Emergence of universality in the transmission dynamics of COVID-19
Ayan Paul, Jayanta Kumar Bhattacharjee, Akshay Pal, Sagar Chakraborty
DDESY 20-234, HU-EP-20/42
Emergence of universality in the transmissiondynamics of COVID-19
Ayan Paul a,b,† , Jayanta Kumar Bhattacharjee c,† , Akshay Pal c,† , and Sagar Chakraborty d,† a DESY, Notkestraße 85, D-22607 Hamburg, Germany b Institut f ¨ur Physik, Humboldt-Universit ¨at zu Berlin, D-12489 Berlin, Germany c Indian Institute for Cultivation of Science, Jadavpur, Kolkata 700032, India d Department of Physics, Indian Institute of Technology Kanpur, Uttar Pradesh 208016, India † e-mail: [email protected], [email protected], [email protected], [email protected] ABSTRACT
The complexities involved in modeling the transmission dynamics of COVID-19 has been a major roadblock inachieving predictability in the spread and containment of the disease. In addition to understanding the modesof transmission, the effectiveness of the mitigation methods also needs to be built into any effective model formaking such predictions. We show that such complexities can be circumvented by appealing to scaling principleswhich lead to the emergence of universality in the transmission dynamics of the disease. The ensuing datacollapse renders the transmission dynamics largely independent of geopolitical variations, the effectiveness ofvarious mitigation strategies, population demographics, etc. We propose a simple two-parameter model—theBlue Sky model—and show that one class of transmission dynamics can be explained by a solution that lives atthe edge of a blue sky bifurcation. In addition, the data collapse leads to an enhanced degree of predictability inthe disease spread for several geographical scales which can also be realized in a model-independent manneras we show using a deep neural network. The methodology adopted in this work can potentially be applied tothe transmission of other infectious diseases and new universality classes may be found. The predictability intransmission dynamics and the simplicity of our methodology can help in building policies for exit strategies andmitigation methods during a pandemic.
Introduction
Despite several warnings of an imminent pandemic, COVID-19 dropped upon the world, quite unexpectedly, out of a blue sky.The spread of SARS-CoV-2 has left significant instabilities in the socioeconomic fabric of the society essentially defenestratingany notion of normality that existed. While the spreading dynamics of the disease is not novel [1], the instabilities it has causedhas made different parts of the society, and notably, different governance, respond to containing its spread in very differentmanners [2–6]. The determination of the optimal strategy, while not dragging what people have known as normality to theground, has been quite a challenge and highly dependent on the socio-economic condition of the country or region [7–10]. Onthe other hand, researchers have been struggling to bring some predictability in the spread of the pandemic since even a fewweeks of foresight can not only save an economy from being jettisoned but also save a considerable number of lives that needn’tbe lost. Moreover, given the predator-prey game that is being played out between the human population and the virus, once thegame starts showing positive outcomes for the human population, deciding on exit strategies become of prime importance as animmediate resurgence of the pandemic is the least coveted outcome.Epidemiologists and mathematicians have constructed and resurrected various kinds of models to describe the spreadof COVID-19 with varying degree of success. These include several variants of the venerable SIR (Susceptible, Infected ,Recovered) model like SEIR [11–14]—E for exposed, SEIRD—D for deceased, and Stochastic SIR [14, 15]. There are morerealistic delay differential equation models [16–20], renormalization group equations [21] and the highly computation intensive(and arguably the most detailed) agent-based network models [22–30]. There are even purely data-driven models [31–33]. Inall these approaches, sets of parameters have to be tuned precisely to analyse a particular country or region to obtain predictions,since every region seems to have its own special time evolution for disease propagation, making this a Herculean task. Althoughall models’ assumptions cannot strictly describe every detail of any real-world system, it is believed that such systems maypossess certain universal behaviours independent of these details [34].We propose to establish that there is a method in the madness of the spreading of this disease which can be revealed bysubjecting the available data of all the countries or regions to a scaling analysis reminiscent of the data collapse in the study ofcritical phenomena [35, 36] in the sixties and seventies of the previous century. Data collapse leading to universality has been a r X i v : . [ phy s i c s . s o c - ph ] F e b ESY 20-234, HU-EP-20/42
Mar2020 Apr May Jun Jul Aug Sep Oct Nov Dec d N ( t ) / d t Phase 1 of Type I
JapanNew ZealandIrelandAustraliaSlovakia
Mar2020 Apr May Jun Jul Aug Sep Oct Nov Dec d N ( t ) / d t Phase 1 of Type II
UKGermanyItalyNetherlandsSpainSouth KoreaQatarRussia
Mar2020 Apr May Jun Jul Aug Sep Oct Nov Dec d N ( t ) / d t States in the USA
New YorkCaliforniaTexasArizonaIllinoisFlorida
Mar2020 Apr May Jun Jul Aug Sep Oct Nov Dec d N ( t ) / d t States in Germany
HamburgBerlinBayernNordrhein-WestfalenBaden-W¨urttemberg
Figure 1.
The variation in the spreading of COVID-19 in various countries and some states of the USA and Germany.
Top left: the countries for which the first phase of spreading falls into the category defined as Type I. Top right: the countriesfor which the first phase of spreading falls into the category defined as Type II. Bottom left: some states in the USA that showvery different spreading patterns for the disease. Bottom right: some states in Germany that show very similar pattern in thespreading of COVID-19. observed in a large variety of systems and we provide a short overview of the literature in the Appendix. We emphasize that theprocedure we follow is model-agnostic and does not require the assumption of a model for the transmission and mitigationdynamics. In addition, this method lies open to model-based interpretations as we later point out. Almost miraculously, wefind a simple yet nontrivial mean-field model where the carrying capacity of infected population depends on the populationsize. The model—that we call Blue Sky model (BSM)—undergoes a blue sky bifurcation [37] (also called the saddle-nodebifurcation or the fold bifurcation) which is a ubiquitous structurally stable bifurcation found in variety of systems, such aspower systems [38], viscous profiles [39], neural networks [40], excited molecules [41], plasmas [42], and the Ginzburg–Landausystems [43]. At the edge of the blue sky bifurcation point, appears a universal data collapse. Furthermore, given that thescaling and data collapse have stood well the test of time for a very diverse set of systems and is grounded in well understoodphysical principles, we are proposing a robust method for making long-term predictions that will help in evaluating diseasemitigation efficacies and planning exit strategies for the future. We hope to convince the readers that we have established amethod which does not require parameter tuning or model selection to gain a handle over the prediction of disease spread.The salient features of our work are:• We show that there exists a universality in the spread of the disease once proper scaling is applied to the data.• Two universality classes emerge that give us a window into understanding how well a country or region performed inmitigating the disease. These universality classes are independent of geographical scales.• We propose the BSM—an extension of the logistic growth model—which explains the existence of these two classes, thefirst, through a solution that lives at the edge of blue sky bifurcation and a second that maps the BSM onto the logisticgrowth model.• We show how predictability can be realized from this analysis in a model-agnostic manner relying only on the inferencedrawn from the first phase of the disease spread. For this we construct a deep neural network (DNN) based predictivemodel. We also show that an equally good prediction can be made using the BSM. We give a few examples from theongoing phase of the pandemic in several countries and states.
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Emergence of Universality in Transmission Dynamics
To bring about some semblance of order in what looks like a disparate set of data from several countries, we first focus entirelyon phase-1 of the pandemic. We denote the total number of cases (individuals who have tested positive for the COVID-19)at time t as N ( t ) with t being measured in days. The derivative dN ( t ) / dt is the number of new cases detected on a givenday. Phase-1 refers to the situation, for a given country or region, where the dN ( t ) / dt increased from zero or a small value,went through a maximum and then decayed to almost zero (Type I transmission) or to a small value (Type II transmission). Itstayed near zero or near the small value for a few weeks before starting to rise again at the onset of the next phase. We need todifferentiate quantitatively between the phrase “near zero for a few weeks” and “a small value for a few weeks”. We do this byinspecting the rise in the total number of cases, N ( t ) , over four weeks past phase-1. If the change is more than 4% of N ( t ) , wemark the transmission as being of Type II, otherwise it is Type I.Taking a more careful look at Figure 1, we note that the total number of cases and the total duration of phase-1 are verydifferent for each country. This is to be expected as these countries have very different population densities and differentpatterns of inhomogeneities in the density of population. Moreover, the policies of the government and the acceptance of theseby the governed affect the implementation of precautionary measures [8–10, 44]. The policies and exit strategies towards theend of a phase in any country also determines whether the spread of the disease lingers on, albeit at a much reduced rate, oris completely rooted. The latter essentially divides the world into two classes in our formalism. However, looking at barecase numbers, one is bound to conclude that each country or region is a unique case study requiring a detailed modeling oftransmission dynamics and incorporating the effects of mitigation and containment measures, a task that seems formidableright from the onset.The top left and right panels of Figure 1 show some examples of countries that went through a phase-1 transmission of TypeI and Type II respectively. The disparity in the spreading patterns is apparent. In the lower panels of Figure 1 we focus on thestates of the USA and Germany. In the USA we see that the spreading pattern in the different states was somewhat decorrelated.This is where our formalism is particularly useful. The physical principles that we advocate being independent of geographicalscales can be applied to countries, states or even counties and districts given they have seen a significant spread of the diseasewithin the region. On the other hand, the spread of COVID-19 in Germany in the various states has been quite correlated.Now, let us see how universality can emerge from a scaling of the COVID-19 confirmed case rate data. We introduce anumber N max which is the total number of infected people at the end of phase-1. For Type I transmission the value of N max iseasy to infer. It is the the total number of cases at which the N ( t ) curve flattens out after the close of the phase. For Type IItransmission, where N ( t ) is never completely flat, we determine the close of phase-1 following the procedure described above.A perfect example is the case of Germany. For the period May 31 st , 2020 to July 17 th , 2020, the value of dN ( t ) / dt hoveredaround 400 which is more than an order of magnitude smaller than the peak value of 6000. In this case N max corresponds tothe number of cases at the end of this period of stagnancy in the growth rate, i.e., the number of cases on July 17 th , 2020. Ofcourse, there are some degree of variability associated with the choice of the end-date for a phase. However, an estimation errorof a few days does not change the onset of universality. Therefore, we choose the end-date for phase-1 as the first day when the N ( t ) curve flattens out for Type I transmission and the last day before the onset of the next phase for Type II transmission.Our conjecture is that universality emerges from the scaling given by˜ t = tt / , and ˜ N ( ˜ t ) = N ( ˜ t ) N max . (1)The ratio ˜ N ( ˜ t ) is relatively insensitive to the testing policies, the inhomogeneity in the population density, the size of the countryor region, the details of the mitigation methods advocated, etc. The rescaling of the time by an appropriate time scale for eachcountry or region, irons out the nuances brought about by the government policies like the duration of lockdown, exit strategies,etc., at the lowest order of approximation. Therefore, for each country or region a natural time scale to choose is t / which isthe time at which N ( t ) = N max / t = t / n corresponding to the time when N ( t ) = N max / n , with n being a positive definite number would not affect the nature of this analysis. However, choosing n = d ˜ N ( ˜ t ) / d ˜ t at t = t / fora simple logistic growth. According to our conjecture, ˜ N ( ˜ t ) plotted against ˜ t should show a data collapse opening up avenuesfor simple predictive modeling and generalization beyond socioeconomic and geopolitical variations. Data-collapse in transmission dynamics of COVID-19
In the light of the preceding discussion we present the ˜ N ( ˜ t ) vs. ˜ t plots for several countries and states in left-most plots ofFigure 2. The data collapse is clearly evident. We see that independent of variabilities in transmission dynamics the universalityin the rescaled data dictates that there are two distinct classes. The plots in the middle of Figure 2 show the functional ESY 20-234, HU-EP-20/42
Type I Type I Type IType II Type II Type II
Data collapse for Type I and Type II transmission Blue Sky model and DNN
Figure 2.
The emergence of universality in the transmission dynamics of COVID-19 . Left panels: The rescaled totalnumber of confirmed cases, ˜ N ( ˜ t ) = N ( t ) / N max , and the its derivative, d ˜ N ( ˜ t ) / d ˜ t, with respect to the scaled time ˜ t = t / t / forType I transmission where the disease spread was almost completely routed after phase-1 and Type II transmission where thedisease spread was not completely routed after phase-1. The data collapse is evident and independent of the country or statebeing considered showing the emergence of two universality classes in the transmission dynamics of COVID-19 independent ofsocial and geopolitical variations. Right panels: The fit to the rescaled data using the Blue Sky model (solid blue line) andthe model agnostic fit (dashed black line) with a deep neural network. Note that ˜ α = . in the rightmost bottom panel ispractically the value expected at the edge of the blue sky bifurcation at / κ = / ( × . ) = . . The good agreementbetween the two approached can be clearly seen. dependence of d ˜ N ( ˜ t ) / d ˜ t on ˜ t . For the Type I transmission one clearly sees that tail of the distribution at higher ˜ t dips to very lowvalues. This implies that after phase-1 ended, the daily case rate actually went down to a negligible number, which, originally,was our basis for identifying Type I transmission. On the other hand, for the same curve in the lower middle plot of Figure 2,the tail of the distribution at higher ˜ t never really goes to as low values as that for Type I transmission. This is because ina transmission of Type II the countries and regions never saw a complete rooting of the spread of the disease. To show theindependence of geographical scales of our formalism, we include New York, USA and Bayern, Germany as two states trackingthem over phase-1 of the spread of COVID-19. We see that both the states show transmission of Type II.It is important to point out here that the data collapse that we see effectively filters out the variances in the dynamics of thedisease spread and the policies for disease mitigation. It also effectively filters out how quickly the response to the pandemicwas orchestrated (a matter of government policy), how drastic these measures were and all other geopolitical demographics thatmake the disease propagation dynamics in each country or state look very different in Figure 1. The key difference between thetwo classes reduces to the effectiveness of the countries in fully containing the disease itself, each using their own bouquet ofmitigation methods and exit strategies. Indeed, this data collapse removes all necessity of fine-tuning a plethora of parametersto examine each country or state separately and shows that there is a common pattern in the spread of COVID-19. Interpretation of t / The parameter t / gives us a distinct insight into how well the spread of the disease was mitigated in a particular phase. Ashorter t / implies a more effective mitigation strategy. For example, from Figure 2 we see that the t / for South Korea is 18days, which means that the spread of the disease was stopped very rapidly as compared with UK for which t / is 57 days.However, despite being effective in slowing down the spread rapidly, South Korea falls into Type II since the spread was notcompletely stopped but lingered on. On the other hand, Ireland, even being of type I, has a relatively longer t / of 39 daysshowing that while the spread of the disease was almost completely rooted it took a longer period of time to do so hinting aspossibilities of drafting better mitigation strategies. ESY 20-234, HU-EP-20/42
Blue Sky model
Among the four broad classes of models (with possibility of overlap among them) of infectious disease spreading—viz,data-driven phenomenological models [45, 46], compartmental or lumped parameter models [47], agent-based models [48], andstochastic differential equation models [49]—while the former two are mathematically more accessible to policy makers, thelatter two are more detailed in their approach. The first two types of models are to the last two types, what thermodynamics is tostatistical mechanics. The most simple lumped parameter model is undoubtedly the well-known one dimensional autonomouslogistic growth model [50]: dN ( t ) dt = β N ( t ) (cid:18) − N ( t ) κ (cid:19) , (2)on which we shall focus. Here, β > κ → ∞ . Mostcommonly, the resource is assumed to be unaffected by the population size, N ( t ) , and hence the population grows to settle ontothe value N ( ∞ ) = κ , which is the carrying capacity. We discuss some variants of the the logistic growth model and their fit tothe data in the Appendix.The system of our interest is a population where N ( t ) represents the number of COVID-19 infected individuals that infecthealthy individuals so that the infected population grows at a rate β . All the depletable conditions that are favourable forthe infected population to grow is quantified through the positive parameter, κ . One could argue that rather than relyingcompletely on the carrying capacity parameter, one could make this model more elaborate by making it higher dimensionalto mathematically incorporate the interaction among infected, susceptible, exposed, asymptomatic, quarantined, recovered,immune, and deceased individuals. However, the fact remains that all such higher dimensional models, despite their share ofsuccess, bring in more parameters and more speculations while the only available data is still mostly that of the infected anddeceased individuals. Thus, in this paper, we are motivated to take the route of simple two-parameter one dimensional modelcomplimented with the information extracted from the real data.It is straightforward to realize that the carrying capacity can easily depend on N ( t ) : e.g., government may be practicallyincapable of quarantining infected individuals as their numbers grow, thereby effectively increasing the carrying capacitybecause the infected individuals would interact with more and more healthy individuals leading to terms which are higher orderin N ( t ) . Therefore, if we were to model the realistic scenarios, we should let κ depend on N ( t ) in nontrivial ways. In thiscontext, we find it useful to define an order parameter, ρ , as ρ ≡ κ (cid:18) d κ dN (cid:19) . (3)Note that lower the value of the order parameter, more is the control on the change of the carrying capacity. The case ofthe order parameter being zero corresponds to the traditional logistic equation governed system where the carrying capacityis independent of N ( t ) . We would like to mention here that past considerations [16, 51–59] of the logistic growth relatedcompartmental models, as applied to the modeling of COVID-19, do not explicitly incorporate the natural notion of thestate-dependent carrying capacity.In order to account for the aforementioned variation of the carrying capacity with the change in the infected population size,we propose a simple, yet non-trivial, functional form for the carrying capacity: κ = κ bsm ≡ κ lgm − ακ lgm N ( t ) , (4)where κ lgm is the constant carrying capacity corresponding to the traditional logistic growth model and α is a non-negativeparameter. In Equation 2 with κ bsm as the carrying capacity, we define the Blue Sky model (BSM) that we use to explain ourfindings in this paper. In the BSM, we note ρ = α which renders a solid physical meaning to the (otherwise ad hoc) parameter α . Following the transformations, ˜ N ( t ) = N ( t ) / N max and ˜ t = t / t / , adopted in the previous discourse, we define ˜ β ≡ t / β ,˜ κ lgm ≡ κ lgm / N max , and ˜ α ≡ N α . Henceforth, we shall consider only the rescaled BSM where these rescaled parameters areused.In the BSM, the blue sky bifurcation occurs at the nonzero fixed point at ˜ α = ˜ α edge ≡ / κ — the edge of blue sky . Beyondthe edge of blue sky, i.e., for ˜ α > / κ , the two nonzero fixed points vanish and ˜ N ( ˜ t ) = α should respect for the model to be physical: one must impose the condition,˜ α ∈ [ , / κ ] , so that a stable attracting nonzero state of the infected population, viz., ˜ N ( ˜ t ) = ( − (cid:113) − α ˜ κ ) / ( α ˜ κ lgm ) exists. The third fixed point, ˜ N ( ˜ t ) = ( + (cid:113) − α ˜ κ ) / ( α ˜ κ lgm ) , is unstable and not of any practical interest to us. ESY 20-234, HU-EP-20/42 - - - - - - - - - - - - - - - - - - - - . . . . . . . N ( t ) × France from2020-8-8after2020-11-15TypeI-DNN[ t / : 89days]TypeI-BSM[ t / : 89days]TypeII-DNN[ t / : 101days]TypeII-BSM[ t / : 103days] - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - . . . . . . . N ( t ) × Germany from2020-8-30after2020-12-25TypeI-DNN[ t / : 109days]TypeI-BSM[ t / : 109days]TypeII-DNN[ t / : 125days]TypeII-BSM[ t / : 126days] - - - - - - - - - - - - - - - - - - - - - - - - - - N ( t ) Ireland from2020-8-15after2020-10-30TypeI-DNN[ t / : 68days]TypeI-BSM[ t / : 68days]TypeII-DNN[ t / : 78days]TypeII-BSM[ t / : 79days] - - - - - - - - - - - - - - - - - - - - - - . . . . . N ( t ) × Italy from2020-9-20after2020-12-15TypeI-DNN[ t / : 57days]TypeI-BSM[ t / : 58days]TypeII-DNN[ t / : 66days]TypeII-BSM[ t / : 67days] - - - - - - - - - - - - - - - - - - - - - - - - . . . . N ( t ) × UK from2020-8-30after2020-11-15TypeI-DNN[ t / : 69days]TypeI-BSM[ t / : 69days]TypeII-DNN[ t / : 79days]TypeII-BSM[ t / : 80days] - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - N ( t ) × Russia from2020-8-20after2021-1-1TypeI-DNN[ t / : 124days]TypeI-BSM[ t / : 125days]TypeII-DNN[ t / : 142days]TypeII-BSM[ t / : 144days] - - - - - - - - - - - - - - - - - - - - - - - - - - - - N ( t ) Arizona from2020-10-1after2020-12-25TypeI-DNN[ t / : 81days]TypeI-BSM[ t / : 81days]TypeII-DNN[ t / : 93days]TypeII-BSM[ t / : 94days] - - - - - - - - - - - - - - - - - - - - . . . . . . N ( t ) × California from2020-11-1after2020-12-25TypeI-DNN[ t / : 54days]TypeI-BSM[ t / : 54days]TypeII-DNN[ t / : 61days]TypeII-BSM[ t / : 62days] - - - - - - - - - - - - - - - - - - - - - - - - N ( t ) Berlin from2020-8-30after2020-12-25TypeI-DNN[ t / : 93days]TypeI-BSM[ t / : 93days]TypeII-DNN[ t / : 107days]TypeII-BSM[ t / : 108days] - - - - - - - - - - Figure 3.
A display of predictability from the emergence of universality classes.
We show how predictions can be madeform scaling data and using knowledge gathered from phase-1 of the pandemic. Once the peak in the daily case rate is reached(or has been assumed to have reached), prediction for several weeks or months can be made. The solid dark blue lines in theinsets show the data being used to find t peak , and from it, t / and N max . The dotted red lines in the insets are the test datawhich lie in the future with respect to the data used to determine the peak. The light blue lines and bands are predictionsfrom the model-dependent fit using the Blue Sky model (BSM) to phase-1 data and the gray lines and bands are those fromthe model-agnostic fit using a deep neural network (DNN). The dashed lines are prediction for transmission of Type I and thedot-dashed lines are for transmission of Type II. All predictions are made till ˜ t = . or t = . t / starting from the beginningof the ongoing phase of disease spread. The top and middle rows show predictions for various countries while the bottom rowshows predictions for two states in the USA and one in Germany. As seen in the rightmost column of Figure 2, the BSM remarkably segregates the COVID-19 transmission dynamics incountries and states into two universality classes—Type I and Type II—characterized by two distinct and well separated valuesof the (rescaled) order parameter, ρ —one close to the logistic growth value ( ˜ α ≈
0) and the other close to the edge of blue sky( ˜ α ≈ / κ ) respectively. The data collapse onto the respective curves is strongly reminiscent of the similar phenomenonobserved in the critical phenomena and their universality across a plethora of physical systems [35, 36]. Thus, we speculatethat it may be a potential hint of a hidden renormalization scheme present in the microscopic details of the correspondingagent-based models. Deep neural network
While the BSM provides a very good and intuitive explanation of the data, we feel that one should not be limited to exploringthe universality in the data through specific models only. Hence, we perform a model-agnostic fit using a universal function
ESY 20-234, HU-EP-20/42generator in the form of a deep neural network (DNN). We use a three-layers deep, each layer 16 nodes wide DNN for thispurpose. The core of the architechture can be found in the Methods and Materials section and a detailed discussion can befound in the Appendix. The fit to the data using the DNN can be seen in the rightmost panel of Figure 2. The back dashed linecorresponds to the DNN fit. We fit to the ˜ N ( ˜ t ) distribution. We see that the fit of Type II transmission is almost identical to theBSM fit showing that the data-driven and model-dependent methods reach the same conclusion. For transmission of Type I theDNN fit is slightly different from the BSM fit and notably so at the center of the distribution where d ˜ N ( ˜ t ) is shown to be higherfrom the DNN fit than from the BSM fit. Predicting of COVID-19 spread from universality
The data collapse that we show leads us to the ability of making predictions in both a model-dependent and a model-agnosticway. While the BSM shows a clear explanation of the data, the advantage of a data collapse lies in the emergence of universalitywhich allows for model-agnostic predictions from the simple understanding that the transmission dynamics described by therescaled variables are the same in every phase of the the disease spread. Hence, we would like to emphasize both possibilitiesalthough they lead to similar results in predictions.It should be noted that the only parameter that needs to be estimated for predictions to be made is t / for the ongoing phasefor a country or region. Once that is determined, the parameters extracted from the fit to the phase-1 data can be used to makepredictions for the ongoing phase. For both the approaches we use a common algorithm except for the functional form that weuse to fit to the data. For the model-dependent approach we use the BSM and extract its parameters from data, whereas in themodel-agnostic approach we use the trained DNN to predict the distribution of ˜ N ( ˜ t ) .The algorithm used to make the predictions is delineated in Materials and Methods section. To show the extent to whichour prediction work, we assume that a country or state is just past the peak of dN ( t ) / dt . The data that we use for determining t peak is represented by the solid dark blue line in the insets of Figure 3. The data past the end of this line, marked with a dottedred line is assumed to be in the future. We use the “past” data to predict for the “future” and then extend our prediction to˜ t = .
5. Since t / is unique for each country or state, each approach and each transmission types, the length of time for whichpredictions are made differ. Model-dependent prediction
The fitting of the BSM to data from phase-1 paves the way for making model-dependent predictions for future phases of thepandemic. This can be done by identifying t / and, from it, N max . This is only possible if, in a current phase of the pandemic,the daily case rate dN ( t ) / dt has already reached a peak. This peak lies at time t = ˜ t peak t / where ˜ t peak can be derived from thefit parameters and is given in Equation 7. By definition, t = t / is the time at which the N = N max / N ( t ) being countedfrom the beginning of the phase. Once these two parameters have been extracted, one can make two separate predictions forany geographical region: the first assuming a spread of Type I and the second assuming a spread of Type II. These predictionsfor Type I (dashed blue line) and Type II (dot-dashed blue line) are shown in Figure 3 for several countries and states. Thebands in blue represent the error in the prediction propagated from the error in the estimation of ˜ t peak . Model-agnostic prediction
We also show the predictions from the model-agnostic approach using a trained DNN in Figure 3. The logic that we follow toidentify the peak of the dN ( t ) / dt distribution is the same as for the model-dependent predictions. However, t / now variesbetween the two methods since ˜ t peak is different for the model-dependent and the model-agnostic fits. The dashed gray linesrepresents transmission of Type I and the dot-dashed gray lines represents transmission of Type II. For Type I transmission, theBSM gives predictions similar to the DNN. However, for Type II the BSM mostly predicts somewhat larger number of cases.This is because ˜ t peak = .
85 from the BSM fit of the phase-1 data is a bit lower than that from the DNN fit of ˜ t peak = .
87. Thegray bands show the error in prediction propagated from the esrror in estimation of ˜ t peak .In the model-agnostic approach the regression is completely data driven, and hence, does not represent any underlyingprinciple of the disease spread. This can be considered as a drawback of this approach. However, we show that the predictionsmade through this approach is very similar to the one made with the BSM. In addition the fit to data is also quite robust as canbe seen from Figure 3. Discussions on the predictions
There are several subtleties about the prediction that we should mention here. Firstly, the most important and sensitive part ofthe prediction is locating the peak of dN ( t ) / dt during an ongoing phase. For some cases t peak is easy to locate, for instance, forItaly and Ireland in Figure 3. However, for some other countries like Germany, it is somewhat more involved. Secondly, justpast the peak it is difficult to gauge whether N ( t ) will be monotonically decreasing or not. Lastly, in cases like Germany andUK we expect a significant margin of error in determining the peak itself since the dN ( t ) / dt is not strictly unimodal. This ESY 20-234, HU-EP-20/42uncertainty in the determination of the peak is what the error bars for the predictions quantify. The error in determining t peak propagates to an error in determining N max . Hence, the error bars are small for Ireland and Italy, while they are quite large forGermany and California—the latter being just past the peak in the dataset we use. The spread of COVID-19 in Russia had justreached the peak while we were doing this analysis and hence, we make pure predictions without any test data.In all the plots in Figure 3, we see that Type II transmission lasts much longer than Type I transmission. For Ireland we seethat it was proceeding along the curve for Type I transmission until the next phase started when it diverged once more. Formthis plot we now get an idea of how the pandemic might have been rooted in Ireland had the next wave not started. UK on theother hand seemed to be following the trajectory taken by Type II transmission until the number of new cases per day startedrising once more. Summary
The surfacing of a worldwide phenomenon such as a pandemic is highly disruptive. The social, political, and economicdisruption that this pandemic has brought resulted in an essentially chaotic response to its mitigation. Everything from testingstrategies to containment procedures followed very different patterns in different nations and variation even at the regional levelswere very palpable. This has made the analysis of the data to model the underlying dynamics of disease spread a formidabletask that has not yielded to any generalization.In this paper we show that the emergence of order from within this chaos is made possible by applying well tested proceduresdrawn from the study of critical phenomenon leading to the formation of universality classes. Such ordering in the data leads tothe possibility of characterizing the transmission dynamics into two classes, distinguished by how well the disease has beencontained by each country or region. This clearly shows that the variation of the transmission dynamics, and the effect ofpolicies related to disease mitigation, testing guidelines and exit strategies can be scaled away to instigate a data collapse thatallows us to bring about remarkable predictability in the spread of the disease.We reemphasize that without having to go into the rigours of extracting epidemiological parameters and having to formulatespecific models for the spread of the disease, the emergence of universality allows for the prediction of future disease spread.The lack of clear understanding of what model to use and how to extract epidemiological parameters has been a majorhurdle to making mathematically sound predictions during the spread of COVID-19. In addition, such methods require that eachcountry and each region be treated separately, quickly leading to an explosion of parameters and systems of equations. Whilebeing model-agnostic certainly has several drawbacks, the universality in the spread of the disease that we bring to surfacehere circumvents many of these problems by exploring the emergence of one single functional form that can be used to makeprediction for the spread of the disease once a region has reached the peak of the spread. This clearly allows policy decisions tobe made around exit strategies and lets the responsible governance understand how long it might take for a spreading wave ofthe disease to be contained given the current measures are kept in place.We would like to point out that the advent of vaccination can be accommodated for in our analysis since it effectivelyreduces the size of the total susceptible population which, in turn, does not affect the rescaled parameters since they are scaledby N max . Therefore, the validity of our work is not short-lived and have general applicability, regardless of the mitigationmethods used.Lastly, we bring to light the proposition of the BSM which explains the data remarkably well. We wish to highlight that it isan intriguing coincidence that Type II dynamics should live at almost the edge of blue sky bifurcation in the BSM. Looking atthe finite nonzero value of the order parameter vis-à-vis the almost vanishing value of order parameter for Type I transmission,one is readily reminded of the phase transition phenomenon in physical systems. It is tempting to interpret the two types oftransmission as two phases quantified by an order parameter. Furthermore, it remains to be seen if the newly introduced BSMfinds applications in other epidemiological systems. Acknowledgements
This research was supported in part through the Maxwell computational resources operated at DESY, Hamburg, Germany. Wewould like to thank Jon Crowcroft, Alejandro de la Puente, Mikhail Prokopenko and Melinda Varga for useful comments.
Appendix: Materials and Methods
All the numerical analysis in this work was performed using Python 3 and several libraries like SciPy, NumPy, Pandas andstatsmodels. All the code necessary for reproducing our results, further exploring the data, making predictions and for makingthe plots are available in a GitHub repository: https://github.com/talismanbrandi/Universality-COVID-19.
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Data sources and curation
In this work we used data from publicly available sources. The two primary sources that we used are:• Data on COVID-19 confirmed case rate obtained from the Johns Hopkins University, Center for Systems Science andEngineering database [60] through their GitHub repository. The time-series for all the countries in the world and for allthe counties in the USA are available from here. For the confirmed case rates in each state of the USA we merged thedata for the counties in each state.• For Germany the data is available through the Robert Koch Institute from the RKI COVID19 database.For all datasets we worked with the seven day averages to remove the fluctuations due to variations in reporting over theperiod of a week. The last date on the time-series used in this work is 31 st of December 2020. Rescaling of the BSM Differential Equations
The rescaled BSM can be explicitly written as: d ˜ N ( ˜ t ) d ˜ t = ˜ β ˜ N ( ˜ t ) (cid:18) − ˜ N ( ˜ t ) ˜ κ lgm + ˜ α ˜ N ( ˜ t ) (cid:19) . (5)For the purpose of prediction, we need to estimate ˜ t = ˜ t peak at the peak of the distribution of the first derivative, d ˜ N / d ˜ t . ForType I transmission, ˜ t peak = t peak = .
98 from the fit shown in Figure 2 for Type Itransmission since the solution lies very close to the pure logistic growth model ( ˜ α = t peak depends on the fit parameters. Since the transmission happens at the bifurcation point — ˜ α = / κ —integration ofEquation 5 at the corresponding value of parameters is needed. Integration yields:˜ t = β (cid:20) ln (cid:12)(cid:12)(cid:12)(cid:12) ˜ N ( ˜ t )( ˜ N − κ lgm ) ˜ N ( ˜ N ( ˜ t ) − κ lgm ) (cid:12)(cid:12)(cid:12)(cid:12) − (cid:18) κ lgm ˜ N ( ˜ t ) − κ lgm − κ lgm ˜ N − κ lgm (cid:19)(cid:21) . (6)where ˜ N = ˜ N ( ˜ t = ) .In general, the BSM has three extrema in d ˜ N / d ˜ t at ˜ N ( ˜ t ) =
0, ˜ N ( ˜ t ) = κ lgm /
3, and ˜ N ( ˜ t ) = κ lgm . For the solution at thebifurcation point, there are only two extrema which are the first two, and the maximum is then at ˜ N ( ˜ t ) = κ lgm /
3. Putting all ofthese together, one finds from Equation 6,˜ t peak = β (cid:20) ln (cid:18) κ lgm − ˜ N N (cid:19) + κ lgm − N κ lgm − N (cid:21) . (7)On using the fit parameters for transmission of Type II shown in Figure 2, we get ˜ t peak = .
85 from Equation 7. Since ˜ N (cid:28) ˜ κ lgm ,we can simplify the expression for ˜ t peak to: ˜ t peak = β (cid:20) ln (cid:18) ˜ κ lgm ˜ N (cid:19) + (cid:21) . (8) Numerical solution of BSM
The fit to the BSM was performed by numerical integration of the solution to the differential equation given by Equation 5. Forthis we used the SciPy package [61].
Deep neural network architecture
To construct the DNN used in the model-agnostic fit we used TensorFlow [62]. The following are the details of the DNNarchitecture:• depth: 3 layers• width: 16 nodes in each layer• optimizer: Adam with a learning rate of 0.01• loss function: mean square error• regularization: early stopping• validation split: 0.2Since the data for ˜ N ( ˜ t ) is already scaled between 0 and 1, no scaling of the data was necessary. The DNN took about100-200 epochs to train with the regression taking only a few seconds to be completed. A more detailed discussion of thearchitechture can be found in the following sections. ESY 20-234, HU-EP-20/42
Algorithm for predictions
Once we have the fit to the BSM or have trained the DNN with phase-1 data we can make predictions for the subsequent phases.The procedure we follow for this is given by:1. Choose a starting date where the cases start to rise (the start-date) from the plot of dN ( t ) / dt for the second or anysubsequent phase.2. Subtract the number of cases up until the start-date from the total number of cases to get the adjusted number of cases.3. Locate the time t = t peak at which the peak of dN ( t ) / dt occurs with t = t peak is proportional to t / with the constant of proportionality being approximately 1 for Type I transmissionand a constant (approximately 0.85) extracted from data for phase 1 of the spread for transmission of Type II.5. Find the adjusted number of cases on the date corresponding to t / . Twice that number is the adjusted N max .6. Use the functions fit from data for phase 1 (the BSM for model-dependent predictions and DNN for model-agnosticpredictions) to estimate the future adjusted number of cases.7. Now add back the number of cases up until the start-date to get the prediction of the disease spread. Appendix: Universality in physical systems . . . . . . . . . . . . . . ρ/ρ c . . . . . . . . . . T / T c NeAKrXeN O COCH
10 1 0 . . qξ Γ ∗ ( ρ c ) CO (coex. curve)CO ( ρ c ) Xe (coex. curve)Xe ( ρ c ) Figure 4.
The emergence of universality in fluid systems .The left panel is a reproduction of figure 2 of Ref. [63] which showsdata collapse in the coexistence of liquid and vapour states for several fluids. The right panel is a reproduction of figure 10 ofRef. [64] and shows dynamical scaling for several fluids.
In this section we discuss some examples of data collapse seen in physical system. The emergence of universalityaccompanied by a data collapse is quite common and has been observed for more than half a century. We begin our expositionwith data collapse in the study of critical phenomenon [35, 36, 65] in fluids. We intend to keep the discussion only at arepresentative level to give some introduction to universality and data collapse and not make it an exhaustive survey which isnot within the scope of this paper.It is well known that if a gas is subjected to isothermal compression, then at a certain pressure it undergoes a phase transitionto the liquid state while the pressure remains constant. Upon completion of the phase transition, the pressure increases verysharply on further compression while the volume increases too. If the temperature is raised and the process repeated, the samephenomenon occurs except that the parameter space over which the pressure and volume simultaneously rise grows smaller andsmaller as the temperature is raised. At a particular temperature, T = T c , this parameter space reduces to a point and at highertemperatures the liquefaction does not occur however high the pressure is. The temperature, T c , is the critical temperature andin its vicinity the fluid shows a large amount of fluctuation as it lies between a state of reasonable order, the liquid state, and astate of disorder, which is the gaseous state.The fluctuations in density very close to the critical point actually mean that some short-lived liquid state bubbles areproduced in the gas. As the critical point is reached these fluctuations become long lived and infinitely long ranged as one has aliquid-vapour coexistence. The infinite lifetime of fluctuations and the infinite range of correlation mean that this phenomenon ESY 20-234, HU-EP-20/42cannot be sensitive to which gas is being considered. Whether it be Carbon Dioxide, Oxygen, Nitrogen, Xenon or SulphurHexafluoride, they will all behave in the same way. This is known as universality and, hence, means that if we draw the equationof state of any gas near the critical point with pressure scaled by the critical pressure and the volume scaled by the criticalvolume, then we will get just one curve. The critical volumes, critical temperatures and critical pressures are all different for thedifferent materials but the scaled equation of state is the same as can be seen from Figure 4. If P , V and T are the pressure,volume and temperature of a fluid with a and b being constants specific to the fluid, then (cid:16) p + aV (cid:17) ( V − b ) = RT −→ (cid:18) π + φ (cid:19) ( φ − ) = τ , (9)where π = P / P c , φ = V / V c and τ = T / T c with P c , V c and T c being the pressure volume and temperature at the critical point.The equation of state thus collapses to a single equation independent of the nature of the fluid.An identical situation holds for the dynamics which probes the lifetime of the fluctuations. The experimental techniquefor probing fluctuations directly is frequency resolved light scattering. The intensity I as a function of frequency, ω , is whatis measured and at a given temperature and has a Lorentzian shape. As the temperature is lowered towards the critical point,both the peak (at ω =
0) and the width at half-maximum increase with decreasing temperature, both headed for infinitely largevalues at the critical point. Different materials show different Lorentz distributions at different temperatures. If we denote theintensity at a particular frequency and a particular temperature, T , by I ( ω , ∆ T ) , with ∆ T = T − T c , then for different materialsand different sets of ∆ T one has different I ( ω , ∆ T ) vs ω distributions. However, if one plots I ( ω , ∆ T ) / I ( , ∆ T ) vs. ω / ω where ω is the characteristic frequency for a given material, then the data for hundreds of individual distributions collapse on asingle distribution. This phenomenon is called dynamic scaling [35, 36, 65] and is displayed in Figure 4.This phenomenon of universal data collapse is ubiquitous: It appears in various forms in network dynamics [66], neuronalnetwork [67], brain [68, 69], supercritical fluids [70], amorphous solids [71], Bose–Einstein condensate [72], jammingtransition [73], granular media [74,75], financial markets [76], metallic liquids [77], etc. Thus, in some sense, it is not surprisingthat this intriguing phenomenon be present in the transmission of infectious diseases as well. Appendix: Comparisons between different models
In this section we compare some models that have been used commonly in epidemiology to quantify transmission dynamics ofdiseases including COVID-19 [53]. All the models that we describe here are variants of the logistic growth model. The logisticgrowth model (LGM) is defined as: dN ( t ) dt = β N ( t ) (cid:18) − N ( t ) κ (cid:19) . (10)We define ˜ t = t / t / and ˜ N ( t ) = N ( t ) / N max , and by setting ˜ β = β t / and ˜ κ = κ / N max , we get the rescaled equation: d ˜ N ( ˜ t ) d ˜ t = ˜ β ˜ N ( ˜ t ) (cid:18) − ˜ N ( ˜ t ) ˜ κ (cid:19) . (11)We see that β scales as the time dimension and κ scales as the number of cases. This makes the dimensional quantities, ˜ β and˜ κ , independent of any characteristic timescale or size of the system. ˜ κ = dN ( t ) dt = β N ( t ) π (cid:18) − N ( t ) κ (cid:19) . (12)Similar rescaling as above leads to the rescaled parameters ˜ β = β t / N π − and ˜ κ = κ / N max leading to the rescaled GLM being: d ˜ N ( ˜ t ) d ˜ t = ˜ β ˜ N ( ˜ t ) π (cid:18) − ˜ N ( ˜ t ) ˜ κ (cid:19) . (13)The generalized Richards model (GRM) is given by: dN ( t ) dt = β N ( t ) π (cid:20) − (cid:18) N ( t ) κ α (cid:19)(cid:21) . (14) ESY 20-234, HU-EP-20/42 . . . . . . ˜ t . . . . . . ˜ N ( ˜ t ) Japan [ N max : N max : N max : N max : N max : x : 0.0027, β : 5.98, κ : 0.87, α : 0.12)logistic growth model( x : 0.0035, β : 5.68, κ : 0.96)generalized logistic model( x : 0.0039, β : 5.72, κ : 0.96, π : 1.01)generalized Richards model( x : 0.0037, β : 5.59, κ : 0.96, π : 1.0, α : 1.02) . . . . . . ˜ t . . . . . . . . d ˜ N ( ˜ t ) / d ˜ t Japan [ t / :
47 days]New Zealand [ t / :
25 days]Ireland [ t / :
39 days]Australia [ t / :
30 days]Slovakia [ t / :
45 days]Japan [ t / :
47 days]New Zealand [ t / :
25 days]Ireland [ t / :
39 days]Australia [ t / :
30 days]Slovakia [ t / :
45 days]
Type I transmission . . . . . . ˜ t . . . . . . ˜ N ( ˜ t ) UK [ N max : N max : N max : N max : N max : N max : N max : x : 0.0027, β : 6.77, κ : 0.52, α : 0.92)logistic growth model( x : 0.0147, β : 4.43, κ : 0.91)generalized logistic model( x : 0.001, β : 3.6, κ : 0.92, π : 0.79)generalized Richards model( x : 0.001, β : 6.0, κ : 0.93, π : 0.89, α : 0.55) . . . . . . ˜ t . . . . . . . . d ˜ N ( ˜ t ) / d ˜ t UK [ t / :
57 days]Germany [ t / :
38 days]Italy [ t / :
46 days]South Korea [ t / :
18 days]Qatar [ t / :
80 days]New York [ t / :
42 days]Bayern [ t / :
37 days]UK [ t / :
57 days]Germany [ t / :
38 days]Italy [ t / :
46 days]South Korea [ t / :
18 days]Qatar [ t / :
80 days]New York [ t / :
42 days]Bayern [ t / :
37 days]
Type II transmission
Figure 5.
A comparison of the variants of the logistic model.
The panels on the left show ˜ N ( ˜ t ) vs. ˜ t distribution for Type Iand Type II transmission dynamics; and the right panels shows the d ˜ N ( ˜ t ) / d ˜ t vs. ˜ t plot for the same. We fit four models to thedata – the logistic growth model, the generalized logistic model, the generalized Richards model and the Blue Sky model. Whilethe fit to data for all the models are the same for Type I transmission where all the models go to the limit of being a logisticgrowth model, for Type II transmission the Blue Sky model can explain the data better than the other models. For π =
1, it reduces to the standard Richards model [78]. We see that the scaling is similar to the GLM with ˜ β = β t / N π − and ˜ κ = κ / N max ; the rescaled GRM is: d ˜ N ( ˜ t ) d ˜ t = ˜ β ˜ N ( ˜ t ) π (cid:34) − (cid:18) ˜ N ( ˜ t ) ˜ κ (cid:19) α (cid:35) . (15)In Figure 5 we show a comparison between all the models we list here and the Blue Sky model (BSM). The two upperpanels are for transmission of Type I. The LGM fits this data quite well and all the other models acquire the parameter valuesthat take them to the limit of being an LGM. The two lower panels are for transmission of Type II. Here we clearly see that theBSM fits the data much better that the other models. Especially for the tail of distribution past the peak of the d ˜ N ( ˜ t ) / d ˜ t vs. ˜ t plot where ˜ t > ˜ t peak , the BSM model allows for larger d ˜ N ( ˜ t ) / d ˜ t compared to the other models hence being able to replicatetransmission of Type II where the spread of the disease lingers on. Appendix: The deep neural network architechture
In our model-agnostic approach we used a deep neural network (DNN) [79, 80] to fit the data. A DNN is a universal functiongenerator. The reason we chose a DNN over other machine learning frameworks (i.e. decisions trees, random forests, SVMetc.) is the ease with which it can be use for a regression to noisy data to extract the underlying function. We will describe inbrief the choice of the network architechture.We started with a small DNN with 1 input node for ˜ t , 2 hidden layers, each with 4 nodes and 1 output layer for ˜ N ( ˜ t ) . Alllayers were fully connected as shown in Figure 6. The activation function used was the sigmoid function. The solution to theLM is a sigmoid function and hence the transmission of Type 1 can be fairly explained with a just one sigmoid function thatconnects the input ˜ t to the output ˜ N ( ˜ t ) . This is not possible though for transmission of Type II which shows clear deviationfrom the LM. However, in the model agnostic approach we did not want to introduce any model biases through assumptionsof functional form. Hence we used a larger (and deeper) neural network with 33 parameters to begin with. We scaled up the ESY 20-234, HU-EP-20/42 ̃"
Activation function: Sigmoid
Figure 6.
A representative architechture of the DNN used for the model-agnostic analysis to fit ˜ N ( ˜ t ) as a function of ˜ t.For the final analysis we used a similar architechture with 3 layers each with 16 nodes to keep the analysis as model-agnostic aspossible. To regulate the training of the DNN we used an early callback procedure which stops the training when the validationloss stops decreasing thus avoiding training biases. network to 3 hidden layers each with 16 nodes to check for the stability of our approach. The regression with the DNN with 2hidden layers with 4 nodes each gave almost the same fit as the one with 3 hidden layers with 16 nodes each having a total of593 parameters. To keep our method as general as possible and not introduce model biases from the activation function wedecided to use the larger DNN with 3 hidden layers each having 16 nodes.In addition, we tested the DNN with other activation function like tanh and relu . For tanh a somewhat wider and deepernetwork was needed to fit to the data. With the relu activation the network needed to be very large. Hence, we used the sigmoid activation function for the final DNN architecture.Even though the DNN we used was not so large we used an early callback algorithm to stop the training of the networkwhen the validation loss stopped decreasing to avoid overtraining and chose the epoch where the validation loss was the least.The loss function that we used was a simple mean square error function which measures the L distance of the fit from the data.The training time was a few seconds with the early stopping coming into effect at about 100-150 epochs. The trained DNN wasused for making prediction for the ongoing phases References
1. Li, Q. et al.
Early Transmission Dynamics in Wuhan, China, of Novel Coronavirus–Infected Pneumonia.
New Engl. J.Medicine , 1199–1207, DOI: 10.1056/NEJMoa2001316 (2020). PMID: 31995857.2. Dehning, J. et al.
Inferring change points in the spread of COVID-19 reveals the effectiveness of interventions.
Science , DOI: 10.1126/science.abb9789 (2020).3. Wilder-Smith, A. & Freedman, D. O. Isolation, quarantine, social distancing and community containment: pivotalrole for old-style public health measures in the novel coronavirus (2019-nCoV) outbreak.
J. Travel. Medicine , DOI:10.1093/jtm/taaa020 (2020).4. Sohrabi, C. et al. World Health Organization declares global emergency: A review of the 2019 novel coronavirus(COVID-19).
Int. J. Surg. , 71 – 76, DOI: 10.1016/j.ijsu.2020.02.034 (2020).5. Zhang, S. et al. Covid-19 containment: China provides important lessons for global response.
Front. Medicine , 215–219,DOI: 10.1007/s11684-020-0766-9 (2020).6. Fisher, D. & Wilder-Smith, A. The global community needs to swiftly ramp up the response to contain COVID-19. TheLancet , 1109–1110, DOI: 10.1016/S0140-6736(20)30679-6 (2020).7. Fraser, C., Riley, S., Anderson, R. M. & Ferguson, N. M. Factors that make an infectious disease outbreak controllable.
Proc. Natl. Acad. Sci. , 6146–6151, DOI: 10.1073/pnas.0307506101 (2004).8. Flaxman, S. et al.
Estimating the effects of non-pharmaceutical interventions on COVID-19 in Europe.
Nature ,257–261, DOI: 10.1038/s41586-020-2405-7 (2020).
ESY 20-234, HU-EP-20/429. Perra, N. Non-pharmaceutical interventions during the COVID-19 pandemic: a rapid review. arXiv e-prints arXiv:2012.15230 (2020). arXiv:2012.15230.10. Kühn, M. J. et al.
Assessment of effective mitigation and prediction of the spread of SARS-CoV-2 in Germany usingdemographic information and spatial resolution. medRxiv
DOI: 10.1101/2020.12.18.20248509 (2020).11. Zhou, Y. et al.
Effects of human mobility restrictions on the spread of COVID-19 in Shenzhen, China: a modelling studyusing mobile phone data.
The Lancet Digit. Heal. , e417–e424, DOI: 10.1016/S2589-7500(20)30165-5 (2020).12. Brett, T. S. & Rohani, P. Transmission dynamics reveal the impracticality of COVID-19 herd immunity strategies. Proc.Natl. Acad. Sci. , 25897–25903, DOI: 10.1073/pnas.2008087117 (2020).13. Gatto, M. et al.
Spread and dynamics of the COVID-19 epidemic in Italy: Effects of emergency containment measures.
Proc. Natl. Acad. Sci. , 10484–10491, DOI: 10.1073/pnas.2004978117 (2020).14. Bertozzi, A. L., Franco, E., Mohler, G., Short, M. B. & Sledge, D. The challenges of modeling and forecasting the spreadof COVID-19.
Proc. Natl. Acad. Sci. , 16732–16738, DOI: 10.1073/pnas.2006520117 (2020).15. López, L. & Rodó, X. The end of social confinement and COVID-19 re-emergence risk.
Nat. Hum. Behav. , 746–755,DOI: 10.1038/s41562-020-0908-8 (2020).16. Shayak, B. & Sharma, M. M. A New Approach to the Dynamic Modeling of an Infectious Disease. medRxiv DOI:10.1101/2020.10.30.20223305 (2020).17. Dellâ Anna, L. Solvable delay model for epidemic spreading: the case of Covid-19 in Italy.
Sci. Reports , 15763, DOI:10.1038/s41598-020-72529-y (2020).18. Vyasarayani, C. & Chatterjee, A. New approximations, and policy implications, from a delayed dynamic model of a fastpandemic. Phys. D: Nonlinear Phenom. , 132701, DOI: 10.1016/j.physd.2020.132701 (2020).19. Young, L.-S., Ruschel, S., Yanchuk, S. & Pereira, T. Consequences of delays and imperfect implementation of isolation inepidemic control.
Sci. Reports , 3505, DOI: 10.1038/s41598-019-39714-0 (2019).20. Kreck, M. & Scholz, E. Studying the course of Covid-19 by a recursive delay approach. medRxiv DOI:10.1101/2021.01.18.21250012 (2021).21. Cacciapaglia, G., Cot, C. & Sannino, F. Second wave COVID-19 pandemics in Europe: a temporal playbook.
Sci. Reports , 15514, DOI: 10.1038/s41598-020-72611-5 (2020).22. Wong, F. & Collins, J. J. Evidence that coronavirus superspreading is fat-tailed. Proc. Natl. Acad. Sci. , 29416–29418,DOI: 10.1073/pnas.2018490117 (2020).23. Chang, S. L., Harding, N., Zachreson, C., Cliff, O. M. & Prokopenko, M. Modelling transmission and control of theCOVID-19 pandemic in Australia.
Nat. Commun. , 5710, DOI: 10.1038/s41467-020-19393-6 (2020).24. Maziarz, M. & Zach, M. Agent-based modelling for SARS-CoV-2 epidemic prediction and intervention assessment: Amethodological appraisal. J. Eval. Clin. Pract. , 1352–1360, DOI: 10.1111/jep.13459 (2020).25. Tatapudi, H., Das, R. & Das, T. K. Impact assessment of full and partial stay-at-home orders, face mask usage, andcontact tracing: An agent-based simulation study of COVID-19 for an urban region. Glob. Epidemiol. , 100036, DOI:10.1016/j.gloepi.2020.100036 (2020).26. Rockett, R. J. et al. Revealing COVID-19 transmission in Australia by SARS-CoV-2 genome sequencing and agent-basedmodeling.
Nat. Medicine , 1398–1404, DOI: 10.1038/s41591-020-1000-7 (2020).27. Aleta, A. et al. Modelling the impact of testing, contact tracing and household quarantine on second waves of COVID-19.
Nat. Hum. Behav. , 964–971, DOI: 10.1038/s41562-020-0931-9 (2020).28. Hinch, R. et al. OpenABM-Covid19 - an agent-based model for non-pharmaceutical interventions against COVID-19including contact tracing. medRxiv
DOI: 10.1101/2020.09.16.20195925 (2020).29. Thurner, S., Klimek, P. & Hanel, R. A network-based explanation of why most COVID-19 infection curves are linear.
Proc. Natl. Acad. Sci. , 22684–22689, DOI: 10.1073/pnas.2010398117 (2020).30. Thomas, L. J. et al.
Spatial heterogeneity can lead to substantial local variations in COVID-19 timing and severity.
Proc.Natl. Acad. Sci. , 24180–24187, DOI: 10.1073/pnas.2011656117 (2020).31. Merow, C. & Urban, M. C. Seasonality and uncertainty in global COVID-19 growth rates.
Proc. Natl. Acad. Sci. ,27456–27464, DOI: 10.1073/pnas.2008590117 (2020).
ESY 20-234, HU-EP-20/4232. Perkins, T. A. et al.
Estimating unobserved SARS-CoV-2 infections in the United States.
Proc. Natl. Acad. Sci. ,22597–22602, DOI: 10.1073/pnas.2005476117 (2020).33. Chatterjee, S. et al.
Evolution of COVID-19 pandemic: Power-law growth and saturation.
J. Indian Stat. Assoc. , 1–3(2020).34. Siegenfeld, A. F., Taleb, N. N. & Bar-Yam, Y. Opinion: What models can and cannot tell us about COVID-19. Proc. Natl.Acad. Sci. , 16092–16095, DOI: 10.1073/pnas.2011542117 (2020).35. Hohenberg, P. C. & Halperin, B. I. Theory of dynamic critical phenomena.
Rev. Mod. Phys. , 435–479, DOI:10.1103/RevModPhys.49.435 (1977).36. Stanley, H. E. Scaling, universality, and renormalization: Three pillars of modern critical phenomena. Rev. Mod. Phys. ,S358–S366, DOI: 10.1103/RevModPhys.71.S358 (1999).37. Abraham, R. & Shaw, C. D. Dynamics: The Geometry of Behavior. Part 4: Bifurcation Behavior (Ariel Press, Santa Cruz,CA, 1988).38. Rao, S. D., Tylavsky, D. J. & Feng, Y. Estimating the saddle-node bifurcation point of static power systems using theholomorphic embedding method.
Int. J. Electr. Power & Energy Syst. , 1 – 12, DOI: 10.1016/j.ijepes.2016.04.045(2017).39. Achleitner, F. & Szmolyan, P. Saddle–node bifurcation of viscous profiles. Phys. D: Nonlinear Phenom. , 1703 – 1717,DOI: 10.1016/j.physd.2012.06.008 (2012).40. Cessac, B. & Samuelides, M. From neuron to neural networks dynamics.
The Eur. Phys. J. Special Top. , 7–88, DOI:10.1140/epjst/e2007-00058-2 (2007).41. Joyeux, M., Farantos, S. C. & Schinke, R. Highly Excited Motion in Molecules: Saddle-Node Bifurcations and TheirFingerprints in Vibrational Spectra.
The J. Phys. Chem. A , 5407–5421, DOI: 10.1021/jp0131065 (2002).42. Cassak, P. A., Shay, M. A. & Drake, J. F. A saddle-node bifurcation model of magnetic reconnection onset.
Phys. Plasmas , 062105, DOI: 10.1063/1.3435269 (2010).43. Descalzi, O., Argentina, M. & Tirapegui, E. Saddle-node bifurcation: Appearance mechanism of pulses in the subcriticalcomplex Ginzburg-Landau equation. Phys. Rev. E , 015601, DOI: 10.1103/PhysRevE.67.015601 (2003).44. Mitze, T., Kosfeld, R., Rode, J. & Wälde, K. Face masks considerably reduce COVID-19 cases in Germany. Proc. Natl.Acad. Sci. , 32293–32301, DOI: 10.1073/pnas.2015954117 (2020).45. Lega, J. & Brown, H. E. Data-driven outbreak forecasting with a simple nonlinear growth model.
Epidemics , 19 – 26,DOI: 10.1016/j.epidem.2016.10.002 (2016).46. Buczak, A. L., Koshute, P. T., Babin, S. M., Feighner, B. H. & Lewis, S. H. A data-driven epidemiological predictionmethod for dengue outbreaks using local and remote sensing data. BMC Med. Informatics Decis. Mak. , 124, DOI:10.1186/1472-6947-12-124 (2012).47. Brauer, F. Compartmental Models in Epidemiology , 19–79 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008).48. Adam, D. Special report: The simulations driving the world’s response to COVID-19.
Nature , 316–318, DOI:10.1038/d41586-020-01003-6 (2020).49. Allen, L. J. S.
An Introduction to Stochastic Epidemic Models , 81–130 (Springer Berlin Heidelberg, Berlin, Heidelberg,2008).50. Verhulst, P.-F. Notice sur la loi que la population poursuit dans son accroissement.
Corresp. Mathématique et Physique ,113–121 (1838).51. Chen, D.-G., Chen, X. & Chen, J. K. Reconstructing and forecasting the COVID-19 epidemic in the United States using a5-parameter logistic growth model. Glob. Heal. Res. Policy , 25, DOI: 10.1186/s41256-020-00152-5 (2020).52. Shen, C. Y. Logistic growth modelling of COVID-19 proliferation in China and its international implications. Int. J. Infect.Dis. , 582 – 589, DOI: 10.1016/j.ijid.2020.04.085 (2020).53. Wu, K., Darcet, D., Wang, Q. & Sornette, D. Generalized logistic growth modeling of the COVID-19 outbreak: com-paring the dynamics in the 29 provinces in China and in the rest of the world. Nonlinear Dyn. , 1561–1581, DOI:10.1007/s11071-020-05862-6 (2020).54. Aviv-Sharon, E. & Aharoni, A. Generalized logistic growth modeling of the COVID-19 pandemic in Asia.
Infect. Dis.Model. , 502 – 509, DOI: 10.1016/j.idm.2020.07.003 (2020). ESY 20-234, HU-EP-20/4255. Jain, M., Bhati, P. K., Kataria, P. & Kumar, R. Modelling Logistic Growth Model for COVID-19 Pandemic in India. In , 784–789, DOI: 10.1109/IC-CES48766.2020.9138049 (2020).56. Malavika, B. et al.
Forecasting COVID-19 epidemic in India and high incidence states using SIR and logistic growthmodels.
Clin. Epidemiol. Glob. Heal. , 26 – 33, DOI: 10.1016/j.cegh.2020.06.006 (2021).57. Zou, Y. et al. Outbreak analysis with a logistic growth model shows COVID-19 suppression dynamics in China.
PLoSONE , e0235247, DOI: 10.1371/journal.pone.0235247 (2020).58. Pelinovsky, E., Kurkin, A., Kurkina, O., Kokoulina, M. & Epifanova, A. Logistic equation and COVID-19. Chaos, Solitons& Fractals , 110241, DOI: 10.1016/j.chaos.2020.110241 (2020).59. Wang, P., Zheng, X., Li, J. & Zhu, B. Prediction of epidemic trends in COVID-19 with logistic model and machine learningtechnics.
Chaos, Solitons & Fractals , 110058, DOI: 10.1016/j.chaos.2020.110058 (2020).60. Dong, E., Du, H. & Gardner, L. An interactive web-based dashboard to track COVID-19 in real time.
The Lancet Infect.Dis. , 533–534, DOI: 10.1016/S1473-3099(20)30120-1 (2020).61. Virtanen, P. et al. SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python.
Nat. Methods , 261–272,DOI: 10.1038/s41592-019-0686-2 (2020).62. Abadi, M. et al. TensorFlow: Large-Scale Machine Learning on Heterogeneous Systems (2015). Software available fromtensorflow.org.63. Guggenheim, E. A. The Principle of Corresponding States.
The J. Chem. Phys. , 253–261, DOI: 10.1063/1.1724033(1945).64. Swinney, H. L. & Henry, D. L. Dynamics of Fluids near the Critical Point: Decay Rate of Order-Parameter Fluctuations. Phys. Rev. A , 2586–2617, DOI: 10.1103/PhysRevA.8.2586 (1973).65. LÜbeck, S. Universal scaling behavior of non-equilibrium phase transitions. Int. J. Mod. Phys. B , 3977–4118, DOI:10.1142/S0217979204027748 (2004).66. Barzel, B. & Barabási, A.-L. Universality in network dynamics. Nat. Phys. , 673–681, DOI: 10.1038/nphys2741 (2013).67. Friedman, N. et al. Universal Critical Dynamics in High Resolution Neuronal Avalanche Data.
Phys. Rev. Lett. ,208102, DOI: 10.1103/PhysRevLett.108.208102 (2012).68. Bhattacharya, J. & Petsche, H. Universality in the Brain While Listening to Music.
Proceedings: Biol. Sci. et al.
True scale-free networks hidden by finite size effects.
Proc. Natl. Acad. Sci. , DOI:10.1073/pnas.2013825118 (2021).70. Ha, M. Y., Yoon, T. J., Tlusty, T., Jho, Y. & Lee, W. B. Universality, Scaling, and Collapse in Supercritical Fluids.
The J.Phys. Chem. Lett. , 451–455, DOI: 10.1021/acs.jpclett.9b03360 (2020).71. Ulrich, S., Upadhyaya, N., Opheusden, B. v. & Vitelli, V. Shear shocks in fragile networks. Proc. Natl. Acad. Sci. ,20929–20934, DOI: 10.1073/pnas.1314468110 (2013).72. Klinder, J., Keßler, H., Wolke, M., Mathey, L. & Hemmerich, A. Dynamical phase transition in the open Dicke model.
Proc. Natl. Acad. Sci. , 3290–3295, DOI: 10.1073/pnas.1417132112 (2015).73. Goodrich, C. P., Liu, A. J. & Sethna, J. P. Scaling ansatz for the jamming transition.
Proc. Natl. Acad. Sci. , 9745–9750,DOI: 10.1073/pnas.1601858113 (2016).74. Manna, S. S. Two-state model of self-organized criticality.
J. Phys. A: Math. Gen. , L363–L369, DOI: 10.1088/0305-4470/24/7/009 (1991).75. Denisov, D. V., Lörincz, K. A., Uhl, J. T., Dahmen, K. A. & Schall, P. Universality of slip avalanches in flowing granularmatter. Nat. Commun. , 10641, DOI: 10.1038/ncomms10641 (2016).76. Denys, M., Gubiec, T., Kutner, R., Jagielski, M. & Stanley, H. E. Universality of market superstatistics. Phys. Rev. E ,042305, DOI: 10.1103/PhysRevE.94.042305 (2016).77. Blodgett, M. E., Egami, T., Nussinov, Z. & Kelton, K. F. Proposal for universality in the viscosity of metallic liquids. Sci.Reports , 13837, DOI: 10.1038/srep13837 (2015).78. Richards, F. J. A Flexible Growth Function for Empirical Use. J. Exp. Bot. , 290–301, DOI: 10.1093/jxb/10.2.290(1959). ESY 20-234, HU-EP-20/4279. Bengio, Y., Courville, A. & Vincent, P. Representation Learning: A Review and New Perspectives.
IEEE Transactions onPattern Analysis Mach. Intell. , 1798–1828, DOI: 10.1109/TPAMI.2013.50 (2013).80. Schmidhuber, J. Deep learning in neural networks: An overview. Neural Networks , 85 – 117, DOI:10.1016/j.neunet.2014.09.003 (2015)., 85 – 117, DOI:10.1016/j.neunet.2014.09.003 (2015).