In complex systems, the statistical variability of the size of a phenomenon carries fundamental information about its underlying dynamics [1] [2] [3] . Recently, Cirillo and Taleb [4] have studied the size of major epidemics in human history, measured in number of fatalities.
Using the figures from 72 epidemics, from the plague of Athens (429 BC) to the COVID-19 (still ongoing), they claim that the resulting fatality distribution is "extremely fat-tailed", i.e., a power law, asymptotically (for very large values of the number of fatalities).
Power-law distributions are an important paradigm in complex-systems science, representing the non-existence of characteristic scales and the divergence of moments. However, the empirical treatment of power-law data is a delicate issue [5] [6] [7] [8] [9] [10] [11] [12] , resulting in that the rigorous evidence in support of these distributions has been rather limited [11] .
Reanalyzing the same epidemic data as Ref. [4] , we find that, although the data may be compatible with an asymptotic power-law, there is no evidence in favor of this behavior, and other distributions, not fat-tailed, could explain the tail equally well or even better.
In concrete, simulation of a tail coming from a truncated log-normal distribution provides data whose statistics are undistinguishable from the epidemic empirical data. Based on this log-normal tail we are able to provide a very rough estimate of the final expected size of the COVID-19 pandemic (which would turn out to be infinite in the case of a power law). We finally discuss under which physical circumstances one could expect a power-law tail for the size of epidemics, and the necessity of much better data.
Most of the events analyzed in Ref. [4] come from the list collected by Wikipedia of the biggest known epidemics caused by infectious diseases [13] . The list was complemented with a few more events, obtained from other sources (mainly Ref. [14] ). Restricting to epidemics causing 1000 or more fatalities, this led to N = 72 events, from 429 BC to 2020 AD (CE); the resulting data is provided in Table 1 of Ref. [4] , in concrete, in the 4th numeric column (average estimate). We will refer to this data as the original data set.
We are not aware of any study of completeness of such data. In order to address this important point, we display the number of fatalities versus time of occurrence (beginning of the outbreak) in Fig. 1(a) , where it can be clearly seen that the data looks certainly inhomogeneous. For example, between 735 AD (Japanese smallpox epidemic) and 1485 AD there is only one event, which happens to be the Black Death (largest event on record, with roughly 140,000,000 fatalities, starting in 1331). confirms that the rate of occurrence of large epidemics seems to be highly inhomogeneous, from roughly 3 per century between 500 and 750 AD to just 1 in seven centuries (up to 1485) and to about 0.5 per year after 2000 AD. We associate this high variability to the large incompleteness of the data and not to a wild variation of the occurrence of epidemics over the centuries.
Incompleteness can be also seen in the uneven distribution of the smallest events considered, i.e., out of the 17 epidemics below 10,000 fatalities, only 2 are contained in the first half of the data (from 429 BC to 1813). Including events below the completeness threshold can lead to important biases. Although in Ref. [4] it is claimed that a jack-knife resampling technique can account for incompleteness, it is clear that resampling does not correct any bias present in the data.
For this reason, in order to evaluate the impact of data incompleteness, in addition to analyze the raw data used in Ref. [4] , we also consider a restricted data set, from 1485 to 2000 AD, which shows a higher homogeneity, with an average of 0.1 epidemics per year.
This restricted data set consists of N = 50 epidemics. Note that the largest event, the Black Death, is not contained in the data set; thus, this provides a way to test the robustness of the results in front of the value of the most extreme event. This also eliminates ongoing epidemics (such as the COVID-19) from the analysis (except the HIV/AIDS pandemic).
A third data set, considered in Ref. [4] , consist of rescaling the number of fatalities (of the original data set) by the whole world population at the time of the epidemic. This represents a procedure to take into account that the world population is far from stationary across history, and it is not the same to have 140 million of fatalities nowadays than in the Middle Ages (where the global population was around 400 million).
It is questionable why one should rescale by the world population and not by the local population (i.e., the population of Greece for the plague of Athens), and also why the scaling is linear with the total world population (for instance, in a simple branching process the size does not scale linearly with the size of the underlying tree, but with the square of the number of generations [15] , and in general one has to take into account the so-called fractal dimension of the avalanches, see e.g. Ref. [16] ). Nevertheless, we will also explore this simple rescaled data set, in order to test the robustness of the results.
The authors of Ref. [4] identify "fat-tailed" distributions with regularly varying distributions [12] , defined by a complementary cumulative distribution function (or survival function [17] , probability of being above x) given by S f at (x) = (x)/x α , with α the exponent (of S f at (x), and ξ = 1/α the tail index, measuring the "fatness" of the tail) and (x) an unspecified slowly varying function (for example, a function that tends to a constant when
x → ∞, but not only). Roughly speaking, a "fat-tailed" distribution becomes a power law asymptotically.
From the terminological point of view, the name "fat tail" becomes unfortunate, as it can be confused with the terms "long tail" and "heavy tail", which denote different classes of probability distributions. It is convenient to remember that fat-tailed distributions are subexponential distributions, which are long-tailed in their turn, which are heavy-tailed, but not the opposite [12] ; in other words, in the previous four categories there is an implication chain that goes from left to right, but not from right to left.
In a nutshell, subexponential means that the probability of the sum of two independent variables is twice the probability of one of them; long-tailed means that the tail is unaffected by finite shifts (S long (x+c) → S long (x)); and heavy-tailed is when the tail decays more slowly than any exponential [12] . A fundamental result of extreme-value theory is that fat-tailed distributions belong to the so-called Fréchet maximum domain of attraction, whereas heavytailed distributions that are not fat-tailed do not belong to such domain [12] .
From an operational point of view, we define a power-law-tailed distribution (or just a power law, pl) as a distribution whose tail, defined by x above a lower-cut-off u (i.e., x ≥ u), is given by the probability density
.
We will consider f pl (x) = 0 for x < u, that is, all empirical data below u need to be disregarded [7] . Note that f pl (x) is the minus derivative of S pl (x) = (u/x) α (for x ≥ u).
Note also the difference between a fat tail, where the power-law arises asymptotically, and a power-law tail, where the power-law arises above the cut-off u.
As an alternative description of the tail, we consider the example given by the truncated log-normal (ln) distribution, which is not fat-tailed but subexponential (and therefore longtailed and heavy-tailed [12] ). Its probability density is The log-normal distribution has been an important competitor of the power law for the size distribution of structures and events in complex systems [11, 18, 19] , due to the fact that it can be described as a sort of power law whose exponent is not constant but changes very slowly with x. This is commonly found in log-log plots of empirical probability densities f (x), which tend to show a slight downwards curvature (convexity for physicists and concavity for mathematicians).
We proceed to the fitting of the power-law tail and the log-normal tail to the epidemic-size empirical data. Given the lower cut-off u, the fitting of the distributions is straightforward by maximum-likelihood estimation. However, the important point is precisely the determination of u, both when fitting the power-law and the log-normal tail. Many different methods have been proposed for this in the case of power-law distributions, including visual methods even (for bibliography, see Ref. [12] ).
We use the fitting procedure developed in Ref. [9] , which is similar in spirit to the verypopular one of Ref. [7] but performing much better under controlled situations [8, 10, 12] . In essence, the procedure tries all possible values of u and selects the smallest one which gives an acceptable p−value in a Kolmogorov-Smirnov goodness-of-fit test. The same method is applied to fit the truncated log-normal distribution [11] .
A visualization of the resulting fits for the original data set in comparison with the empirical estimations of the probability density (using logarithmic binning [9] ) and of the survival function is provided in Fig. 2 . It is apparent how both fits are very close to each other (the difference being much smaller than the uncertainty in the empirical value of the density) and also that the power law gives a higher probability than the log-normal for Table I . The log-normal fit also leads to values of the parameters close to those of the original data set, as also shown in the table. In addition, for the rescaled data set we find the same overall behavior (see the table once more) . Thus, we conclude that the properties of the epidemic-size distribution are roughly the same for the original data set, for the restricted data set, and for the rescaled data set. This does not mean that incompleteness is not relevant, only that the degree of incompleteness is similar for the three data sets.
In this way, for the rest of the paper we concentrate on the original (not rescaled) data set for a more in-depth analysis, which will compare the statistical behavior of this empirical data with that of the simulation of a truncated log-normal distribution (with N = 72 events and the values of the parameters u, µ, and σ given above and also in Table I ). Reference [4] proposes two main ways to check fat-tailness. One of this uses the meanexcess function (u) = x − u|x ≥ u , where the brackets denote expected value and the vertical bar conditioning. This is the same as the expected residual size [20] (see also
Ref. [21] ) used in reliability theory and survival analysis, which characterizes a probability distribution in a way totally equivalent to f (x) or S(x), provided that the first moment of the distribution ( x = (0)) is finite, which for a power-law tail happens when α > 1. In this case, (u) should increase linearly with the lower cut-off u. However, for a power-law tail with α < 1 the first moment does not exist (is infinite), which makes the use of this method void of theoretical support. The results, displayed at Fig. 3(b) , show again that the behavior of synthetic log-normal data is very close to that of the empirical data, with the max-to-sum ratio decreasing with the number of data but without reaching zero. Thus, although the theory teaches us that the maximum-to-sum ratio tends to zero when N → ∞ if the distribution has a finite mean, this convergence can be rather slow, as it happens with the log-normal distribution for the parameter values that describe the epidemic data.
In summary, the two methods used on Ref. [4] are not able to rule out a log-normal tail for the description of the epidemic data.
Now we provide complementary evidence that the truncated log-normal distribution provides a good description for the tail of the epidemic data of Ref. [4] . In fact, the power law can be considered a particular case of the truncated log-normal (in the same way that the exponential is a particular case of the truncated normal distribution when σ 2 → ∞ and µ → −∞ [22] ). In this sense, a log-normal tail cannot provide a worst fit than a power law (it will be the same, or better). However, on the other hand, it may happen that this improvement in the fit is not significant, and then the power-law tail suffices for describing the data (as it has one parameter less that the log-normal). This is something that can be evaluated by a likelihood-ratio (LR) test [23] .
Taking advantage of the fact that the LR between both distributions is a decreasing function of the logarithmic coefficient of variation (CV) [22] , this provides a very simple way to perform the LR test (without the need of performing maximum-likelihood fitting):
critical values of the LR translate into critical values of the logarithmic CV. When this quantity is close enough to one, the power-law hypothesis cannot be rejected, and when it departs significantly from one (from below), the power law is rejected in favor of the lognormal. This simple procedure provides a uniformly most powerful unbiased test between a log-normal tail and a power-law tail [18, 22] .
The test is performed for different values of the lower cut-off u, and the results, for the original data set, are displayed on Fig. 3(c) . This shows that only for the 21 largest epidemics the power-law tail is not rejected in favor of the log-normal. The corresponding cut-off u turns out to be at about 350,000 fatalities. In other words, the 21 epidemics with more than 350,000 casualties are well described by a power law (the improvement brought by the log-normal is not significant), but, including events below 350,000, the log-normal fit is significantly better (for the full range).
Applying the same procedure to the log-normally simulated data reproduces again the pattern obtained for the empirical data, as also shown in Fig. 3(c) . Notice then how the most extreme events of log-normally simulated data are indistinguishable from a power-law tail, and therefore the power-law is preferred, despite the simulated data are log-normal (by construction). This is an unavoidable consequence of the similarity between truncated log-normals and power laws, and implies that the fact that for the 21 largest epidemics the preferred fit is power law does not guarantee that the underlying distribution is not log-normal.
Assuming that a power law can describe the largest epidemics (in terms of fatalities), which would be the value of the corresponding power-law exponent α? Above, for u 33, 000, we report α = 0.34, but for u 350, 000 the value is larger: α = 0.45. In fact, the value of α is not stable at all, growing when the lower cut-off u increases (reaching α > 1 for the highest values of u, see Figure 3 (d)); that is, the fatness of the tail ξ = 1/α decreases systematically with u (this is already apparent in the results of Ref. [4] ), which prevents that one can establish a well-defined exponent [24] .
Notice that even though ξ is the parameter separating fat tails from light (exponential) tails in the generalized Pareto distribution, the fact of obtaining a positive value of ξ does not mean that the data are fat tailed, as the calculation of ξ (or α) by maximum-likelihood estimation assumes that ξ is positive (in other words, one never can obtain a zero or negative value of ξ from Hill estimator). 
Having in mind the limitations of the current study (incompleteness of historical records and information limited to just one value of a random variable per epidemic), an estimation of the expected final death toll from the current COVID-19 pandemic has to be understood as a mathematical exercise. Nevertheless, this exercise can be very illuminating to understand the counterintuitive properties of fat-tailed processes. Given that, at the time of submitting this article, the number of fatalities of the COVID-19 pandemic is u 1, 000, 000, which will its expected final size x be? We are asking for x|x ≥ u , which is directly related to the mean-excess function (u) by x|x ≥ u = (u) + u.
For a power-law distribution, this is given by x|x ≥ u pl = αu/(α − 1); however, this is only valid for α > 1 [21] . For the mentioned values of α below one, x|x ≥ u pl becomes infinite (although finite for a finite sample and strongly dependent on N ). In contrast, for the log-normal distribution, we obtain (from Fig. 3(a) ) x|x ≥ 1, 000, 000 ln 1.1 × 10 8 , which seems exaggeratedly large, but at least it is not infinite. In any case, this calculation clearly shows that one should not disregard the risk posed by a log-normal tail.
These results highlight the importance of the underlying statistical model, as the results may depend more on the assumptions contained in the model than on the empirical data.
The estimations are crude because are based on crude data, just a one-dimensional random variable. Knowledge of the dynamics of the growth of the death toll with time until its final value x (on a daily, or monthly basis, etc.) for the historical data (in other words, knowledge of the "avalanche profile") would provide more valuable information to improve the current estimation.
The fact that the log-normal distribution fits well the epidemic data can be extended to other distributions, fat-tailed or not. That is, there are more distributions that can reproduce the empirical results well enough. This means that the "true" probability distribution describing the number of fatalities of epidemics cannot be established from a purely statistical analysis.
Fortunately, physical insights can shed light on this problem. Assuming a simple (meanfield) model in which infections propagate following a Galton-Watson stochastic branching process [25] [26] [27] , and if the number of fatalities is a fixed fraction of the number of infections, identifying the branching ratio with the basic reproductive number, R 0 , it is immediate to see that R 0 < 1 leads to rather small epidemics (few number of fatalities), whereas for R 0 > 1 two scenarios are possible starting from a single individual, again, few fatalities (as given by an exponential tail for x), or an infinite number of fatalities (in an infinite system).
It is only at the critical point, R 0 = 1, where the number of fatalities is fat tailed, with an exponent α = 1/2 (and this is known at least since the 1940s [25] , provided that the distribution of contagions arising directly from one individual has a finite second moment).
For a sequence of historical epidemics, the distribution of the resulting number of fatalities will be a mixture of subcritical (R 0 < 1), critical, and supercritical (R 0 > 1) distributions, weighted by the distribution of R 0 , whose density is denoted here by ρ(R 0 ); thus,
For large x, only the critical and supercritical regimes need to be taken into account (as in subcriticality large values of x are totally negligible). Moreover, in a supercritical situation, whenever x reaches large enough values, one expects that social interventions to fight the epidemic are implemented; if these are effective the value of R 0 should decrease, slightly below one in the ideal case. In this way, the contention of the epidemic triggers a feedback mechanism that, when x is large, sets the value of R 0 close to the critical point of the model. The situation is that of self-organized critical phenomena [1, 28] . Under these circumstances, one would expect a power-law tail for large x, with an exponent α = 1/2 [27] .
This shows in a simple scenario how a power-law tail is feasible, in agreement with Cirillo and Taleb [4] . Nevertheless, the situation just described is highly idealized, and the precise results would depend on the "natural" distribution of R 0 for contagious diseases and the dynamics of R 0 under mitigation measures. Finite-size effects should also be taken into account properly [15] . And, needless to say, the Galton-Watson model is too simplistic, and other models, beyond mean field, may lead at least to a different value of the exponent α. For example, superspreading phenomena for which the number of contagions triggered directly by a single individual were power-law distributed would lead, counterintuitively, to larger values of α (still fat tails, but thinner than in the Galton-Watson model [29] ).
We have shown that there is not enough empirical evidence that the fatalities caused by epidemics along history follow a fat-tailed distribution. A log-normal tail (lacking fattailness) is able to replicate all sort of metrics used before [4] to support fat-tailness in the empirical record. For sure, other not fat-tailed distributions could be the data similarly well as the log-normal. What our work shows is the importance of considering alternative probability models when fitting heavy-tailed distributed data (which is different from fattailed data [12] ), as well as the key role of computer simulations to contrast the validity of theoretical results when the number of data is not infinite.
Summarizing, log-normal tailed simulated data, in the same way as the empirical data, has: a maximum-to-sum ratio that does not tend to zero as the number of events increases (up to N = 72); a mean excess size (u) that increases with a lower cut-off in size u;
and a power-law tail exponent α that also increases with u (and which can be erroneously associated to a positive tail index ξ, despite the fact the log-normal belongs to the Gumbel maximum domain of attraction and should have ξ = 0 [30] ; obviously, the calculation of ξ from the power-law exponent implicitly assumes ξ > 0). The agreement between empirical data and the log-normal simulations is not only qualitative but quantitative for the three metrics.
Moreover, the uniformly most powerful unbiased test based on the logarithmic coefficient of variation [18] shows that the power-law fit is preferred for the top 21 events of the empirical data, but this preference for the power law is also shown when the most extreme log-normal synthetic events are used; this is due to the well-known fact that the power law can be considered as a special case of a log-normal tail [18] . So, and this is something that it is not so well known, the log-normal distribution, belonging to the Gumbel maximum domain of attraction, has a special power-law limit, belonging to the Fréchet maximum domain of attraction. In other words, when it is commonly stated that "log-normal is Gumbel" the special power-law limit of the truncated log-normal (σ 2 → ∞ and µ → −∞) is implicitly disregarded.
One may argue that from the point of risk management it is more conservative to take the power law (which gives a larger probability for the most extreme events) than the log-normal tail. This is true, but constitutes a different problem, which could be addressed even without any statistical modelling. For example, using the empirical data of Ref. Ref. [4] ), but also from the incompleteness of the data (with a bias in favor of very large events that resampling [4] cannot correct) and from the lack of homogeneity in time (with just one event, the Black Death, between 750 and 1450, and 11 events since 2008 in Ref. [4] ).
Notice that the data are inhomogeneous in additional way: epidemics in the Middle Ages and in the 21st century are not comparable in the sense that they propagate differently and that the measures implemented for their contention should be more effective nowadays. If one mixes historical epidemic data with contemporary data what one obtains is, obviously, a mixture of distributions. This is not per se wrong, but one needs to have in mind for which reason one needs such knowledge in order to interpret the results properly.
In any case, one can dig a little in the existing records and find many more historical events. As an example, Villalba [31] reported several epidemics in Spain with more than 10,000 fatalities that are not considered in the data of Ref. [4] (these missing Spanish epidemics took place in 1283, 1394, 1490, 1564, 1589, 1637, 1726, 1741, 1784, and 1800).
For sure, there is nothing special about Spain and other countries can contribute more or less in the same way with more "hidden" epidemics. Nevertheless, the compilation of a reliable record for historical epidemics is something that should not be done by probabilists, statisticians, or physicists, and needs to be carefully undertaken by true epidemiologists.
We urge here for the necessity of such an important endeavor. 