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Epidemic spreading in an expanded parameter space: the supercritical scaling laws and subcritical metastable phases

So far most of the analysis of coronavirus 2020 epidemic data has been focusing on a short-time window and consequently a quantitative test of statistical physical laws of Coronavirus Epidemics with Containment Measures (CEwCM) is currently lacking. Here we report a quantitative analysis of CEwCM over 230 days, covering the full-time lapse of the first epidemic wave. We use a 3D phase diagram tracking the simultaneous evolution of the doubling time Td(t) and reproductive number Rt(t) showing that this expanded parameter space is needed for biological physics of CEwCP. We have verified that in the supercritical [Rt(t)>1, Td(t)<40 days] regime i) the curve Z(t) of total infected cases follows the growth rate called Ostwald law; ii) the doubling time follows the exponential law Td(t)=A exp((t-t0)/s) as a function of time and iii) the power law Td(t)=C(Rt(t)-1)^-n is verified with the exponent n depending on the definition of Rt(t). The log-log plots Td(t) versus (Rt-1) of the second 2020 epidemic wave unveil in the subcritical regime [Td(t)>100 days] arrested metastable phases with Rt>1 where Td(t) was kept constant followed by its explosion and its containment following the same power law as in the first wave

CEwCM using a new 3D expanded parameter space, T d (t, R t ): where T d (t) and R t (t) are the timedependent doubling and reproductive number.
In CEwCM time evolution three main regimes are clearly identified: supercritical, critical, and subcritical phase. In the supercritical phase the extrinsic effects control the characteristic time s in the exponential law of the time-dependent doubling time T d (t)=Ae (t-t0)/s . We have verified here that in the first wave this phase is characterized by the T d (t)=C(R t (t)-1) -ν power law function of the variable doubling time T d (t) versus the reproductive number R t (t).
A key result of this work is the use of the log-log plots of T d (t) versus (R t (t)-1) to understand the time evolution of coronavirus first wave which is used to characterize the time evolution of epidemics in different countries. The results provide a quantitative comparison of the Covid-19 first wave evolution resulting from different choices of containment policies and the evolution of the second wave in Italy compared with South Korea.

Results and Discussion
The data for each country have been taken from the recognized public data base OurWorldInData 31 .
We have extracted, first, the time-dependent doubling time T d (t) from the curve of total infected cases, Z(t), and, second, the time-dependent reproductive number R t (t) from the curve of active infected cases, X(t), using the methodological definition provided by the Koch Institute 31 . The qualitative results of this approach have been verified by producing the log-log plots of T d (t) versus (R e (t)-1) where the effective reproductive number R e (t) and T d (t) have been extracted from joint Z(t) and X(t) curves by using the inverted SIR model.   (2) has been quickly extracted by several groups showing that it is in the range 2<T d0 <2.8 days with 2<R 0 <3, as reported in several works. 19

Time evolution law of the time-dependent doubling time T d (t) and reproduction number R t (t) in the 3D phase diagram T d (t,R t )
As the epidemic goes forward, the total number of cases will include both the active cases and The time-dependent variable reproductive number R t can be measured by different methods 2-7,32-34 .
We have first used the model independent procedure proposed by the Koch Institute 33 , where R t is the ratio of the actually infected individuals X(t) at time t divided by the actually infected individuals at the time (t -Δt): The relationship between the time-dependent doubling time T d and the reproductive number R t as defined by the Koch Institute 33 can be derived by considering that where the equality holds true if ∆ < 0 .
Hence, we find that and with K = ∆ 2 and the exponent = 1 under the hypothesis that ∆ is small.
We have used in this work ∆ = 5 days, which gives a values of R t in qualitative agreement with other methods. 34 Fitting the data where the empirical R t value calculated by the Koch method using the time interval, Δt=5 days, we have found the exponent ≈ 0.7 ± 0.05 in the three studied countries. The gray critical zone separates the supercritical regime from the subcritical regime.
The supercritical regimes occurs in the lower part of follows the universal exponential law 10,27 T d (t)=Ae (t-t 0 )/s (8) where t 0 is the time of the onset of the outbreak and s (called the s-factor) is the characteristic time of the applied containment policy 10,27 . The upper panel shows that the time lapse of the supercritical phase t*-t0 =24 days for the yellow region of South Korea is related with the small s-factor (6-7 days) which are both a factor 2.5 shorter than the s-factor of Italy and USA where 60<(t * -t0)<70 days, which imply a time duration of the first wave lockdown a factor 2-3 longer.
The arrested phase occurs in the light blue areas where T d(t) becomes larger than 100 and R t(t) drops below 1 (see green areas in Figure 1d).
Italy reached a well defined arrested phase extending in the green area due to the enforced strict rules of the imposed lockdown. On the contrary, the plot shows that in USA, where loose rules for the lockdown have been applied, the transition to the subcritical regime has been stopped, and the country remained for a very long time in the critical phase. This approach shows that the arrested phase in South Korea remained close to the critical phase and faced a second Covid-19 very short wave indicated by the yellow area. We underline that the second wave in South Korea lasted only 20 days and with a doubling time longer than 40 days. The shape of the second wave in South Korea clearly shows the efficiency of the applied contact tracing to keep the spreading under control, to reduce the lockdown to a minimum time and to reduce the number of fatalities.

Infection and removal time from inverted SIR model: the R e (t) time-dependent reproduction time in the T d (t, R e ) phase diagram
We move now to describe the T d (t',R e ) phase diagram, where the effective reproduction time, namely R e (t), is extracted by the inverted SIR theory 25,26 .
In the framework of the SIR mean-field theory, the individuals in a population are classified as individuals, respectively can be written as: The effective reproduction number obtained by inverting the SIR equations is called \ which is defines as under the hypothesis that the relative change in the number of Susceptibles is negligible, therefore S can be considered almost constant and equal to N. Hence the second equation simplify to: pointing out that the rate of changes of the Infected is related to the ratio of U to T that is called here the effective reproduction number \ .
The values of \ for the three countries calculated with Eq. 11 are shown in Fig. 2.
The amount of currently infected or active positive cases (I) and the total number Z = X + Y of individuals that are affected, at any time, by the infection are available on the data base. The rate equation for Z is hence, starting from the pandemic data for X and Z and using the approximation S(t) ≈ N, it is possible to invert the SIR model to evaluate U and T which is a mixed exponential and power-law behavior determined by nucleation of different phases and ordering phenomena in complex multiphase systems out of equilibrium [35][36][37] . The onset of this behavior, obtained by best curve fitting, is reported by the dashed lines in the central panels of Fig.   2.
The doubling time T d (t) calculated using Eq.17, has been plotted in Figure 2 with the effective At variance the R e (t) curve exhibits only a qualitative agreement with the corresponding R t (t) curve estimated with the Koch Institute method shown in Fig.1. Nevertheless, both methods agree on the time where the critical regime occurs, i.e., when both R t (t) and R e (t) are equal to one.

The universal power law relation between time-dependent doubling time, T d (t), and reproduction number, R t (t), in the supercritical phase: prediction of the onset of new epidemic
waves.
Log-log plot of the doubling time T d as a function of R t -1 and R e -1 in the three countries are compared in the upper and lower panels of Fig. 3. The R t values are higher than R e calculated with inverted SIR method. although the difference, the T d (t) vs R t (t)-1 behavior in the two approach appears qualitatively similar. We get the critical exponent ν=1, using R e -1 extracted by the meanfield inverted SIR model which is different from the exponent ν=0.7 obtained using Rt defined by the Koch method.
The exponent ν=1 was predicted for the Koch R t definition in the limit Δ → 0 by Eq.7 but this limit cannot be reached since the time interval cannot be smaller than 3 days. We have estimated R t with the Koch Institute method 33   The rapidly aborted second wave in South Korea is well depicted by the green dots forming the loop in Fig.3, where the containment policy succeeded to push up the doubling-time avoiding its following down toward the universal power law growth rate of the first Covid-19 wave.

Metastable phases and the second wave in Italy
In Figure 3 the phase diagram of epidemic spreading in Italy during the first wave shows the grey strip corresponding to the critical regime occurring for T d (t) between 40 and 100 days above the supercritical orange area, and the arrested subcritical regime, characterized by for T d (t)>100 days.

In this arrested subcritical phase the T d (t) vs R t (t) values show a non-analytical disordered
distribution (see also Figure 1c), while in the supercritical regime it is described by the analytical universal power law of Eq. (7). The green dots in the T d vs. R t plots in Fig. 3 show that the arrested phase occurs also for cases where T d (t)>100 although R t (t)>1. When put on a lattice, the SIR model depends on the geometry of contacts, and is in the same universality class as ordinary percolation.
R t is essentially the percolation threshold for the SIR mean-field model, in fact infection can grow without bound where R t is greater than 1 on a Erdos-Reny network or a Bethe lattice 38 , while there is no percolation for R t less than 1. However, making the network composed of closed loops which attenuate the probability of an epidemic, in different complex geometries and in presence of long range interactions the percolating epidemic threshold could be larger than 1 (in fact it can be as high as 1.2). 39 The plot of the Italian case unveils the unexpected metastable phases in the arrested regime with is about two times larger than in the South Korea second wave shown in Fig. 3.

Conclusions
The results of this work provide an original quantitative approach for understanding the time evolution of the Covid-19 pandemic. We show that it is necessary to expand the parameter space,    In the middle panels, we report the curves of cases per million population, of infected (X) (red), removed Y (blue) and total cases (Z=X+Y) (black) as a function of time. The Z(t) curve follows the Ostwald law 10,27 (dashed line) characteristic of ordering growth in heterogeneous systems [35][36][37] . In the lower panels we show T d and R t (R e ) as extracted by the inverted SIR model in analogy with Figure 2c. The shaded light blue area indicates the arrested regime in the subcritical phase separating the first and the second Covid-19 wave. In Italy this arrested phase occurs for 60<t'<230 days. In USA there is no evidence for the presence of a subcritical regime, the yellow supercritical phase is followed by the critical phase in the gray strip extending over a large time interval.

Figure 3
Log-log plot of doubling time T d in three countries as a function of R(t)-1 extracted by the inverted SIR model (upper panels) and by the Koch Institute empirical method (lower panels). Black dots in the yellow region indicate T d in the supercritical regime which follows the universal analytical power law function [T d =C(R t -1) -ν ] of a divergent T d approaching R t =1 fitted by the red dashed lines. The critical exponent ν is 1 in the inverted SIR model and 0.7 using Koch prescriptions with 5 days interval. The critical phase is described by the horizontal grey strip where 40<T d <100 days. The green dots in the blue light area indicate the metastable states in the subcritical regime where the epidemic spreading growth is arrested T d >40 days also if R t >1. While the metastable state in Italy in the subcritical (green dots) with T d =constant indicates a precursor of the second wave regime in South Korea, the loop of the curve (green dots) in the critical regime (gray strip) is determined by the second wave shown in the lower panel of Fig. 2a. In USA the epidemic spreading never entered in the arrested phase and reversible oscillations (R t decreasing is inducted by red dots) and are observed in the critical regime.