A fractal viewpoint to COVID-19 infection
Oscar Sotolongo-Costa, José Weberszpil, Oscar Sotolongo-Grau
aa r X i v : . [ phy s i c s . s o c - ph ] J u l A FRACTAL VIEWPOINT TO COVID-19 INFECTION
Oscar Sotolongo-Costa, ∗ José Weberszpil, † and Oscar Sotolongo-Grau ‡ Cátedra "Henri Poincaré" de sistemas complejos.Universidad de La Habana, Habana 10400 Cuba. Universidade Federal Rural do Rio de Janeiro,UFRRJ-IM/DTL; Av. Governador Roberto Silveiras/n- Nova Iguaçú, Rio de Janeiro, Brasil 695014. Alzheimer Research Center and Memory Clinic,Fundaci´0 ACE, Institut Catalá de Neurociéncies Aplicades;08029 Barcelona, Spain. (Dated: July 16, 2020)
Abstract
One of the central tools to control the COVID-19 pandemics is the knowledge of its spreadingdynamics. Here we develop a fractal model capable of describe this dynamics, in term of dailynew cases, and provide quantitative criteria for some predictions. We propose a fractal dynamicalmodel using conformed derivative and fractal time scale. A Burr-XII shaped solution of the fractal-like equation is obtained. The model is tested using data from several countries, showing that asingle function is able to describe very different shapes of the outbreak. The diverse behavior ofthe outbreak on those countries is presented and discussed. Moreover, a criterion to determine theexistence of the pandemic peak and a expression to find the time to reach herd immunity are alsoobtained.
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Covid-19; Nonlinear Fractal-Like Kinetics; Analogy Relaxation Model; NonlinearEquations; Herd Immunity; Deformed Derivatives. ∗ [email protected] † [email protected] ‡ [email protected] . INTRODUCTION The worldwide pandemic provoked by the SARS-CoV-2 coronavirus outbreak have at-tracted the attention of the scientific community due to, among other features, its fastspread. Its strong contamination capacity has created a fast growing population of peopleenduring COVID-19, its related disease, and a non small peak of mortality. The temporalevolution of contagion over different countries and worldwide brings up a common dynamiccharacteristic, in particular, its fast rise to reach a maximum followed by a slow decrease(incidentally, very similar to other epidemic processes) suggesting some kind of relaxationprocess, which we try to deal with, since relaxation is, essentially, a process where the pa-rameters characterizing a system are altered, followed by a tendency to equilibrium values.In Physics, clear examples are, among others, dielectric or mechanical relaxation. In otherfields (psychology, economy, etc.) there are also phenomena in which an analogy with "com-mon" relaxation can be established. In relaxation, temporal behavior of parameters is ofmedular methodological interest. That is why pandemics can be conceived as one in whichthis behavior is also present. For this reason, we are interested, despite the existence of sta-tistical or dynamical systems method, in the introduction of a phenomenological equationcontaining parameters that reflect the system´s behavior, from which its dynamics emerges.We are interested in studying the daily presented new cases, not the current cases by day.This must be noted to avoid confusion in the interpretation, i.e. we study not the cumulativenumber of infected patients reported in databases, but its derivative. This relaxation processin this case is, for us, an scenario that, by analogy, will serve to model the dynamics of thepandemics. This is not an ordinary process. Due to the concurrence of many factors thatmake very complex its study, its description must turn out to non classical description. So,we will consider that the dynamics of this pandemic is described by a "fractal" or internaltime [1]. The network formed by the people in its daily activity forms a complex field oflinks very difficult, if not impossible, to describe. However, we can take a simplified modelwhere all the nodes belong to a small world network, but the time of transmission from onenode to other differs for each link. So, in order to study this process let us assume thatspread occurs in "fractal time" or internal time [1, 2]. This is not a new tool in physics.In refs. [3–5] this concept has been successfully introduced and here, we keep in mind thepossibility of a fractal-like kinetics [6], but generalizing as a nonlinear kinetic process. Here2e will follow to what we refer as a "relaxation-like" approach, to model the dynamics ofthe pandemic and that justify the fractal time. By analogy with relaxation, an anomalousrelaxation, we build up a simple nonlinear equation with fractal-time. We also regain theanalytical results using a deformed derivative approach, using conformable derivative (CD)[7]. In Ref. [8] one of the authors (J.W.) have shown intimate relation of this derivativewith complex systems and nonadditive statistical mechanics. This was done without resortto details of any kind of specific entropy definition.Our article is outlined as follows: In Section 2, we present the fractal model formulatedin terms of conformable derivatives, to develop the relevant expressions to adjust data ofCOVID-19. In Section 3, we show the results and figures referring to the data fitting alongwith discussions. In section 4, we finally cast our general conclusions and possible paths forfurther investigations.
2. FRACTAL MODEL
Let us denote by F ( t ) the number of contagions up to time t .The CD is defined as [7] D αx f ( x ) = lim ǫ → f ( x + ǫx − α ) − f ( x ) ǫ . (1)Note that the deformation is placed in the independent variable.For differentiable functions, the CD can be written as D αx f = x − α dfdx . (2)An important point to be noticed here is that the deformations affect different functionalspaces, depending on the problem under consideration. For the conformable derivative [8–12], the deformations are put in the independent variable, which can be a space coordinate,in the case of, e.g, mass position dependent problems, or even time or spacetime variables, fortemporal dependent parameter or relativistic problems. Since we are dealing with a complexsystem, a search for a mathematical approach that could take into account some fractalityor hidden variables seems to be adequate. This idea is also based in the fact that we donot have full information about the system under study. In this case, deformed derivativeswith fractal time seems to be a good option to deal with this kind of system. Deformed3erivatives, in the context of generalized statistical mechanics are present and connected[8]. There, the authors have also shown that the q − def ormed derivative has also a dualderivative and a q − exponential related function [13]. Here, in the case under study, thedeformation is considered for the solutions-space or dependent variable, that is, the number F ( t ) of contagions up to time t . One should also consider that justification for the useof deformed derivatives finds its physical basis on the mapping into the fractal continuum[8, 14–16]. That is, one considers a mapping from a fractal coarse grained (fractal porous)space, which is essentially discontinuous in the embedding Euclidean space, to a continuousone [9]. In our case the fractality lies in the temporal variable. Then the CD is with respectto time.A nonlinear relaxation model can be proposed here, again based on a generalization ofBrouers-Sotolongo fractal kinetic model (BSf) [3, 4, 17], but here represented by a nonlinearequation written in terms of CD: D αt F = 1 τ α F q , (3)where τ is our "relaxation time" and q and α here are real parameters. We do not imposeany limit for the parameters. Equation (3) has as a well known solution a function with theshape of Burr XII [18], with : F = F [1 + (1 − q ) ( t α − t α ) τ α αF − q ] − q . (4)The density (in a similar form of a PDF, but here it is not a PDF) is, then: f ( t ) = F q τ α [ C + (1 − q ) t α τ α αF − q ] q − q t α − , (5)where C = 1 − (1 − q ) t α τ α αF − q , which can be expressed as: f ( t ) = A ′ [ C + B ′ t α ] − b t a − (6)where the parameter are A ′ = F q τ α , B ′ = (1 − q ) τ α αF − q , b = qq − , a = α. Or, in a simpler form for data adjustment purposes f ( t ) = A [1 + Bt α ] − b t a − , (7)with A = A ′ C b , B = B ′ C . A, B, C, b , and a as parameters to simplify the fitting, the true adjustment constants are, clearly, q, τ and α. Note that we do not impose any restrictive values to the parameters.There is no need to demand that the solution always converge. The equation to obtainBurr XII has to impose restrictions but this is not the case. In Burr XII the functionwas used as a probability distribution. But here the function describes a dynamic, whichcan be explosive, as will be shown for the curves of Brazil and Mexico. Therefore, if weconsider infinite population, a peak will never be reached unless the circumstances change(treatments, vaccines, isolation, etc.). Our model does not impose finiteness of the solution.The possibility for a decay of the pandemic in a given region in this model requires thefulfillment of the condition a (1 − b ) − < , (8)what expresses the property that lim t →∞ f ( t ) = 0 , (9)what means that the function has a local maximum. If this condition is not accomplished,the pandemic does not have a peak and, therefore, the number of cases increases forever inthis model.In this case there is, apart from the change of propagation and development conditions,the possibility for a given country that does not satisfies condition (8), to reach "herdimmunity", i.e., when the number of contagions has reached about 60% of population, inwhich case we may calculate the time to reach such state using (4), assuming t = 0 : T hi = [(0 . P ) / (1 − b ) − /B ] /a . (10)We will work with what we will call T ahead and that seems to make more sense andbring more information.
3. DATA FITTING
With eq. (7) let us fit the data of the epidemic worldwide. The data was extracted fromJohns Hopkins University [21] and the website [22] to process the data for several countries.5e covered the infected cases taken at Jan 22 as day 1, up to June 13. The behavior ofnew infected cases by day is shown in figure 1. The fitting was made with gnuplot 5.2. Asit seems, the pandemic shows some sort of "plateau", so the present measures of preventionare not able to eliminate the infection propagation in a short term, but it can be seen thatcondition (8) is weakly fulfilled.
100 1000 10000 100000 20 40 60 80 100 120 140 d a il y i n f ec t e d t (days) Figure 1. Worldwide infections from Jan, 22 to June 13 and fitting with eq. (7). Thebehavior fits well with parameters in Table I. Condition (8) is satisfied.
In the particular case of Mexico the fitting is shown in figure 2. In this case condition (8)is not fulfilled. In terms of our model this means that the peak is not predictable within thepresent dynamics. Something similar occurs with Brazil, as shown in figure 3. The data forBrazil neither fulfill the condition (8). In this case there is neither the prevision of a peakand we can say that the data for Mexico and Brazil reveals a dynamics where the peak seemsto be quite far if it exists. But there are some illustrative cases where the peak is reached.Progression of the outbreak in Cuba and Iceland are shown in Figure 4 and 5 respectively.Condition (8) is satisfied for both countries and we can see that the curve of infection ratedescends at a good speed after past the peak. Now let us take a look at United States data,shown in Figure 6. The USA outbreak is characterized by a very fast growth until the peak6nd, then, very slow decay of the infection rate is evident. As discussed above, the outbreakwill be controlled for almost infinite time in this dynamics. There is also some intermediatecases as Spain and Italy, shown in Figures 7 and 8. In this case the data exhibits the samebehavior as in USA, a fast initial growth and a very slow decay after the peak. However, theoutbreak is controlled in a finite amount of time. In Table I we present the relevant fittingparameters, including herd immunity time, T hi and T , the time to reach a rate of 1000infections daily. This, for countries that have not reached the epidemic peak, Mexico andBrazil. We also include the population, P ; of each country. d a il y i n f ec t e d t (days) Figure 2. Daily infections in Mexico and fitting with eq. (7) for parameters in Table I. T hi = 778 days. Condition (8) is not satisfied. d a il y i n f ec t e d t (days) Figure 3. Evolution of daily cases in Brazil and fitting with eq. (7) for parameters inTable I. T hi = 298 days. Condition (8) is not satisfied. d a il y i n f ec t e d t (days) Figure 4. Daily infections in Cuba. The theoretical curve fits with data though with apoor correlation due to the dispersion. See fitting parameters in Table I. Condition(8) issatisfied. d a il y i n f ec t e d t (days) Figure 5. Daily infections in Iceland, where the pandemic seems to have ceased. Hereagain, in spite of the relatively small correlation coefficient, the behavior of the pandemicin this country looks well described by eq. (7). See fitting parameters in Table I. Condition(8) is satisfied. d a il y i n f ec t e d t (days) .Figure 6. Daily infections in USA, where the peak looks already surpassed. Here again, the behaviorof the pandemic in this country looks well described by eq. (7). See fitting parameters in Table I.Condition (8) is satisfied. d a il y i n f ec t e d t (days) Figure 7. Daily infections in Spain. the data shows a large dispersion but the curvedescribes well the behavior. See fitting parameters in Table I. Condition (8) is satisfied. d a il y i n f ec t e d t (days) Figure 8. Daily infections in Italy. See fitting parameters in Table I. Condition (8) issatisfied. ountry A B a b P T hi (days) T Brazil 0.0152828 0.0104434 4.31197 0.0671095 212559417 298 36Cuba 1.80E-05 3.30E-09 5.31906 1.40779 11326616 - -Iceland 6.08E-05 1.69E-09 5.09845 4.94326 341243 - -Italy 1.20E-07 1.85E-13 7.50956 1.32858 60461826 - 34Mexico 0.0541958 0.0104956 3.60005 0.0641971 128932753 778 56Spain 0.000170317 2.75E-10 6.31706 1.35476 46754778 - 19USA 1.09E-13 5.99E-20 11.5099 0.973087 331002651 - 34Worldwide 3.18E-06 4.65E-13 6.84834 0.816744 7786246434 - 29Table I. Relevant fitting parameters for each country.
As can be seen from fitting coefficients, the exponent b drives the behavior of infections inevery country. Those countries that manage well the disease expansion have b values widelarger than . Countries with b values close to one, as Italy and Spain, have managed thepandemics but poorly and at high costs. The recovery in both countries will be long. Thesame is valid for USA, that manage poorly the outbreak and its struggling with an evenlonger recovery to normal life. Even worst scenario is taken place in Mexico and Brazil, withvery low values of b . Those countries are experiencing a big outbreak where even can getherd immunity. This, however, implies very high values of infections and mortality for thenear future.But let us briefly comment about herd immunity. Those countries that have managed tostop the outbreak, even with relative high mortality as Spain and Italy, will not reach theherd immunity. As a matter of fact, This can not be calculated for those countries. Then,we can see countries like Brazil where, if the way of deal with the outbreak do not change,the herd immunity will be reached. Even when it seems desirable, the ability to reach theherd immunity brings with it a high payload. That is, for a country like Brazil the herdimmunity would charge more than 100 million of infected people. That is, much the sameas if a non small war devastates the country. There is an alike scenario in Mexico, but thedifference here is that the value for T hi is so high that SARS-CoV-2 could even turn intoa seasonal virus, at least for some years. We can expect around the same mortality butscattered over a few years. 13 special observation deserves USA, where T hi tends to infinity. Here we can expect acontinuous infection rate for a very long time. The outbreak is controlled but not enoughto eradicate the virus. Virus will not disappear in several years but maybe the healthcaresystem could manage it. The virus will get endemic, and immunity will never be reached.However the infections and mortality rate associated with it, can be, hypothetically, smallif compared with Mexico and Brazil. We can also compare the speed of the outbreak indifferent countries. As we already said in Table I we calculated T for some countries.However, it should be noticed that this time is not calculated from day 0, which is alwaysJanuary 22, but for the approximated day when the outbreak began in the correspondentcountry. By example, in Brazil there was no cases at January, 22 but the first cases weredetected around March, 10. So both, data fitting and T , were calculated from March,10.
4. CONCLUSIONS AND OUTLOOK FOR FURTHER INVESTIGATIONS
In this work, for the first time, we presented a model built using the method of analogy,in this case with a nonlinear relaxation-like behavior. With this, a good fitting with the ob-served behavior of the daily number of cases with time is obtained. The explicit expressionsobtained may be used as a tool to approximately forecast the development of the COVID-19pandemic in different countries and worldwide. In principle, this model can be used as a helpto elaborate or change actions. This model does not incorporate any particular property ofthis pandemic, so we think it could be used to study pandemics with different sources. Withthe collected data of the pandemics at early times, using this model, it can be predicted thepossibility of a peak, indefinite growth, time for herd immunity, etc.What seems to be clear from the COVID-19 data, the fitting and the values shown inthe Table I, is that SARS-CoV-2 is far from being controlled at world level. Even whensome countries appear to control the outbreak, the virus is still a menace for its healthsystem. Furthermore, in the nowadays interconnected world it is impossible for any countryto keep closed borders and pay attention to what happens only inside. All isolation measuresshould be halted at some time and we can expect new outbreaks in countries like Spain orItaly even after the current one could be controlled. The only way to control the spread ofSARS-CoV-2 seems to be the development of a vaccine that provides the so much desired14erd immunity. Indeed, the model made possible to make an approximate forecast of thetime to reach the herd immunity. This may be useful in the design of actions and policiesabout the pandemic. We have introduced the T , that gives information about the earlyinfection behavior in populous countries. A possible improvement of this model is the formalinclusion of a formulation including the dual conformable derivative [12, 13]. This will bepublished elsewhere. ACKNOWLEDGMENTS
We acknowledge Dr. Carlos Trallero -Giner for helpful comments and suggestions
CONFLICT OF INTEREST
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