law growth between these countries are highly correlated, even between north and south hemispheres. The high correlation between the power-law curves of different continents strongly suggest that government strategies can be applied with success around the whole World.

Furthermore, we propose a variation of the well known SEIR epidemic model Li et al. (1999); Wu et al. (2020) for predictions using (or not) distinct government strategies applied on March 28${}^{\mbox{\footnotesize th}}$, 2020. We apply numerically distinct strategies to flatten (or not) the power-law curves. Even though the social isolation, a well know benefit, is very powerful to flatten the curves, we found other strategies which lead to comparable results. For Italy and France, for example, the best scenario was obtained when reducing the social interaction to $\theta=0.8$ (light blue curves). However, if $5\%$ of the asymptomatic and mild symptomatic individuals are identified and isolated every day ($\kappa_{a}=0.05$), asymptotic values of the same order of magnitude for the symptomatic cases were obtained even if $\theta=1.0$ (magenta-dashed curve). On the other hand, for Brazil and USA, our simulations confirm that to keep the social distance is essential to decrease the asymptotic number of infected individuals, and even better results could be obtained by increasing the tests to identify and isolate asymptomatic and mild symptomatic individuals (compare light blue and orange curves). The above combination between social interaction and the huge degree of isolation of infected individuals could be implemented to prevent economic catastrophes because people are not working. In other words, let some essential individuals go back to work (increasing $\theta$) and, simultaneously, increase by a huge amount the number of daily tests and isolation of infected individuals. This could furnish an efficient scenario to flatten the power-law.

Nevertheless, we point out again that our main results confirm that the social isolation of individuals is by far the best efficient strategy to flatten the curves.