The spread of COVID-19 in all countries is being reported with a massive wealth of data. Although data are collected by different sources, at different stages and with heterogeneous protocols (different national policies, etc.), this huge amount of information allows detailed statistical analysis of the process (see for example [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] ) and enables robust tests on the universal behaviour of the epidemic dynamics through different communities.
Simple models as the susceptible-infectious-recovered (SIR) scheme [15] historically have been used to describe the salient properties of the spreading statistics by relying on an economic amount of parameters and a mean field approach based on a system of nonlinear ordinary differential equations. The SIR model is especially suited to describe closed, spatially homogeneous communities [16] . Deviations from the simple SIR dynamics may reveal changes in the social behaviour that eventually reduce the spreading of the disease.
The SIR model may be cast in terms of fractions of population at a given time (day) t: S t , I t , and R t , represent respectively the fraction of susceptible, infected, and recovered people, so that S t + I t + R t = 1 (note that in R t we are not distinguishing the kind of exit from the infectious state). Its characteristic feature is that susceptibles are infected at a rate proportional both to their number and to the fraction of infected people. By indicating time derivatives by dots, e.g.Ṡ t = dS t /dt, the SIR evolution sketched in figure1(a) is described by three coupled differential equations,
where β is a constant determining the strength of the spreading in the transition rate from S to I. The healing rate µ, according to the literature, should be related to a healing time 1/µ of the order of two weeks (reported recovery times range from a few days [17] to almost four weeks [18] ). If β > µ, the SIR model predicts an exponential explosion of I t ∼ e (β−µ)t in the early stages of the epidemic. To a good degree, this is the kind of scaling that one could expect in all countries before their lockdown was Figure 1 . Scheme of each of the models considered in this work: (a) classic SIR, (b) SHIR [12] , and (c) our SS'HIR. eventually issued. However, we will show that this is not always the case.
In this paper we collect data for some countries where the statistics is significant and sufficiently regular, especially in the first stage of the pandemic expansion, and we introduce a non-standard way of describing its time evolution, by defining a "velocity" v t of the spreading and its time derivative, or "acceleration" a t . A v-a diagram is useful for comparing the various stages of the epidemic as it shows at a glance not only the number of new cases per day, but also the trend in its variation, in a range of unities for v and a.
From the v-a diagrams of several countries it emerges that the COVID-19 spreading was decelerating already before the application of the lockdown. Since the SIR model predicts a constant acceleration, it cannot explain this observed scaling. However, this is fairly well reproduced by a simple extension called SHIR (susceptible-hidden-infectiousrecovered) model, recently introduced by Barnes [12] . In this model one includes a fraction of "hidden" people H t , who either decide or are forced to isolate from the rest of the susceptible community S at hiding rate σ, see figure 1 (b). The time evolution of SHIR is then governed by the following set of differential equations:
In this picture the deceleration of the epidemic spreading before a lockdown is simply due to a steadily increase of the subset H the population that, although susceptible, is becoming aware of the potential danger of the virus and changes its social behaviour accordingly. We show below that the SHIR model has indeed an acceleration a t with an exponential decrease down to an asymptotic value. However, although data (before and) after lockdowns comply fairly well with its prediction, we find that a simple extension of the SHIR model, where only part of susceptible population can access the hidden state, can improve significantly the fit of the available data. This holds both for countries where the lockdown reduced the epidemic spreading to a negative acceleration (e.g. Germany and Italy) and for countries where the lockdown only reduced the acceleration to a positive value smaller than the initial one (e.g. Brazil and India, during the initial months of the epidemic).
Since universal features of disease spreading should better emerge from relative figures, we perform the analysis on the number of confirmed cases over the total population of a given country. Data were downloaded from the repository for the 2019 Novel Coronavirus Visual Dashboard operated by the Johns Hopkins University Center for Systems Science and Engineering (JHU CSSE) [19] . The dataset include the cumulative number F t of cases at date t = 0, 1, 2, . . ., i.e. the total number of people tested to have been infected by the virus up to time t, out of N people in the ensemble. In the notation of the SIR model, this number F t = N (I t + R t ) corresponds to the current count of infectious people N I t (databases might add tested asymptomatic people in this counting) plus the number N R t of previously infected people.
The new cases per day, n t = F t − F t−1 Ḟ t are a manifestation of the epidemic spreading speed. Since this quantity is noisy due to statistical fluctuations and other factors (variable medical protocols, number of tests, weekly periodicity, etc.) we consider instead the number of new cases occurred in a period ending at time t and averaged (smoothed) over a window of L days:
This quantity is reported in figure 2 for different countries. The figure shows that, over a time scale of two to three weeks, after lockdowns (black dots) there has been either a decrease of the number of cases per day or a decrease in the acceleration of the spreading (smaller positive slope of curves). In later periods, when the lockdowns were removed in the European counties, one notes the restart of second and further waves of the epidemic in those countries. However, maybe less evident, there also appears to be some negative curvature in the plots before lockdowns. This feature is better analysed with the following procedure. A comparison to a time-shifted n t−z gives the variation
where z = 1 is the minimum value. In presence of an exponential growth, we thus get a smoothed estimate of the daily rate of increase ∆n (L,1) t e β−µ , which is the exponential of the infection rate of the SIR model. In general we should have ∆n
From now on we consider L = 14, z = 3, and we drop them from the notation.
The typical exponential trends of the disease evolution suggest to consider log scales for a better visualisation and characterisation of the stage of the spreading. Hence, we portray the trajectory of the disease (parameterised by time t) by its velocity v t in log scale and by the rate of its variation, or "acceleration" a t . These are defined by their proxies
where N is the total population of a given country. The numerical prefactor ln 10 in the definition of the acceleration simplifies formulas later. Figure 3 shows the trajectories of the epidemic spreading in the (v, a) plane for the six selected countries: one can notice that all trajectories start from the upper left corner, they continue with increasing v (a > 0) and eventually cross the a = 0 axis. From there, typically a becomes negative indicating a period of recession of the viral disease. However, this occurs with such a slowly receding velocity (small negative values of a) that random fluctuations in the population of S or I may bring the system back to the dynamics characterised by a > 0, i.e v increasing phase. Note that for some countries as Brazil and India, the period of positive acceleration has been longer than in other countries and recession has been, so far, negligible. It is particularly for these cases that an extension of the SHIR model is needed (see next section). Clearly all trajectories show a consistent drop in a t after the application of the national lockdowns (see black point along the trajectory). Nevertheless, there are countries in which deceleration sets in several days before their lockdowns due to the application of some preventive not draconian measures. In Germany, for instance, the cancellation of several public events started before the official lockdown and has probably slowed down the spreading of COVID-19. In Italy, the national lockdown was preceded by the isolation of so called local "red zones" (where initial cases were detected) from the rest of the community. This measure would be placed at the beginning of the trajectory in figure 3, suggesting that it was effective in inducing the strong initial decrease of the spreading acceleration in Italy.
Next, we analyse this non trivial behaviour analytically for the SIR and SHIR model, then we perform fits to data to assess the performance of the SHIR model, Figure 3 . Acceleration vs velocity "falling star" trajectories for six large countries with major COVID-19 outbreaks: time parameterizes trajectory with a colour code from red to blue, black dots mark the major lockdown dates during Spring 2020, and an empty diamond indicates the last day of a trajectory. Starting from the upper left corner, trajectories continue to increase the spreading velocity v t as long as a t remains positive while decreasing. When a t turns to negative values, trajectories start to regress the velocity to lower values. It is visible that when social measures are loosened (green turning to blue region for Italy, Germany, and UK), fluctuations seem to bring back the trajectories to the a t > 0 phase, leading to oscillations around the a ≈ 0 region or to eventual further outbreaks.
finally we extend this model to improve the agreement with data and we discuss the implications of our results.
Since the simplest part of the epidemic evolution ranges from its initial rise to the end of the lockdowns, we have isolated the portion of the time series within this time window. For each country this is done by shifting the time t − t 0 → t to set t = 0 to a day (t 0 in the original scale) close to the first maximum of a t and by keeping a period of about two to three months since then. In this way we exclude the possibility of including in the analysis the onset of potential second waves of the spreading.
Let us start by considering the SHIR dynamics (2), of which the standard SIR model is a special case with σ = 0. For this model the confirmed cases have a time derivativeḞ t = N (İ t +Ṙ t ) = N βS t I t (7) and the velocity becomes v = log 10Ḟ t /N = log 10 β + log 10 S t + log 10 I t .
Since the fraction of infected people has been always at most of the order of 10 −4 one can safely assume that the decreasing of the susceptible population S t is mainly due to an increase of the hidden fraction of responsible people, H t . If the rate σ is large enough, i.e. so that σS t βS t I t , from (2) it follows that S t decays exponentially as S t = S 0 e −σt . Using this scaling, we have a velocity v t log 10 β + log 10 S 0 − σt ln 10 + log 10 I t ,
and acceleration
Hence, according to the SIR model, since in most countries the condition F t N is still fulfilled and S t 1, one would expect a spreading with constant acceleration a β − µ from (10). This contrasts with data shown in figure 3 and figure 4: even at initial stages (before lockdowns), in Germany, Italy, and UK it is readily seen a natural decrease of a t . In this respect the SIR model is too simple to describe the initial trend of the COVID-19 spreading, as highlighted also by Barnes [12] by simply looking at the ratio of new cases over the total ones.
On the other hand the SHIR model explains economically the deceleration by allowing a fast discharge of S t via a sufficiently large hiding rate σ; this is a sensible assumption, as it does not take many days to organise a reduced level of contact between people.
In figure 4 we show the time series of the epidemic acceleration for different countries. The dotted curves represent the fits of data based on (10) where we have assumed S 0 1, µ = 1/14 day −1 and let the parameters σ and β vary freely (the tiny initial I 0 value is also left as a free parameter). Note that it is not possible to consider µ as a further free parameter since the number of new cases per day is not very sensible to its value. Fortunately, this also means that the fits are quite independent on the chosen value of µ. By assigning a weight w t = n t to each data point, the best fit gives the estimates of σ and β reported in table 1. In figure 4 one notes that the SHIR model (see dotted curves) gets sufficiently close to the observed trends of the acceleration; there are however some visible deviations at short times and, most importantly, in some cases also at long times. Furthermore, some very small estimates of σ reported in table 1 corroborate the idea that some extension of the SHIR model may improve the agreement with the real data. . Time series of the virus spreading "acceleration" a t for six countries with major COVID-19 outbreaks, in the period before eventual second waves and wuth red dots marking the lockdown dates, and fits according to SHIR and SS'HIR models, both performed with dedicated python libraries for nonlinear fits. For Italy, Germany, UK, and Russia one notes that the asymptotic value of a t is not correctly recovered by the SHIR model. This value is only approximated for India and Brazil by introducing a tiny value σ for the transition rate to the hidden state (see table 1 ). The SS'HIR variant, on the other hand, can fit the long time value as well as the early stage of a t by yielding reasonable values of σ. For these reasons, we test a model in which the long time limit of S t is not negligible even if σ = 0. This is obtained by dividing the fraction of susceptible population in two groups: a fraction S can access freely the hidden state, as sketched in figure 1(c) , while a fraction S could not access the isolated condition (for reasons as work duties, poverty, etc.). The evolution equations of this SS'HIR model become
To partition the initial fraction of the susceptibles we consider a parameter p ≤ 1 such that S 0 = p(1 − I 0 ) p and S 0 = (1 − p)(1 − I 0 ) 1 − p. The number of new cases per day in this model is given bẏ
and the velocity becomes v t = log 10Ḟ /N = log 10 β + log 10 (S t + S t ) + log 10 I t .
By assuming S t S 0 and S t S 0 e −σt we thus get v t log 10Ḟ /N = log 10 β + log 10 (S 0 + S 0 e −σt ) + log 10 I t (14)
Sinceİ t /I t = β(S t + S t ) − µ, the acceleration reduces to
By fitting data with this formula we get the values listed in table 1, which are more realistic than those obtained from the fits based on the SHIR model. For instance, the unrealistic value σ 0.001 day −1 now becomes σ 0.049 day −1 , in line with other values. The fits in figure 4 show that the SS'HIR model can correctly capture the trend of the acceleration in a long time span, including also long time values of a t ≈ 0 (Russia) or a t > 0 (Brazil and India) ‡ Finally, we note that values p 0.8 for Brazil and India mean that, in the SS'HIR modelling, a fraction S 0 1 − p ≈ 0.2 of the population cannot isolate itself from the rest of the community. A classic SIR dynamics on this 20% of the population, with β(1−p)−µ 0 might be the reason for which the epidemic has been continuing to accelerate for some months in those countries.
We have presented a velocity-acceleration diagram that visualises the epidemic state and its trend. In the v-a diagrams of COVID-19 evolution, illustrated in this work for six large countries, we note that the acceleration is not constant often in periods including days before national lockdowns. The observed deceleration cannot be explained by a bias introduced by a variable number of tests, which were in general following an opposite trend increasing with time. This suggests that social distancing, either introduced by local lockdowns, personal choices, or cancellation of public events due to the news from Asia, was already effectively reducing to some degree the spreading of the virus.
The simplest effective explanation of the observed deceleration in the number of new COVID-19 cases per day comes from assuming that the fraction of susceptible population is reduced over a time scale 1/σ of tens of days/weeks by the isolation imposed by the national lockdowns. This confirms the findings by Barnes' with the SHIR model. Building upon this model, we have introduced a simple modification in which only part of the population can comply with the enforcement of strict social distancing. The remaining part obeys the usual rules of the SIR model. Our modification better fits the data from several countries, especially those where the acceleration of the spreading has remained positive for months, where our fits suggest that about 20% of the population effectively was never socially isolated.
There remains to understand how much the timescale 1/σ that emerges for most of the isolation dynamics is actually representative of a more complex mixture of effects, eventually including the personal evolution in the stages of the illness (as for exposed compartments in the SEIR model [11] ), which is not considered in the basic SIR model. However, the SEIR and similar variants of the SIR model, or simple models including an asymptomatic fraction of the population [7, 8] (undetected infected cases; their role for COVID-19 spreading has been studied for example in [1] [2] [3] ), so far in our tests (not shown) could not fit quantitatively the deceleration of the pandemic spreading if the hiding of people in the isolated state was not considered.