COVID-19 scaling dynamics in growth and decline phases

7 The definition of optimal COVID-19 mitigation strategies remains worldwide on the top of public 8 health agendas, particularly when facing a second wave. It requires a better understanding and a 9 refined modelling of its dynamics. We emphasise the fact that epidemic models are 10 phenomenologically based on the paradigm of a cascade of contacts that propagates infection. 11 However, the introduction of ad-hoc characteristic times and corresponding rates spuriously break 12 their scale symmetry. 13 Here we theoretically argue and empirically demonstrate that COVID-19 dynamics, during both 14 growth and decline phases, is a cascade with a rather universal scale symmetry whose power-law 15 statistics drastically differ from those of an exponential process. This involves slower but longer 16 phases which are furthermore linked by a fairly simple symmetry. These results explain biases of 17 epidemic models and help to improve them. Due to their generality, these results pave the way to a 18 renewed approach to epidemics, and more generally to growth phenomena.

led to qualitatively change the type of spread: from normal diffusion (e.g. for the Middle Age plague 19 ) to anomalous diffusion.Normal diffusion does provide characteristic times due to the fact that the generating Gaussian distribution has a fall-off faster than an exponential (e.g., the probability to exceed 3 root mean squares (RMS) is only of the order of 10 $% , i.e., a Gaussian variable is never too far from the mean).For instance, the spread radius () can be measured by the RMS () of particles emerging at the same time from the same source, which grows only like  &/( , i.e. much slower than a ballistic diffusion (() ≈ ).On the contrary, anomalous diffusion is generated by Levy stable distributions that are devoid of characteristic times due to their power-law fall-off , i.e., the probability to exceed a distance () on a given time falls off as () $(&*+) , with 0 <  < 2 20,21 .As a consequence, the RMS () does not exist.Furthermore, the probability to exceed  times a given distance () decreases only by the factor  $+ , which shows that we can easily be far away from the mean… which furthermore exists only for  > 1!Such anomalous diffusion due to human mobility has received some empirical support from a large scale experiment ironically nicknamed "where is George".This experiment corresponds to first tracking via an internet site, over ten millions of displacements of about half a million of one dollar bills (bearing the image of George Washington) then to analyse them 22 .The important result obtained was that this diffusion was anomalous and nearly isotropic in space and time with a common exponent  ≈ 0.6.It therefore corresponds to a strong super diffusivity.It was also shown that a random walker in a Chernobyl contaminated field accumulates radiation doses that also anomalously fluctuate, contrasting sharply with the usual average estimates 23 .Refined small scales statistics of time between contacts could be obtained with the deployment of mobile tracking applications, which nevertheless should protect privacy 24 .Another important aspect to be taken into account is the fractality of measuring networks 25 .

(a) (b) (c)
Fig. 1: Sub-exponential growth phase: log-linear plots of (a) confirmed, (b) death, (c) recovered cumulative incidences () of the following entities: Hubei, France, Germany, Iran, Italy, Japan, Korea, Spain and United Kingdom, over the period 22 January -04 April.An exponential behaviour would correspond to straight lines like those drawn for characteristic times  = 4; 7 days, and therefore to doubling times  !=  ln(2) ≈ 3; 5 days (respectively dashed grey and dotted black straight lines).Due to their (strong) concavity, all the trajectories are sub-exponentials, i.e., grow slower than any of their local approximations by an exponential that corresponds to a tangent to these log-linear graphs.Characteristic times  and the corresponding "doubling times"  ( are related to differential equations of the type: and their exponential solution () ≈ ( !)exp (( −  !)/) , where the symbol ≈ denotes an equivalence that can be more general than a strict deterministic equality, e.g., asymptotic equivalence on a given range of times or/and up to a slowly varying factor, or/and an equality in probability distribution, etc..The SIR model provides again a rather simple example of this behaviour.For a constant population ( () + () + () = () =  =  ) it yields the following evolution equation for the infected population: The second r.h.s.approximation is valid during the initial outbreak (() ≈ (0)), whereas the awaited "flattening of the epidemic wave" ( -.(/) -/ = 0) occurs in the SIR model only when the susceptible population () has been significantly reduced with respect to the total population , hence it occurs at the time  0 such that: ( 0 )/ =  !$& or (( 0 ) + ( 0 ))/ = 1 −  !$& .Database We will not discuss either the interests (e.g., analytically solvable 3 ) or limitations (e.g., non-accounted spatial and intra-compartmental heterogeneities) of the SIR model that we only used to introduce basic concepts needed to explore the database of the Center for Systems Science and Engineering at Johns Hopkins University 26 , more precisely its convenient github web site https://github.com/CSSEGISandData/COVID-19/tree/master/csse_covid_19_data.This database provides daily estimates of the "confirmed", "deaths" and "recovered" cumulative incidences for 187 countries.These data are refined for some regions, e.g.: the data on China are distributed according to all the provinces, metropolitan France is distinct from the overseas departments and the American data are provided according to 3261 urban agglomerations and dependencies.For presentation clarity, results are only displayed for the following entities: Hubei, France, Germany, Iran, Italy, Japan, Korea, Spain and United Kingdom.The choice of these time series is based on their length, as well as their relative high counts.Both factors could presumably reduce artefacts such as incomplete counts or even possible offsets for various reasons, including political ones.However, important changes in tests or/and count methodologies or simply data transmission delays did introduce non negligible, artificial fluctuations in theses series.For instance, important changes on COVID-19 infection evaluations such as +15100 in the Hubei province occurred on 13 February and +26800 in France on 12 April.An additional cumulation of 325 cases in the Hubei province was added to the datasets on ad-hoc dates, a month later the official ending of the epidemic spread in this area.Often, the deaths in rest homes began to be included only at a later stage of the epidemic.Furthermore, data were not always daily provided, therefore introducing a zero increment between two consecutive days, followed by an artificially enlarged increment for the second day.However, our data analysis is rather robust with respect to this sort of time translation, contrary to the direct analysis of the time evolution of the cumulative incidence ().There are other disrupting examples of data adjustments, such as the cumulation of the "deaths" incidences was decreased by 1918 cases in Spain on 25 May."deaths", (c) "recovered" cumulative incidence () of the following entities: Hubei, France, Germany, Iran, Italy, Japan, Korea, Spain and United Kingdom, over the period 22 January-04 April.Graphs should be read from left to right to follow the time arrow (() being non decreasing with time) and the almost vertical parts on the righthand side corresponds to compressed views of decline phases to be expanded in Fig. 3.These graphs display a rather universal behaviour, especially a common scaling/power-law behaviour ∆( " ) ≈ ( " ) # , a discrete version of the differential equation Eq. 4, over a non negligible range of couples ((), ∆()) and therefore of time, with only a few exceptions.The best linear fits are represented by the dashed lines (of the same colour as the graph of this entity), with high determination coefficients  ! and limited dispersion of the estimates of the feedback exponent  (see Supplementary Information).The black dashed line indicates the average estimates over the period of 22 January -04 April.However other entities, and other epidemic data as well, display similar graphs that have called attention to power-law growth, including with the help of log-log curve fittings 27,28,29 .But, these fits are sensitive to the choice of the initial time and data artefacts.We therefore developed a specific data analysis presented below.We analysed data from 22 January till 07 June hence containing both the end of the crisis in Hubei and its beginning in Europe to enable to analyse both stages of the epidemic.Two intermediate dates were used for a comparative analysis: 4 April as a tentative date for the well-advanced growth phase in Europe (about 3 weeks after the pandemic announcement by WHO), and 16 April as the ending date of the epidemic for the Hubei province).Nevertheless, tentative explorations until the beginning of November have confirmed the interest of a follow-up paper focused on the beginning of the second wave.

Growth phase analysis
Figure 1 shows that the time evolution of the cumulative incidence () of the aforementioned categories and entities are sub-exponential.Hence, we introduce a scaling dependency (i.e. a powerlaw) of their characteristic times () (and doubling time  ( ()) with respect to the current scale (): where the exponent  can be close to unity, but nevertheless distinct of it, and the prefactor  is no longer the inverse of a (characteristic) time except for  = 1.The presence of the prefactor () &$1 in the local time () (for  ≠ 1 in Eq. 3) makes all the difference (except at the initial time  ! of the contamination ( ! ) = 1) with respect to the exceptional case  = 1 (Eq.1): the local times () and  ( () are no longer invariants of the contamination dynamics, but increase with  (for  < 1).This merely corresponds to the fact that the system became less and less efficient due to its larger and larger outer scale, which rather makes sense.As expected, this increase of () yields a slower growth of () with respect to the exponential model.Injecting the time () in Eq.1 yields its scaling generalisation: where  can be now be understood as the feedback exponent from the cumulative incidence () onto the elementary increment () = () − ( − ), both are thus non stationary.This is a common feature for growth processes 29 , but not for cascades, which are usually supposed to have a stationary outer scale.We therefore need to proceed to a "cumulative-incremental analysis" to quantify this feedback, mainly to determine its exponent .The simplicity of this analysis might help to change the human behaviour with respect to epidemic risks, whose importance has been often underlined 14 .The very first step (see Fig. 2) corresponds to simply graph ∆( .) vs. ( .) in a log-log plot, where ∆( .) = ( .) − ( .$&).It is rather surprising to see how much the curves (corresponding to the various entities recalled above) collapse together providing a first indication of a given universality of the virus dynamics over a range of time scales (see Supplementary Information).Furthermore, the range over which this universality is better observed corresponds to power-laws ∆( .) ≈ ( .) 1 , i.e., a discrete versions of Eq.3.This scaling behaviour is supported by least square fits of the cumulative-incremental incidence curves (dashed straight lines in the log-log plot of Fig. 2) with high determination coefficients  ( and limited dispersion of the estimates throughout the time scales (see Supplementary Information)., where  7 denotes the average value of the feedback exponents 's of the growth phase analysis (Fig. 2).Hubei (dark blue) and Korea (light blue) rather clearly confirm a feedback exponent ′ ≈  7 , while other time series were beginning their decline phase.The loop exhibited by the time series of Spain in the category "deaths" is due to a sharp decrease (−1918 cases) of this cumulation on 25 May.All the posterior daily cumulation remain inferior to the cumulation of 24 May.There are similar behaviours, although less obvious due to smaller adjustments, for some other countries.
The most important result is that the estimates of the feedback exponent  are clearly below 1 for all types of time series over the period 22 January to 04 April: 0.81 ± 0.09 (confirmed cases), 0.68 ± 0.25 (deaths), 0.78 ± 0.13 (recovered).These estimates are mainly obtained on growth phases, however as displayed by Fig. 3, their mean values fit rather well the corresponding decline phases (see Supplementary Information).The local time () (Eq.3) increases linearly with respect to time, with the prefactor (1 − ).Indeed, a straightforward integration of Eq.4 yields: The empirical, local times () (defined by Eq.3 and normalised by (1 − )) are in agreement with this relation in the growth phase (see Supplementary Information).

Decline phase analysis
The second stage of the cumulative-incremental analysis is to explore the decline phase.By reversing the process () from its maximal value  234 = ( 234 ), we obtain a process ′( 5 ): whose growth phase corresponds to the decline phase of () and vice-versa.As / = ′/′, the scaling decline phase is ruled by a slight modification of Eq.4: Thanks to this symmetry, the cumulative-incremental analysis remains rather the same: it analyses the (discrete) graph ∆( .) vs. ( 234 − ( .)), seeking a power law relation for large times  .'s.
Figure 3 shows an overall agreement with ′ ≈ , particularly for the Hubei province that has a fully developed decline phase.For other countries the total cumulation is weakly approached by the empirical  234 , resulting in a temporary flattening of last points (see Supplementary Information).

Discussion and conclusions
So far, we have been using only deterministic calculus, whereas Figs. 2, 3 display non negligible fluctuations with respect to a power-law.Not all of these fluctuations result either from data uncertainty, or from quasi-Gaussian perturbations.These deviations can be easily accounted for in the framework of stochastic multiplicative cascades 30 .These processes are exponentials of a stochastic generators Γ() that broadly generalise ln( −  ! ) /( − 1).However, their stochastic differentiation remains rather close to the deterministic one, mostly adding a supplementary term  to the differential of the generator: () ≈ () Γ() +   (8)    When the generator Γ() is Gaussian,  is the "quadratic variation", extensions to Lévy stable generators, which are strongly non-Gaussian, have been also considered 8 .Therefore, the next stage of the cumulative-incremental analysis is to investigate the stochastic nature of the cascade generator, as well as how it respects the aforementioned symmetry between the growth and decline phases (Eq.6).Universal multifractals 30 can provide some preliminary, universal scaling characterisation, despite the present sample limitations (see Supplementary Information).The present results paved the way for joint scaling analysis of the vector-valued time series ( & (),  ( (),  6 ()), instead of separately analysing the time series corresponding to the three categories, and to introduce location and other dependencies.This would correspond to enlarge the domain, on which the process and its generator are defined, as well as their co-domain, on which they are valued 8 .

Figures
Figure 1 Sub Cumulative-incremental analysis of the growth phase: log-log plots of the couples ((), Δ()), where Δ() = () − ( − Δ)) is the increment (the time increment Δ being one day) of (a) "con rmed", (b) "deaths", (c) "recovered" cumulative incidence () of the following entities: Hubei, France, Germany, Iran, Italy, Japan, Korea, Spain and United Kingdom, over the period 22 January-04 April.Graphs should be read from left to right to follow the time arrow (() being non decreasing with time) and the almost vertical parts on the righthand side corresponds to compressed views of decline phases to be expanded in Fig. 3.These graphs display a rather universal behaviour, especially a common scaling/power-law behaviour Δ(i) ≈ (i), a discrete version of the differential equation Eq. 4, over a non negligible range of couples ((), Δ()) and therefore of time, with only a few exceptions.The best linear ts are represented by the dashed lines (of the same colour as the graph of this entity), with high determination coe cients 2 and limited dispersion of the estimates of the feedback exponent (see Supplementary Information).The black dashed line indicates the average estimates over the period of 22 January -04 April.
Cumulative-incremental analysis of the decline phase: log-log plots of the couples (max − (), Δ()) to analyse the decline phase of (a) con rmed, (b) deaths, (c) recovered incidences cumulative X(t) of the aforementioned entities (Hubei, France, Germany, Iran, Italy, Japan, Korea, Spain and United Kingdom), over the period 22 January -07 June, with an exception of shorter period for Hubei till 16 April.Contrary to those of Fig. 2, these graphs should be read from right to left to follow the time arrow (max − () being non increasing with time) and the almost vertical parts on the right hand side corresponds to compressed views of growth phases, which were expanded in Fig. 2. The reference curve (dashed black straight line) corresponds to the equation Δ() = (max − ()) , where denotes the average value of the feedback exponents 's of the growth phase analysis (Fig. 2).Hubei (dark blue) and Korea (light blue) rather clearly con rm a feedback exponent ′ ≈, while other time series were beginning their decline phase.The loop exhibited by the time series of Spain in the category "deaths" is due to a sharp decrease (−1918 cases) of this cumulation on 25 May.All the posterior daily cumulation remain inferior to the cumulation of 24 May.There are similar behaviours, although less obvious due to smaller adjustments, for some other countries.

Supplementary Files
This is a list of supplementary les associated with this preprint.Click to download. COVID19cascadeSupplementaryInformation20201106.pdf

Fig. 2 :
Fig.2: Cumulative-incremental analysis of the growth phase: log-log plots of the couples ((), ∆()), where ∆() = () − ( − ∆)) is the increment (the time increment ∆ being one day) of (a) "confirmed", (b) "deaths", (c) "recovered" cumulative incidence () of the following entities: Hubei, France, Germany, Iran, Italy, Japan, Korea, Spain and United Kingdom, over the period 22 January-04 April.Graphs should be read from left to right to follow the time arrow (() being non decreasing with time) and the almost vertical parts on the righthand side corresponds to compressed views of decline phases to be expanded in Fig.3.These graphs display a rather universal behaviour, especially a common scaling/power-law behaviour ∆( " ) ≈ ( " ) # , a discrete version of the differential equation Eq. 4, over a non negligible range of couples ((), ∆()) and therefore of time, with only a few exceptions.The best linear fits are represented by the dashed lines (of the same colour as the graph of this entity), with high determination coefficients  ! and limited dispersion of the estimates of the feedback exponent  (see Supplementary Information).The black dashed line indicates the average estimates over the period of 22 January -04 April.

Fig. 3 :
Fig.3: Cumulative-incremental analysis of the decline phase: log-log plots of the couples ( $%& − (), ∆()) to analyse the decline phase of (a) confirmed, (b) deaths, (c) recovered incidences cumulative X(t) of the aforementioned entities (Hubei, France, Germany, Iran, Italy, Japan, Korea, Spain and United Kingdom), over the period 22 January -07 June, with an exception of shorter period for Hubei till 16 April.Contrary to those of Fig.2, these graphs should be read from right to left to follow the time arrow ( $%& − () being non increasing with time) and the almost vertical parts on the right hand side corresponds to compressed views of growth phases, which were expanded in Fig.2.The reference curve (dashed black straight line) corresponds to the equation ∆() = ( $%& − ()) # ' , where  7 denotes the average value of the feedback exponents 's of the growth phase analysis (Fig.2).Hubei (dark blue) and Korea (light blue) rather clearly confirm a feedback exponent ′ ≈  7 , while other time series were beginning their decline phase.The loop exhibited by the time series of Spain in the category "deaths" is due to a sharp decrease (−1918 cases) of this cumulation on 25 May.All the posterior daily cumulation remain inferior to the cumulation of 24 May.There are similar behaviours, although less obvious due to smaller adjustments, for some other countries.
-exponential growth phase: log-linear plots of (a) con rmed, (b) death, (c) recovered cumulative incidences () of the following entities: Hubei, France, Germany, Iran, Italy, Japan, Korea, Spain and United Kingdom, over the period 22 January -04 April.An exponential behaviour would correspond to straight lines like those drawn for characteristic times = 4; 7 days, and therefore to doubling times = ln(2) ≈ 3; 5 days (respectively dashed grey and dotted black straight lines).Due to their (strong) concavity, all the trajectories are sub-exponentials, i.e., grow slower than any of their local approximations by an exponential that corresponds to a tangent to these log-linear graphs.