Non-integer (or fractional) power model of a viral spreading: application to the COVID-19
Alain Oustaloup, François Levron, Stéphane Victor, Luc Dugard
aa r X i v : . [ q - b i o . P E ] F e b Non-integer (or fractional) power modelof a viral spreading: application to the COVID-19
Alain Oustaloup a , Franc¸ois Levron b , St´ephane Victor a, ∗ , Luc Dugard c a Univ. Bordeaux, CNRS, IMS UMR 5218, Bordeaux INP / ENSEIRB-MATMECA,351 Cours de la Lib´eration,33405 Talence CEDEX, France b Univ. Bordeaux, CNRS, IMB UMR 5251, Bordeaux INP / ENSEIRB-MATMECA,351 Cours de la Lib´eration,33405 Talence CEDEX, France c Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-Lab, 38000 Grenoble, France
Abstract
This paper proposes a very simple deterministic mathematical model, which, by using a power-law, is a non-integer power model (or fractional power model (FPM) ). Such a model, in non-integer power of time, namely t m up to constants, enables representingat each day, with a good precision, the totality of the contaminated individuals. Despite being enriched with knowledge throughan internal structure based on a geometric sequence “with variable ratio”, the model (in its non-integer representation) has onlythree parameters, among which the non-integer power, m , that determines on its own, according to its value, an aggravation or animprovement of the viral spreading. Its simplicity comes from the power-law, t m , which simply expresses the singular dynamics ofthe operator of non-integer di ff erentiation or integration, of high parametric compactness, that governs di ff usion phenomena and,as shown in this paper, the spreading phenomena by contamination. The proposed model is indeed validated with the o ffi cial dataof Ministry of Health on the COVID-19 spreading. Used in prediction, it well enables justifying the choice of a lockdown, withoutwhich the spreading would have highly worsened. The comparison of this model in t m with two known models having the samenumber of parameters, well shows that its representativity of the real data is better or more general. Finally, in a more fundamentalcontext and particularly in terms of complexity and simplicity, a self-filtering action enables showing the compatibility betweenthe internal complexity that the internal structure and its stochastic behavior present, and the global simplicity that the model in t m o ff ers in a deterministic manner: it is true that the non-integer power of a power-law is well a marker of complexity. Keywords:
COVID-19, viral spreading, modeling, prediction; fractional (or non-integer) power model (FPM), a FPM model as aconvexity or concavity model; power-law, non-integer (or fractional) di ff erentiation or integration, self-filtering internal structure.
1. Introduction
The most common approach for modeling an epidemic is abehavior model, originally proposed by Kermack and McKendrick(1927). The population is divided into several categories andmathematical rules dictate how many people move from onecategory to another.First, everyone is
Susceptible (S) to become contaminated.Then some people become
Infected (I) individuals who havebeen infected and are capable of infecting susceptible individ-uals. Finally, individuals, who have been infected enter the
Removed (R) compartment, through either recovery or death.These SIR models can be improved by including
Exposed (E)people but not yet contagious, leading to SEIR models. If post-recovery immunity is temporary, recovered people can go back ∗ Corresponding author
Email addresses: [email protected] (AlainOustaloup), [email protected] (Franc¸ois Levron), [email protected] (St´ephane Victor ), [email protected] (Luc Dugard) to S, thus leading to SEIRS models. The equations that deter-mine how people move from one category to the next dependon a wide variety of parameters drawn from biology, behav-ior, politics, economy, weather, and more. Some of the ex-tended SEIRS models can be found in Satsuma et al. (2004);Kwuimy et al. (2020); He et al. (2020); Ivorra et al. (2020); Dell’Anna(2020); Demongeot et al. (2020); Guan et al. (2020); Efimov and Ushirobira(2020). Other extensions of the SEIRS models have been car-ried out by using fractional (or non-integer) order derivatives(Xu et al. (2020); Lu et al. (2020); Arfan et al. (2020)), as thenon-integer operator is known to well model propagation phe-nomena such as biological systems, thermal di ff usion systems,electro-chemical ones (fuel cells or batteries), etc.Other kinds of spreading models exist in the literature toavoid sorting out people per categories, namely data-driven mo-dels. Black box models are used but the complexity and hugenumber of parameters render these models disconnected fromreality and beyond understanding. On one hand, these mod-els, as being more tied to data, can provide more accurate pre-dictions in the short-term, but on the other hand no guaranteeis proven for long-term predictions. Some data-driven mod- Preprint submitted to Elsevier March 1, 2021 ls use neural networks, deep learning, machine learning, YYGmodel , particle swarm optimization, etc. (see for exampleLiu et al. (2020); Huang et al. (2020); Zhao et al. (2020a); De-Leon and Pederiva(2020)).Infection, death, and hospitalization numbers a ff ect theirusefulness not only as inputs for the model but also as outputs.The true number of infections is hard to evaluate when not eve-ryone is tested. Deaths are countable, but they are the conse-quence of infections after some time delay. Therefore, otherkinds of models have emerged by proposing transmission, net-working and control measures to reduce the epidemic spreadingsuch as the agent-based models, the sliding mode control (asgovernmental control input) (Rohith and Devika (2020)), maskwearing e ff ects (Li et al. (2020)), human mobility control (Iacus et al.(2020)), lockdown e ff ect, etc. (Huang and Qi (2020); De Visscher(2020); Atangana (2020); Tuan et al. (2020); Chen et al. (2020)).Ferguson et al. (2005) have developed a stochastic modelfor the spread of infectious diseases, which also takes into ac-count the geographic spreading. Simple models such as SEIRSones are not su ffi ciently accurate to evaluate the spreading of anepidemic. There is a need for simple models that are su ffi cientlyaccurate for spreading predictions, so that quick and real-timedecisions can be made on how to respond to epidemics. TheBelgian Verhulst (1838) established a model of dynamical be-havior of a population, named logistic growth model. Manystudies use this model with three parameters for modeling thespreading of COVID-19 (Zhou et al. (2020); Kyurkchiev et al.(2020); Zhao et al. (2020b)). There are several possible gener-alizations of the Verhulst model such as proposed by Roosa et al.(2020) who give two well-known generalizations with four pa-rameters: the generalized logistical growth model (Viboud et al.(2016); Ganyani et al. (2018)) and the generalized Richards model(Richards (1959); Wang et al. (2012); Wu et al. (2020)). The specific contributions of the article turn on a very sim-ple deterministic mathematical model, in non-integer power oftime, namely t m up to constants, which enables representing, ateach day, the totality of the contaminated individuals (which,from now on, will be named contaminated), with an absolutelysatisfactory precision regarding the complexity of contamina-tion phenomena. The model is characterized by three parame-ters, the non-integer power, m , as well as two constants, one be-ing multiplicative of t m and the other simply being additive. Thenon-integer power, m , determines on its own a viral spreadingthat worsens or gets better according to whether m is greater orlesser than one: indeed, the spreading graphical representationpresents, for m greater than one, a convexity that expresses aspreading increasing more and more, therefore a viral aggrava-tion ; while for m lesser than one, the curve presents a concavitythat expresses a spreading increasing less and less, therefore aviral improvement.The simplicity of the model in t m (FPM) is due to the factthat t m simply expresses a characteristic response, namely the https: // covid19-projections.com developed by Youyang Gu step response, therefore the dynamics, of the non-integer dif-ferentiation or integration operator. Applicable in real-time bymeans of Oustaloup’s approximation (Oustaloup et al. (2000)),this operator is known for its parametric compactness and its ca-pacity to govern complex phenomena (Oustaloup (1995, 2014)),including the di ff usive phenomena and, as shown in this paper,the contamination spreading phenomena.In order not to limit this model to a simple behavior model,it is enriched by knowledge (of countable nature) through an internal structure , in accordance with the daily recording of thenew contaminated, and founded on a geometric sequence “withvariable ratio”, since the internal structure is none other thanthe sum of the k + k , this sum achieves or synthe-sizes point by point (in this case day after day) a non-integertime function representing the evolution of the new contami-nated number: this three parameter function (section 5) has thesame form (therefore the same degree, m ) as the model in t m ,even in its complete version that represents the evolution of thenumber of all the contaminated, new and former ones (section5.5).The internal structure, through which the model wins inknowledge, very simply enables formalizing the process of dailycensus of the new contaminated, and to represent day after daythe whole of the new contaminated. To that e ff ect, the internalstructure is conceived for introducing ratios, q k , between newcontaminated between two consecutive days k − k (sec-tions 3 and 4), then used without seeking to capture or representthe unpredictable, therefore the hazards of reality, in this casethe random fluctuations of these ratios, which occur from day today in reality given its complexity. Adopted for obvious reasonsof simplicity, such a strategy is all the more justified as the fluc-tuations in question, besides interpretable as an internal noise,have only a negligible e ff ect on the dynamics to be modelled,slower, as being expressed on several days. What follows is li-able to precise the content of these words, and to reinforce theidea of not systematically seeking to represent the whole com-plexity of reality, particularly all these random fluctuations.The internal structure of the model in t m presents a remark-able property of filtering, through a double e ff ect inherent to theform itself of the structure such as conditioned by the geometricsequence “with variable ratio”, namely the sum of the productsof the di ff erent ratios, q , . . . , q k (sections 3 and 4): • the first e ff ect of filtering finds its cause in the terms of thesequence, that is to say the products of the di ff erent ratios,products that can filter by themselves the fluctuations ofthese ratios to only present residual fluctuations; • the second e ff ect of filtering finds its cause in the sumof the sequence terms, namely the non weighted sum ofthe products of the di ff erent ratios, a sum that thereforepresents a (discrete) integral action that can filter at itsturn the residual fluctuations of the products.The specificity of such a property, that confers to the inter-nal structure a self-filtering character with double e ff ect, standsout by the joint e ff ect of a sum and of products, thus overtaking2he filtering e ff ect of a simple integrator, naturally limited to theone of a sum.This self-filtering double e ff ect is validated in section 5.4,through real curves (figure 8) which prove that the stochasticbehavior due to the random fluctuations of the q k , has globallyno incidence or practically not, notably on the sum, s k , of thenew contaminated at day k . The modeling of this sum (or moreprecisely its result), which is an objective in epidemiology andthen in this paper, therefore needs only a deterministic model ,in this case the model in t m , thus expressing the compatibilitybetween internal complexity and global simplicity .In other words, this self-filtering double e ff ect liable to glob-ally denoise the internal structure achieving (or synthesizing)the model in t m , well expresses a coherent physical phenomenon,in the sense that the model in t m is not noisy, even if it is noisyinternally. This phenomenon assuredly is likely to remove theparadox between the simplicity of a model in t m and the com-plexity of a noisy reality, thus reinforcing the interest of theparametric compactness of a model in t m in the modeling ofcomplex systems and phenomena (including among others thoseof finance (see Appendix B)).
2. A geometric sequence and the associated sum
Such as lead in this article, establishing a viral spreadingmodel, calls, so it seems, a targeted mathematical recall as re-gards sequences, in order to better understand the geometricsequence “with variable ratio”, introduced here so that the pro-posed model (via its internal structure) well represents the phe-nomenon of spreading by contamination.In mathematics, a sequence , denoted ( u k ) k ≥ , is written asthe sequence of terms : u , u , u , . . . , u k , . . . ;of positive or null subscript k which indicates the rank, the general term (of rank k) , u k , defines the sequence on its own.Concerning the sum of the terms of the so defined sequence,in particular the first ones, the sum, s k , of the k + u k ) k ≥ , is expressed according to the sum ofterms: s k = u + u + . . . + u k . For the most initiated ones, this sum s k , associated with thesequence ( u k ) k ≥ , is not without recalling the general term, s k ,of the sequence of the partial sums of the terms of ( u k ) k ≥ , thatdefines the series , ( s k ) k ≥ , knowing that if a sequence is a se-quence of terms, a series is a sequence of term sums .A geometric sequence (or geometric progression ) is charac-terized by a constant ratio , q , between two consecutive terms,namely: u k + u k = q ∀ k ≥ u , q is called common ratio of the sequence.The term u k + with k ≥
0, can be written under the form ofa recurrence relation , namely : u k + = u k q , which expresses that each term of the sequence is the product of the precedent with the common ratio. Thus: u = u q , u = u q = u q , u = u q = u q , . . . , the general term of the sequence then being: u k = u q k ∀ k ≥ . The sum s k associated with the so defined geometric se-quence, is given by the sum of terms: s k = u + u + . . . + u k with u k = u q k , namely, as one can show it: s k = u (1 + q + . . . + q k ) = u q k + − q − q , , the case where q = u , immediately lead to s k = ( k + u .
3. Geometric sequence with variable ratio and associatedsum
In a simple geometric sequence, the common ratio is con-stant, in the sense that the ratio between two consecutive termsis the same whatever the rank k (as being independent of it),namely: u k + u k = q ∀ k ≥ u , . In the geometric sequence considered here, said “with vari-able ratio”, the ratio is no more constant, in the sense that theratio between two consecutive terms may be di ff erent accordingto rank k (as being dependent on it), thus varying with the rank ,namely: u k + u k = q k + ∀ k ≥ u , , the term u k + , k ≥
0, being then defined by the recurrence rela-tion: u k + = u k q k + . Thus: u = u q , u = u q = u q q , u = u q = u q q q , . . . , the general term of the sequence then being: u k = u q . . . q k ∀ k ≥ . The sum s k associated with the so defined geometric se-quence “with variable ratio” , is given by the sum of terms: s k = u + u + u + . . . + u k with u k = u q . . . q k , k ≥ , s k = u (1 + q + q q + . . . + q . . . q k ) , k ≥ . By taking into account the geometric means ( ρ , ρ , . . . , ρ k , . . . )of the di ff erent ratios ( q , q , . . . , q k , . . . ), it is then possible towrite: ρ = q , mean of q , namely q ,ρ = ( q q ) / , mean of q and q ,ρ = ( q q q ) / , mean of q , q and q ,. . . ,ρ k = ( q . . . q k ) / k , mean of q , . . . , q k ,. . . , from where one draws, by squaring, cubing, ... : q = ρ , q q = ρ , q q q = ρ ,. . . , q . . . q k = ρ kk . . . . The terms of the sequence, expressed as functions of thedi ff erent ratios, namely u , u q , u q q , . . . , u q . . . q k , . . . , can then be expressed as functions of the geometric means ofthe di ff erent ratios, namely u , u ρ , u ρ , . . . , u ρ kk , . . . , the sum s k associated with this sequence, thus admitting an ex-pression as a function of these means: s k = u (cid:16) + ρ + ρ + . . . + ρ kk (cid:17) , k ≥ .
4. Why this kind of sequence with variable ratio
We have introduced this kind of sequence “with variableratio”, as well as the associated sum, in order to conceive a viralspreading model whose internal structure can be in conformitywith the daily census process of the new contaminated.In the census of the new contaminated by region and by day,that gives a space and time image of the spreading by contam-ination, there is no reason for the ratio of new contaminatedbetween two consecutive days to be constant in time and there-fore to be independent of the rank k . Indeed, the ratio betweenthe new contaminated counted at day k + k , assuredly depends on k , thus expressingthat this ratio well varies day after day.That amounts to saying that such a census cannot be for-malized by a simple geometric sequence, which is with con-stant ratio, whereas it can be well formalized by a geometric sequence “with variable ratio”, whose ratio between two con-secutive terms varies with the rank k .Moreover, given the complexity of contamination pheno-mena, and particularly the stochastic aspects that highly con-tribute to this complexity, there is no reason for, day after day,the ratio of new contaminated between two consecutive daysto increase or to decrease in a monotonic manner. It is indeedvery likely that this kind of variation can be obtained in mean ,in accordance with smoothing this ratio: hence, the geometricmeans of the di ff erent ratios, considered in our developments togo in this direction.Such expectations, linked to our knowledge of complex phe-nomena, thus raise the points to be discussed, after confrontingthe proposed model to o ffi cial data of the French Ministry ofHealth , which, from now on, will be named Ministry.That being said, it is now possible to formalize the dailycensus process of the new contaminated, and this, of course, inconformity with the geometric sequence “with variable ratio”(in the case of a spreading aggravation): • at day 0 (which corresponds to k = u , represents the number of new con-taminated individuals who have been counted this day; • at day 1 (which corresponds to k = u , greater than u , is the numberof new contaminated individuals who have been countedthis day, this number being written as u = u q , thus ex-pressing that at day 1, there are q times more new con-taminated than the day before (day 0); • at day 2 (which corresponds to k = u , greater than u , is the number ofnew contaminated individuals who have been recordedthis day, this number being written as u = u q , thusexpressing that at day 2, there are q times more newcontaminated than at day 1; • at day k , the term of rank k , u k , greater than u k − , can bewritten as u k = u k − q k , thus expressing that at day k , thereare q k times more new contaminated than at day k − s k associated with the sequence, is written, as afunction of the di ff erent ratios, according to s k = u (1 + q + q q + . . . + q . . . q k ) for k ≥ , or, as a function of the geometric means of the di ff erent ratios,according to s k = u (cid:16) + ρ + ρ + . . . + ρ kk (cid:17) for k ≥ . This sum s k , expressed by the one or the other of thesetwo equations, represents the sum of the new contaminated whohave been censed from day 0 to day k (0 and k included). Thetotal sum of the contaminated at day k , is then obtained byadding the former contaminated before day 0. https: // / fr / datasets / donnees-relatives-a-lepidemie-de-covid-19-en-france-vue-densemble / . Proposition and validation of a time model of viral sprea-ding: the FPM model Given that the non-integer di ff erentiation or integration op-erator governs di ff usion phenomena (thermal for example), onecan legitimately suppose that this operator of non-integer order, − m in di ff erentiation or m in integration, with m > t m (Appendix A and figure 1), also governsspreading phenomena by contamination. It is true that thesephenomena, as the di ff usion phenomena, fall under complexphenomena , for which the non-integer operator or its dynamicsin power-law turn out to be particularly appropriated modelingtools, notably by the parametric compactness they o ff er in com-parison with complexity. A newsworthy illustration that widensthis context, incontestably lies in the field of huge networks andparticularly of Internet network: Internet is indeed a paradigmof complex network, whose “degree distribution”, which is in-herent to the di ff erent average numbers of connections per nodeand which thus evokes a “diversity degree” (Oustaloup (2014)),follows a non-integer power law such as N ( k ) = k − γ , N ( k ) rep-resenting the number N of nodes endowed with k connections,and γ being a positive non-integer comprised between 2 and3 (Santamaria (2017) and Mendes and da Silva (2009)). Otherapplicative fields in which the power-law dynamics occurs arepresented in Appendix B. time (s) PSfrag replacements t m , m = t m , m = . t m , m = . Figure 1: Step response in t m , whose non-integer power, m , determines the cur-vature, namely a concavity for m < m >
1: for m << t m = e ln t m = e m ln t , is reduced to 1 + m ln t , whose variation in ln t is represen-tative of a long memory phenomenon To properly verify such an hypothesis of non-integer expo-nent, it su ffi ces to directly check the conformity of a variationin t m with the o ffi cial data of Ministry, besides according to thecomparative analysis presented further and which well confirmsthis conformity (section 6).But it is possible to go beyond this direct verification, bydetermining the di ff erent ratios and the geometric means of thedi ff erent ratios, inherent to the hypothesis of a variation in t m , and this, to see if, themselves, are well in conformity with theo ffi cial data of the Ministry. This is a way to attempt to validatethe internal structure proposed for the viral spreading model.To that e ff ect, a first step consists in imposing, to the sumof the new contaminated, s k , a time variation in t m (up to amultiplying constant and an additive constant), which amountsto simply writing, by introducing a positive constant, c : s k = u (1 + ct m ) , namely, for t = kh where h is the sampling period: s k = u (1 + c ( kh ) m ) , therefore, given the expressions of s k such as already deter-mined: u (1 + q + q q + . . . + q . . . q k ) = u (1 + c ( kh ) m ) , k ≥ u (cid:16) + ρ + ρ + . . . + ρ kk (cid:17) = u (1 + c ( kh ) m ) , k ≥ , or, after simplifying by u and 1: q + q q + . . . + q . . . q k = c ( kh ) m with k ≥ ρ + ρ + . . . + ρ kk = c ( kh ) m with k ≥ . (2)From the two equations so obtained, a second step consistsin determining the di ff erent ratios, q , q , . . . , q k , as well as thegeometric means of the di ff erent ratios, ρ , ρ , . . . , ρ k , for asampling period of one day, namely h =
1, and for the di ff erentvalues of k , namely 1, 2, 3, . . . ff erent ratios For h =
1, the first equation (1) becomes: q + q q + . . . + q . . . q k = ck m ∀ k ≥ . For k =
1, a term by term identification immediately gives: q = c m = c . For k ≥
2, let us rewrite the equation in conformity with: q + q q + . . . + q . . . q k − + q . . . q k = ck m , (3)then let us replace k by k − q + q q + . . . + q . . . q k − + q . . . q k − = c ( k − m . (4)The di ff erence of equations (3) and (4) leads to: q . . . q k = c ( k m − ( k − m ) , (5)or, by replacing k by k − q . . . q k − = c (( k − m − ( k − m ) , (6)5he ratio of equations (5) and (6) then leading to: q k = k m − ( k − m ( k − m − ( k − m ∀ k ≥ , (7)a result that expresses the general term of the sequence ( q k ) k ≥ .For k >>
1, it is convenient to write q k under the form: q k = k m − h k (cid:16) − k (cid:17)i m h k (cid:16) − k (cid:17)i m − h k (cid:16) − k (cid:17)i m = k m − k m (cid:16) − k (cid:17) m k m (cid:16) − k (cid:17) m − k m (cid:16) − k (cid:17) m , or, by simplifying by k m : q k = − (cid:16) − k (cid:17) m (cid:16) − k (cid:17) m − (cid:16) − k (cid:17) m , namely, for k su ffi ciently high: q k ∼ − (cid:16) − mk (cid:17)(cid:16) − mk (cid:17) − (cid:16) − mk (cid:17) = m / km / k , therefore, finally: q k ∼ ∀ k >> , a result, not only remarkable of simplicity, but also very inter-esting with regard to the o ffi cial data of the Ministry.Figure 2 illustrates the variations of q k given by (7) for dif-ferent values of m ( m = . , . . k PSfrag replacements m = . m = . m = . Figure 2: Variations of q k versus k for three values of m ff erent ra-tios For h =
1, the second equation (2) becomes: ρ + ρ + . . . + ρ kk = ck m ∀ k ≥ . For k ≥
1, let us rewrite this equation under the form: ρ + ρ + . . . + ρ k − k − + ρ kk = ck m , (8)then let us replace k by k − ρ + ρ + . . . + ρ k − k − + ρ k − k − = c ( k − m . (9)The di ff erence of equations (8) and (9) leads to: ρ kk = c ( k m − ( k − m ) , from where one draws: ρ k = (cid:16) c k (cid:17) ( k m − ( k − m ) k ∀ k ≥ , (10)a result that expresses the general term of the sequence ( ρ k ) k ≥ .For k >>
1, it is appropriated to write: k m − ( k − m = k m − " k − k ! m = k m − k m − k ! m = k m " − − k ! m , namely, for k su ffi ciently high: k m − ( k − m ∼ k m (cid:20) − (cid:18) − mk (cid:19)(cid:21) = k m mk .ρ k is then written: ρ k = (cid:16) c k (cid:17) ( k m − ( k − m ) k ∼ (cid:16) c k (cid:17) (cid:18) k m mk (cid:19) k , or: ρ k ∼ (cid:16) c k (cid:17) k mk (cid:18) mk (cid:19) k = (cid:16) c k (cid:17) k mk k k m k = ( cm ) k k m − k , or even, knowing that k m − k = e ln (cid:18) k m − k (cid:19) = e m − k ln k = e ( m − ln kk : ρ k ∼ ( cm ) k e ( m − ln kk , or else, given that for k su ffi ciently high,1 k ∼ kk ∼ ρ k ∼ ∀ k >> , a result, not only remarkable of simplicity, but also very inter-esting with respect to the o ffi cial data of the Ministry.Figure 3 presents, for c =
1, the variations of ρ k given by(10) for di ff erent values of m ( m = . , . . k PSfrag replacements m = . m = . m = . Figure 3: Variations of ρ k versus k for c = m PSfrag replacements theoretical q k real q k Figure 4: Comparative illustration of the theoretical and real q k for m = . q k smooth the real q k ; mean( E r ) = . The random fluctuations of the ratios, inherent to the realitycomplexity, are, properly, interpretable as fluctuations that aresuperimposed to a smoothing whose graphical representation isa curve representative, no more of the variations of these ratios,but of the variation mean.So, the internal structure of the model in t m , such as elab-orated by disregarding the unpredictable, (therefore these ran-dom functions) can only represent the smoothing such as de-fined.That is to say that the ratios q k must be in conformity withthe smoothing of the ratios stemming from the o ffi cial data ofMinistry. Thus, the validation of the proposed internal struc-ture, can be carried out by the verification of this conformity.But this verification can be reinforced by benefiting fromthe study of the geometric means ρ k of the di ff erent ratios q k ,notably by comparing the geometric means obtained, theoret- k PSfrag replacements theoretical ρ k real ρ k Figure 5: Comparative illustration of the theoretical and real ρ k for m = . c = . ρ k smooth the real ρ k ; mean( E r ) = − . − ically, with the internal structure and, really, with the o ffi cialdata of Ministry.If a k denotes the total number of contaminated counted atday k by the Ministry, the real q k and ρ k ∀ k ≥
1, stemmingfrom the o ffi cial data, are computed in conformity with: q k = a k + − a k a k − a k − and ρ k = ( q . . . . q k ) / k , where: q . . . q k = a − a a − a a − a a − a . . . a k − a k − a k − − a k − a k + − a k a k − a k − = a k + − a k a − a . To evaluate the quality of the modeling, the maximum ofthe modulus of the absolute or relative errors, or the mean of theabsolute or relative errors can be computed, namely ( a k beingthe real data and b k the theoretical values stemming from themodeling): max ( | E a | ) = max {| a k − b k | , ≤ k ≤ n } or mean ( E a ) = n n X k = ( a k − b k )or max ( | E r | ) = max ((cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a k − b k b k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , ≤ k ≤ n ) or mean ( E r ) = n n X k = a k − b k b k ! . To lead the verification of the conformity of q k and ρ k withthe smoothing of the corresponding quantities stemming from7
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PSfrag replacements theoretical q k real q k Figure 6: Comparative illustration of the theoretical and real q k for m = . q k smooth the real q k ; mean( E r ) = . PSfrag replacements theoretical ρ k real ρ k Figure 7: Comparative illustration of the theoretical and real ρ k for m = . c = . ρ k smooth the real ρ k ; mean( E r ) = . − the o ffi cial data, we have taken two values of m stemming fromthe identification, by the model in t m , of the COVID-19 spread-ing in France (section 6): the first value of m , greater than 1,namely m = . m , less than 1, namely m = . c , that takes partin the general expression of ρ k , its determination according tothe relation c = B / u drawn from (12), is not pertinent in thesense that, if B is known (section 6), u is known only througha noisy data: so, c is determined by optimization, so that thearithmetic mean of the absolute gaps, mean ( E a ), should be the-oretically null. Thus, we get c = . c = . k , figures 5 and 7 well show that the geometric mean of noisy data is not a very significative mean.Note that the data can present some aberrations, particularlythose collected at the weekend.From the whole of the curves represented by figures 4 to7, it turns out that the conformity to be verified is truly e ff ec-tive, thus validating the internal structure of the model in t m ,and consequently, the model itself (more precisely the model in t m up to constants) as the internal structure is inherent to thismodel. The validation presented in the previous section 5.3 is en-riched by the validation of the self-filtering action of the inter-nal structure, such as described at the end of section 1.2 on thespecific contributions of the article. This action, that makes themodel in t m not noisy even if it is internally noisy, indeed turnsout to be graphically validated by figure 8, obtained with thereal data of period 1 which corresponds to a convex growth.The curves of this figure well convey two successive reduc-tions of the random fluctuations of the ratios q k between newcontaminated between two consecutive days: it is, on one hand,a reduction between the q k and the u k (e ff ect of a first filtering)and, on the other hand, a reduction between the u k and the s k (e ff ect of a second filtering), the u k denoting the new contami-nated day after day, and the s k denoting the daily sums of thesenew contaminated. Actually, through all these curves, figure 8speaks by itself: indeed, this figure, which well illustrates twosuccessive improvements of the signal-to-noise ratio, is su ffi -ciently convincing for not being more commented. PSfrag replacements q k q k u k u k s k s k k Figure 8: Illustration of the self-filtering action of the internal structure, fromthe real data, a k , k ≥
0, corresponding to period 1. For k ≥ q k = ( a k + − a k ) / ( a k − a k − ); u k = u q . . . q k = a k + − a k with u = a − a ; s k = u + u + . . . + u k ff ect. At section 1.2, we have at-tributed this origin to the form of the sequence terms, thereforeto the products of the di ff erent ratios, q . . . q k , given the con-sidered geometric sequence “with variable ratio”. But if we hadconsidered an arithmetic sequence “with variable di ff erence”,the origin of the first filtering would have been attributable tothe sums of the di ff erent di ff erences, r + · · · + r k , stemming fromthe general term of the sequence, namely u k = u + r + . . . + r k , ∀ k ≥
1. These sums of di ff erences, that present a discrete in-tegral action, and which are thus likely to filter themselves therandom fluctuations of the di ff erences, o ff er an explanation, asor even more perceptible as the one founded on the geomet-ric sequence “with variable ratio”. Note that this sequence waspreferred knowing that the q k are always positive, contrary tothe r k .Thus, more generally, for an internal structure based ona geometric or arithmetic sequence, its self-filtering propertyresults from the form itself of the structure, which (in real-ity) globally eliminates the random fluctuations of its elements,thanks to the change from the local defined by the q k or r k , tothe global defined by the s k , namely the sums of the u k . Sucha property is admittedly likely to explain the compatibility be-tween the internal complexity that the internal structure and itsstochastic behavior present, and the global simplicity that thedeterministic model o ff ers through its power-law, t m . But thissimplicity form is not usual, in the sense that t m presents, notonly a mathematical structure remarkable of simplicity, but alsoa great richness by expressing very simply the complexity of re-ality, as if t m , through its non-integer power m , implicitly had acomplexity of another kind. Actually, the change to the globalis accompanied by a transformation of the complexity nature:indeed, the internal complexity globally comes down to a non-integer power. It is true that the non-integer character of thepower expresses or represents the complexity of systems andcomplex phenomena. That is to say that the non-integer powerof a power-law is a marker of complexity. Given this context, itis convenient to recall that the power-law t m is none other thanthe solution of a di ff erential equation ( Appendix A) whosenon-integer order, m , expresses the character of infinite dimen-sion (therefore the complexity) of the non-integer di ff erentia-tion or integration operator, an operator which is itself (in theoperational domain) a power-law, but of the operational vari-able, s . After validating the internal structure inherent to the modelin t m , thus validating the model itself, it only remains to test theprecision of this model, therefore its representativity, with theo ffi cial data provided by the Ministry on the COVID-19 spread-ing. Knowing that these data relate to all the (new and former)contaminated, it is convenient to complete the model so that itrepresents the whole set of the contaminated, that is to say thetotal sum of contaminated at day k , S k , resulting from the sumof new contaminated between days 0 and k , s k , and the sum of the former contaminated before day 0, noted here C f , namely: S K = s k + C f , or, given the expression of s k for t = kh and h = S k = u (1 + ck m ) + C f , therefore, finally: S k = A + Bk m , (11)with: A = u + C f and B = u c . (12)
6. Time identification of the COVID-19 spreading by appli-cation of the FPM model to the o ffi cial data on the con-taminated This section is dedicated to the time identification, by a non-integer model, of the COVID-19 spreading in France, based ondata between March 1 and October 1 2020, namely on 7 monthsor 215 days.More precisely, the identification method that uses the leastsquares, is founded on an identification model whose structureis given, namely A + Bt m , (13)and whose three parameters, to be determined in an optimalmanner and without constraint, are respectively the additiveconstant, A , the multiplicative constant, B , and the non-integerexponent, m , which makes the specificity of the model.The data of the Ministry on the total number a i of the con-firmed cases at day i , begin at March 1 2020 and are gathered infigure 9. This figure presents several time periods for which thecurve form is di ff erent: the curve seems convex for the first 32days, then concave for the following 33 days (from day 33 today 66), then quasi-linear from day 67 to day 134, and finallyconvex. If the curve is convex, m is greater than 1: the dailynumber of confirmed new cases increases and the epidemic pro-gresses. If the points ( k , a d + k ) are aligned, then m =
1. If thecurvature is concave, then 0 < m <
1: the daily number ofconfirmed new cases decreases and the epidemic regresses. Ifthe daily number of confirmed new cases is null, m = d + d + n , where d denotesthe day preceding this period and where n is the number of daysof this period, a time modeling is sought under the form (13).That is to say that a d + k is approximated by an expression of theform A + Bk m where k varies from 1 to n .To that e ff ect, the least squares are used to determine thethree parameters A > B > m > A , B and m arecalculated at best by optimization, on each period, the arith-metic mean of the absolute errors, mean ( E a ) is theoreticallynull. More precisely, if we operate with 20 significative dig-its, we find, for example, mean ( E a ) = − . − for period 1,and mean ( E a ) = − .
10 10 − for period 2.9 PSfrag replacements period 1 period 2 period 3 period 4 period 5
Figure 9: Evolution of the confirmed cases a i between March 1 and October 12020, on 215 days For period 1, between March 1 and April 1 2020, namely32 days, where d = n =
32 and 1 ≤ k ≤
32, the identificationprocedure leads to: A = . B = . m = . . For period 2, between April 2 and May 5 2020, namely 34days, where d = n =
34 and 1 ≤ k ≤
34, we get: A = . B = . m = . . For period 3, between May 6 and July 22 2020, namely 78days, where d = n =
78 and 1 ≤ k ≤
78, we get: A = . B = . m = . . For period 4, between July 23 and September11 2020, namely51 days, where d = n =
51 and 1 ≤ k ≤
51, we get: A = . B = . m = . . For period 5, between September 12 and October 1 2020,namely 20 days, where d = n =
20 and 1 ≤ k ≤
20, weget: A = . B = . m = . . The following figures, from 10 to 14, present, for each of thefive periods so defined, the comparative evolution of the numberof contaminated recorded by the Ministry and the number ofcontaminated obtained with the model in t m .
7. On the prediction by the FPM model of the cases suscep-tible to be a ff ected by the virus The model in t m which is used here as a predictor, is ob-tained, by identification, from the o ffi cial data collected be-fore the lockdown, notably between March 1 and March 16 k PSfrag replacements A + Bt m O ffi cial data Figure 10: Comparative evolution, during period 1, of the contaminatedrecorded by the Ministry and obtained with the model in t m :whereasmean( E r ) = − . k , mean( E r , k ≥ = − . included, March 17 being the first day of the lockdown. Themodel parameters respectively admit for numerical values: A = . B = . m = . . Using this model until the last day of lockdown, May 10,enables comparing, during the lockdown, between March 17and May 10 included, the evolution of the prediction and theone of the data provided by the Ministry (figure 15).The comparison which is assessed on May 10, namely 492 274cases predicted without the lockdown and 139 063 cases con-firmed with the lockdown, turns out to be very eloquent in thesense that it well expresses the interest of the lockdown.Concerning the precision associated with such a prediction,it is di ffi cult to provide an estimation, as the prediction model isthe subject of an exploitation that is demanding to say the least:indeed, it is established on a period limited to 16 days, whereasit is used with a prediction horizon of 55 days; the oppositewould have been more in favor of the precision.
8. Comparison of the representativity of the FPM modeland of two known models with three parameters
For a model, representativity (measured by the degree ofrepresentativity), is a characteristic that conveys the model ca-pacity to represent reality. In this section, reality is materializedby the evolution of all the contaminated counted each day dur-ing period 1; the representativity of this reality is illustrated,in a comparative way, for the model in t m and for two knownmodels having also three parameters. If the model in t m canbe interpreted as a “convexity or concavity model”, the twoknown models considered here can be interpreted as a “con-vexity model” for the one and as a “convexity and concavitymodel” for the other, a model that we have rewritten under aform in m t , dual to the one of the model in t m .10 k PSfrag replacements A + Bt m O ffi cial data Figure 11: Comparative evolution, during period 2, of the contaminatedrecorded by the Ministry and obtained with the model in t m : mean( E r ) = . − Taking into account a convex then concave growth, the Ver-hulst model and its generalizations are liable to represent, bothfirst and second growth phases of the epidemics.Around 1838, the Belgian P.F. Verhulst (Verhulst (1838,1845)) has established a model of dynamical behavior of a popu-lation, named logistic growth model. If a k is the size of thepopulation at day k , for k ≥
0, the growth rate of this popula-tion at day k + a k + − a k ) / a k .This rate is equal to the di ff erence between the birth rateand the death rate at day k . Assuming that the daily birth rateand daily death rate are a ffi ne functions of the population size,it then comes: a k + − a k a k = N ( a k ) − M ( a k ) , with: N ( x ) = n + n x and M ( x ) = m + m x . This expression is none other than the discretization of atime di ff erential equation in f ( t ), where f ( k ) = a k , namely: f ′ ( t ) = f ( t ) (cid:2) N ( f ( t )) − M ( f ( t )) (cid:3) with f (0) = a , a di ff erential equation in which: N ( f ( t )) = n + n f ( t ) and M ( f ( t )) = m + m f ( t ) , therefore: f ′ ( t ) = f ( t ) (cid:2) n − m + ( n − m ) f ( t ) (cid:3) , or: f ′ ( t ) = ( n − m ) f ( t ) " − m − n n − m f ( t ) , k PSfrag replacements A + Bt m O ffi cial data Figure 12: Comparative evolution, during period 3, of the contaminatedrecorded by the Ministry and obtained with the model in t m : mean( E r ) = . − or even: f ′ ( t ) = a f ( t ) " − f ( t ) K , by posing a = n − m and K = n − m m − n = am − n . The solution of this di ff erential equation, f ( t ), admits anexpression with three parameters, K , a and a : f ( t ) = K + (cid:16) Ka − (cid:17) e − at , (14)an expression which is easier to verify than to establish. It istrue that to verify it, it su ffi ces to di ff erentiate f ( t ) after writingit under the form f ( t ) = K / D ( t ) = KD − ( t ), namely f ′ ( t ) = KaD ( t ) Ka − ! e − at = a f ( t ) D ( t ) Ka − ! e − at , then to express f ′ ( t ) versus f ( t ), by using1 D ( t ) = f ( t ) K and Ka − ! e − at = Kf ( t ) − , to which case: f ′ ( t ) = a f ( t ) " − f ( t ) K . In the case of an epidemic, f ( t ) represents the number ofcontaminated cases at instant t . Many articles use this model with three parameters for theCOVID-19.11 k PSfrag replacements A + Bt m O ffi cial data Figure 13: Comparative evolution, during period 4, of the contaminatedrecorded by the Ministry and obtained with the model in t m : mean( E r ) = − . − Zhou et al. (2020) use the di ff erential equation: P ′ ( t ) = rP ( t ) − P ( t ) K ! , and its solution: P ( t ) = KP (0) e rt P (0)( e rt − + K . Kyurkchiev et al. (2020) use a model of the form: y ( t ) = r + e a + a ( t ) , which is analogous to Verhulst model and is generalized to: y ( t ) = r + e a + a t + a t + ... . Zhao et al. (2020b) use the model: f ( t ) = a f (0) b f (0) + ( a − b f (0)) e at . R´emond and R´emond (2020) use the model: E ( t ) = L + e − k ( t − τ ) . There are several possible generalizations of Verhulst modelsuch as proposed by Roosa et al. (2020) who give two well-known generalizations with four parameters: • the generalized logistic model (GLM), namely dC ( t ) dt = rC ( t ) p − C ( t ) K ! , used, among others, by Viboud et al. (2016) and Ganyani et al.(2018), the four parameters being r , p , K and C (0); k PSfrag replacements A + Bt m O ffi cial data Figure 14: Comparative evolution, during period 5, of the contaminatedrecorded by the Ministry and obtained with the model in t m : mean( E r ) = . − • the Richards model, namely dC ( t ) dt = rC ( t ) − C ( t ) K ! a ! , used, among others, by Wang et al. (2012) and Richards(1959) in 1959, the four parameters being r , a , K and C (0).Richards model is found under a generalized form with fiveparameters, called generalized Richards model (GRM), namely(Wu et al. (2020)): dC ( t ) dt = rC ( t ) p − C ( t ) K a ! , the five parameters being r , p , a , K and C (0), with p ∈ [0 , The model presented in this section is used to represent thefirst growth phase of the epidemics. Its use is indeed kept forconvex growths.By only taking the first factor of the GLM and GRM mod-els, these models are reduced to a same model such as presentedin Wu et al. (2020), Viboud et al. (2016) and Tolle (2003), namely: dC ( t ) dt = rC ( t ) p with r > p ∈ [0 , , (15)a di ff erential equation that defines a model with three param-eters, r , p and C (0), and that admits three di ff erent solutionsaccording to whether p = p = < p <
1: an a ffi nesolution for p =
0, an exponential solution for p = < p < p =
0, the solution is of constant slope, r , as C ( t ) = rt + C (0), for p =
1, the di ff erential equation then being linear,its solution is here the one of a system of first order, namely: C ( t ) = C (0) e rt .
10 20 30 40 50 60 70 k PSfrag replacements A + Bt m O ffi cial data Figure 15: Comparative evolution, during the lockdown, of the model predic-tion and the data provided by the Ministry
For 0 < p <
1, the di ff erential equation then being nonlin-ear, the establishment of its solution requires a rewriting of theequation in conformity with: C ( t ) − p C ′ ( t ) = r , namely, given the formula u ( t ) α u ′ ( t ) = u ( t ) α + α + ! ′ ∀ u ( t ) and ∀ α , − C ( t ) − p − p ! ′ = r , therefore: (cid:16) ( C ( t ) − p (cid:17) ′ = r (1 − p ) , or, by integrating, thus introducing an integration constant C : C ( t ) − p = r (1 − p ) t + C , or even: C ( t ) = ( r (1 − p ) t + C ) − p , or else, knowing that C (0) = C − p hence C = C (0) − p : C ( t ) = (cid:16) r (1 − p ) t + C (0) − p (cid:17) − p , a solution with non-integer exponent, that we rewrite here un-der a form comparable to the one of the model in t m , A + Bt m ,namely: C ( t ) = (cid:0) A ′ + B ′ t (cid:1) m ′ , with: A ′ = C (0) − p , B ′ = r (1 − p )and m ′ = − p > < p < . Given the two model structures, if the model in t m can beinterpreted as a “model with explicit power of time”, the othermodel can be interpreted as a “model with implicit power oftime”, a power which is non-integer for both models.Let us express the derivatives of order 1 and 2 of both mod-els in question: • for the model C ( t ) = A + Bt m , C ′ ( t ) = mBt m − and C ′′ ( t ) = m ( m − Bt m − ; • for the model C ( t ) = ( A ′ + B ′ t ) m ′ , C ′ ( t ) = m ′ B ′ ( A ′ + B ′ t ) m ′ − and C ′′ ( t ) = m ′ ( m ′ − B ′ ( A ′ + B ′ t ) m ′ − . If all parameters are considered positive, it is then possibleto write: • for the first model, C ′ (0) = m > C ′ (0) = ∞ for 0 < m <
1, nullor infinite initial slope, C ′ ( t ) >
0, increasing C ( t ), C ′′ ( t ) > m >
1, increasing C ′ ( t ) according to aconvexity of C ( t ), C ′′ ( t ) < < m <
1, decreasing C ′ ( t ) according to aconcavity of C ( t ); • for the second model, C ′ (0) = m ′ B ′ A ′ m ′ − >
0, positive initial slope, C ′ ( t ) >
0, increasing C ( t ), C ′′ ( t ) > m ′ >
1, increasing C ′ ( t ) according to aconvexity of C ( t ), C ′′ ( t ) < < m ′ <
1, decreasing C ′ ( t ) according toa concavity of C ( t ).Even if the model represented by the di ff erential equation(15) imposes, on its solution C ( t ), an exponent m ′ greater than1, we have wished to find out more, by studying this solution foran exponent m ′ comprised between 0 and 1, which correspondsto p < ff erential equation and its solution.Thus, to lead a comparative study between the parametersof both models for m and m ′ comprised between 0 and 1, andthis for simply understanding why the “convexity model” is notproposed for concave growths, it is convenient to approximatelyexpress A ′ and B ′ versus A , B , m and m ′ by considering theidentity of the models at an initial instant, t =
0, and at a finalinstant, t = t n , then to analyze A ′ and B ′ for the small values of m ′ . Let be the two models, denoted here by C ( t ) and C ( t ): C ( t ) = A + Bt m , with A , B > < m < , and C ( t ) = (cid:0) A ′ + B ′ t (cid:1) m ′ , with A ′ , B ′ > < m ′ < . t = C (0) = A and C (0) = A ′ m ′ , then, given that C (0) = C (0): A ′ = A m ′ . (16)For t = t n : C ( t n ) = A + Bt mn and C ( t n ) = ( A ′ + B ′ t n ) m ′ , then, given that C ( t n ) = C ( t n ): A ′ + B ′ t n = ( A + Bt mn ) m ′ , from where one draws: B ′ = t − n h ( A + Bt mn ) m ′ − A m ′ i , namely, by posing Btn m = α A with α > B ′ = t − n A m ′ h (1 + α ) m ′ − i . (17)Given that A , α and m ′ are positive, the relations (16) and(17) well show that A ′ and B ′ tend towards infinity when m ′ tends towards zero. This phenomenon is in conformity with thehuge values of A ′ and B ′ that we have obtained, even for m ′ = .
308 which is far from zero, in the case of an identificationattempt of the second period with m ′ between 0 and 1. Indeed,whereas A = .
42 and B = .
54 ( m being 0 . A ′ and B ′ reach respectively, for a similar precision, A ′ = .
106 10 and B ′ = .
123 10 (for m ′ = . k PSfrag replacements ( A ′ + B ′ t ) m ′ A + Bt m O ffi cial data Figure 16: Comparative evolution, during period 1, of the contaminatedrecorded by the Ministry, and obtained with the model in t m , C ( t ), and the“convexity model”, C ( t ), such that A ′ = . B ′ = . m ′ = . C ( t ), mean( E r , k ≥ = − . E r , k ≥ = − . | E a | , k = = C ( t ), mean( E r , k ≥ = . E r , k ≥ = − . | E a | , k = = Such a numerical explosion of the parameters for low valuesof m ′ , therefore well illustrates the di ffi culty of the model C ( t )to naturally capture the concave growths.On the other hand, for a convex growth such as the one forperiod 1, the model C ( t ), with m ′ >
1, is absolutely repre-sentative of the evolution of o ffi cial data corresponding to thisperiod. Indeed, figure 16 seems to show that the model C ( t ) is,for a convexity, as representative as the model C ( t ). More pre-cisely, with respect to the latter, it o ff ers a comparable precisionto say the least, since its precision is better at the start (on threedays) and similar beyond, as shown, with more nuance, by thenumerical errors associated with figure 16.Appropriated to the representation of the complexity thatinvolves high numbers of cases, the model in t m does not havethe vocation to represent small numbers of cases inherent to theshort times: in other words, in reality, where the non-integeroperator has only a sense at medium frequencies, the power-law dynamics, t m , has only a sense at medium times, thereforebeyond the short times. Besides, in the real case of the epidemicstart, the nil slope of the initial behavior of t m , is not stranger tothe representativity lack of the FPM model at the short times.As for the precision of the FPM model in relation to thenumber of cases, it can be illustrated with the confirmed casesof the COVID-19 in Switzerland, given that these ones havebeen provided from a first case, recorded on February 25 2020.Between week 1 starting at this date with one confirmed case,and week 10 starting at April 28 2020 with 29264 confirmedcases, the maximum of the relative errors goes from 0.816 to0.000976, by taking the intermediate values 0.321 and 0.0164for the weeks 2 and 5. That is to say that the precision is clearlyimproved by the case number (therefore by complexity). t whose properties are presented The Verhulst model (14) admits an immediate rewriting: f ( t ) = K + K − a a e − at , or, by multiplying the numerator and the denominator by a / ( K − a ): f ( t ) = Ka K − a a K − a + ( e − a ) t , namely, by posing A = Ka K − a , B = a K − a and m = e − a : f ( t ) = AB + m t , a model in m t , with three parameters, A , B and m , dual to themodel in t m , A + Bt m .Used as an identification model, the function f ( t ) = AB + m t , is applied in a time interval such that 1 ≤ t ≤ n .14his function is a particular solution of the di ff erential equa-tion f ′ ( t ) = − ln( m ) (cid:16) f ( t ) − B f ( t ) / A (cid:17) , f (1) = A / ( B + m ) . Its derivative is f ′ ( t ) = − A ln( m ) m t ( B + m t ) . The function f ( t ) is increasing if A > < m <
1, orif A < m >
1, and decreasing otherwise.Its second derivative is f ′′ ( t ) = A ln( m ) m t (cid:0) m t − B (cid:1) ( B + m t ) . f ( t ) can have an inflexion point in t if f ′′ ( t ) =
0. This isonly possible if t = ln B ln m and if, moreover, 1 < t < n .Such a t exists if 0 < B < < m <
1, or if B > m > < B < < m <
1, then 1 < t < n if m n < B < m .If B > m >
1, then 1 < t < n if m < B < m n .If t is not between 1 and n , then f ( t ) is concave or convexif t varies from 1 to n .If t is between 1 and n , then the curve of f ( t ) is a sigmoid,with two convex and concave parts. t Given a finite sequence ( a k ), k varying from 1 to n , where a k is a daily epidemic datum, and the model in m t with threeparameters, f ( A , B , m , t ), these parameters need to be calculatedso that b k = f ( A , B , m , k ) is as close as possible to a k , for all k between 1 and n .Even if the model in m t is a fraction, the calculus can besimply carried out by minimizing the square sum of the absoluteerrors such as: E ( A , B , m ) = n X k = (cid:16) a k (cid:16) B + m k (cid:17) − A (cid:17) . To minimize E ( A , B , m ), it su ffi ces to cancel the partial deriva-tives with respect to these three parameters.So determined, the model in m t is now applied, as an iden-tification model, to period 1 which is the beginning of the epi-demic in France. This application leads to the following numer-ical values: A = . B = . m = . . Whereas period 1 presents a convexity, the value of B be-longs to an interval corresponding to an inflexion point, thusexpressing that the model in m t is here inappropriate, as figure17 shows it.In conclusion, figure 17 conveys an eloquent result: thecomparative illustration of the representativity of the models in t m and m t , clearly proves to be in favor of the model in t m . Butthis drawback of the model in m t is due to its strategy require-ment, in the sense that this model aims, with a unique parame-terization, to represent both convex and concave phases of theepidemic growth. k PSfrag replacements A / ( B + m t ) A + Bt m O ffi cial data Figure 17: Comparative evolution, during period 1, of the contaminatedrecorded by the Ministry and obtained with the models in t m and m t : the com-parison well illustrates the representativity di ff erence of the models
9. Conclusion
This paper develops an original approach of a very sim-ple deterministic model with three parameters, the FPM model.The non-integer power of time (significative to the curvatureof the viral spreading evolution), enables representing at eachday the totality of the contaminated, and the internal structure,founded on a geometric sequence “with variable ratio”, enables,on one hand, to formalize the daily censing of the new contam-inated and, on the other hand, to represent day after day thewhole of the new contaminated.The simplicity of the model is due to its variation in t m which is a response to the non-integer di ff erentiation or inte-gration operator, an operator known for its capacity to simplyrepresent complex phenomena such as di ff usion phenomena (ofthermal origin (Battaglia et al. (2000)) or fluidic one (Oustaloup(2014))).Founded on a known structure and unknown parameters, q k , to be determined, the construction point by point of a non-integer time function of the same form as the model in t m , canbe interpreted as a time synthesis in non-integer. This synthesisis not without evoking an analogy with the frequency synthesis of the non-integer di ff erentiation or integration operator, from asynthesis network having a known structure and unknown pa-rameters to be determined. Whilst there is indeed an analogythrough a common context of synthesis, this analogy ends here,in the sense that both syntheses in question are not of the samenature. On one hand, the first synthesis turns on a time responseof a non-integer operator, while the second one turns on the fre-quency response of the operator itself. On the other hand, thefirst synthesis uses an evolutive structure of synthesis, due to an evolutive dimension that increases with time, whereas the sec-ond one uses a fixed structure of synthesis with a fixed givendimension.The model internal structure is conceived so as to intro-duce ratios likely to represent those of the new contaminated15etween two consecutive days, then used without seeking torepresent the unpredictable inherent to reality hazards, partic-ularly the random fluctuations of these ratios. It is true that,as shown in the paper, these fluctuations have a negligible ef-fect on the slower dynamics to be modelled by the model in t m ,and this, because of the self-filtering character of the internalstructure, highlighted through a filtering double e ff ect. Moregenerally, this self-filtering character removes the paradox be-tween the simplicity of a model in t m and the reality complexity,thus consolidating the interest of this type of model in modelingcomplex systems and phenomena.Finally, if for the sake of simplicity, the model internal struc-ture only represents the reality mean day after day, namely an“averaged reality”, this average proves to be su ffi cient. Indeed,on one hand, it is in conformity with the model in t m which,on the other hand, is in conformity with reality it is supposed torepresent, being well representative of this reality. These words,whose content deserves some thinking and some hindsight, arelargely validated by a comparative study of a set of graphics,whose curves are obtained, in practice, with the o ffi cial dataof Ministry and, theoretically, with the internal structure (equa-tions (7) and (10)) then with the model in t m (equation (13)).Used as a prediction model, to predict the number of casesliable to be a ff ected by the virus, this model highly justifies thelockdown choice, without which the number of contaminatedwould have really exploded.Even if the results are presented briefly in annex for the pa-per length sake, the model in t m has also been applied with suc-cess to the hospitalized, with or without intensive care, and evento the deceased.In order to better measure the contribution of the article, theproposed model in t m is successively compared to two knownmodels having the same number of parameters, namely a “con-vexity model” and a “convexity and concavity model”, the Ver-hulst model, which has been rewritten under the form of a modelin m t so as to introduce a duality with the proposed model. Thecomparison is eloquent, in the sense that a quick analysis of thecomparative illustration of the representativity of the models,clearly proves to be in favor of the model in t m , whose repre-sentativity is better or more general according to the comparedmodel.Furthermore, in a context that scientifically involves com-plexity and simplicity, the self-filtering action of the internalstructure, makes it possible to show the compatibility betweenthe internal complexity (that the internal structure and its stochas-tic behavior present) and the global simplicity (that the determi-nistic model o ff ers via its power-law t m ): it is true that the non-integer character of the power very simply expresses or repre-sents complexity. Appendix A. Impulse and step responses of non-integer in-tegration operator and corresponding power-law
This appendix shows that the power-law, t m , can be definedas the step response of a non-integer order integrator, or as thesolution of a non-integer di ff erential equation. Impulse response
The unit impulse response, or response to the Dirac func-tion, of the non-integer integration operator, of positive non-integer order, m , admits an expression of the form (Liouville(1832); Erd´elyi (1962); Oustaloup (2014)): y imp ( t ) = t m − Γ ( m ) u ( t ) , u ( t ) being the Heaviside function and Γ ( m ) the gamma functiondefined by: Γ ( m ) = Z ∞ x m − e − x dx . Step response
The unit step response, or response to the Heaviside func-tion, of the non-integer integration operator, of positive non-integer order, m , is given by the integral of the unit impulseresponse, namely, for t > y step ( t ) = Z t y imp ( θ ) d θ, or, given the expression of y imp ( t ): y step ( t ) = Z t θ m − Γ ( m ) d θ = Γ ( m ) Z t θ m − d θ, or else: y step ( t ) = Γ ( m ) " θ m m t = m Γ ( m ) t m = t m Γ ( m + , a result that enables concluding that the step response (unit ornot) of the integration operator of non-integer order, m > t m . A non-integer di ff erential equation governing the power-law t m Given the step response so obtained, the power-law t m ap-pears as a function f ( t ) such that: y step ( t ) = f ( t ) Γ ( m + , or, in Laplace transforms: Y step ( s ) = F ( s ) Γ ( m +
1) with F ( s ) = L (cid:2) f ( t ) = t m (cid:3) , a symbolic equation that enables writing, knowing that Y step ( s ) = s m U ( s ) where U ( s ) = L [ u ( t )] : F ( s ) Γ ( m + = U ( s ) s m , namely: s m F ( s ) = Γ ( m + U ( s ) , from where one draws the concrete equation: ddt ! m f ( t ) = Γ ( m + u ( t ) , a relation that expresses a linear di ff erential equation of non-integer order, m , whose solution is nothing but the power-law f ( t ) = t m .16 ppendix B. Some applicative fields in which power-lawdynamics occurs Power-law dynamics such as expressed by t m can be foundin many applicative fields.For example, in cosmology, the expansion of the universeis explained in Frieman et al. (2008) through the cosmic scalefactor, a ( t ), which is controlled by the dominant energy form a ( t ) ∝ t + w ) for a constant w .Also in fluid dynamics, the Tanner law, which was estab-lished in Tanner (1979) for explaining the spreading of siliconeoil drops, enables explaining the spreading dynamics of fluiddroplets according to R ( t ) ∝ t m where R ( t ) is the radius of thedroplet (see Liang et al. (2009)).And even in neurobiology, the dynamics of biological sys-tems appear to be exponential over short-term courses and tobe in some cases better described over the long-term by power-law dynamics. A model of rate adaptation at the synapse, be-tween inner hair cells and auditory-nerve fibers, is presented inZilany et al. (2009).A non-exhaustive list of application fields that use the po-wer-law dynamics, can be cited: in biochemical systems (Savageau(1970)), in DNA dynamics (Andreatta et al. (2005)), in semi-conductor nanocrystals (Sher et al. (2008)), in water ressources(Harman et al. (2009)), in oscillators (Korabel et al. (2007)) andeven in financial market (Gabaix et al. (2003)), etc. Appendix C. Application of the FPM model to the o ffi cialdata on the hospitalized and the deceased The proposed model in t m , A + Bt m , has also been appliedwith success to the hospitalized, with or without intensive care,and even to the deceased cases for the first two periods, that isto say the first convex period and the first concave one. Thecomparative evolutions are illustrated on figure C.18, where theo ffi cial data are plot in solid lines, the models in the convexparts in circle symbols and the models in the concave parts arein star symbols. As expected, the provided models well fits theo ffi cial data. Moreover, the parameters obtained after identi-fication for the di ff erent models for each period are providedin table C.1, and the maximum relative error is provided, thusvalidating the proposed models. References
Andreatta, D., P´erez Lustres, J. L., P´erez Lustres, J. L., Kovalenko, S. A., Ernst-ing, N. P., Murphy, C. J., Coleman, R. S., Berg, M. A., May 2005. Power-lawsolvation dynamics in dna over six decades in time. Journal of the AmericanChemical Society 127 (20), 7270—7271.Arfan, M., Shah, K., Abdeljawad, T., Mlaiki, N., Ullah, A., 2020. A caputopower law model predicting the spread of the covid-19 outbreak in pakistan.Alexandria Engineering Journal.Atangana, A., 2020. Modelling the spread of covid-19 with new fractal-fractional operators: Can the lockdown save mankind before vaccination?Chaos, Solitons & Fractals 136, 109860.Battaglia, J.-L., Le Lay, L., Batsale, J.-C., Oustaloup, A., Cois, O., 2000. Heatflux estimation through inverted non integer identification models. Interna-tional Journal of Thermal Science 39 (3), 374–389.Chen, T., Rui, J., Wang, Q., Zhao, Z., Cui, J.-A., Yin, L., 2020. A mathematicalmodel for simulating the transmission of wuhan novel coronavirus. bioRxiv.De-Leon, H., Pederiva, F., 2020. Particle modeling of the spreading of coron-avirus disease (covid-19). Physics of Fluids 32 (8), 087113. PSfrag replacements k hospitalized A + Bt m convex A + Bt m concavehospitalized without Intensive Care A + Bt m convex A + Bt m concavehospitalized in Intensive Care A + Bt m convex A + Bt m concavedeceased A + Bt m convex A + Bt m convex Figure C.18: Comparative evolution, during periods 1 and 2, of the hospital-ized (––), hospitalized without intensive care (––), hospitalized in intensivecare (––), and deceased (––) recorded by the Ministry and obtained with themodels in t m , one for the convex part (o) and one for the concave part ( ∗ )De Visscher, A., 2020. The covid-19 pandemic: model-based evaluation of non-pharmaceutical interventions and prognoses. Nonlinear Dynamics 101 (3),1871–1887.Dell’Anna, L., 2020. Solvable delay model for epidemic spreading: the case ofcovid-19 in italy. Scientific Reports 10 (1), 15763.Demongeot, J., Griette, Q., Magal, P., 2020. Si epidemic model applied tocovid-19 data in mainland china. medRxiv.Efimov, D., Ushirobira, R., 2020. On interval prediction of covid-19 develop-ment based on a seir epidemic model. In: Journ´ees Nationales Automtiquede la SAGIP - Groupe de Travail Identification et COVID-19. Lille, France.Erd´elyi, A., 1962. Operational Calculus and Generalized Functions. Holt, Rine-hart and Winston, New-York.Ferguson, N. M., Cummings, D. A. T., Cauchemez, S., Fraser, C., Riley, S.,Meeyai, A., Iamsirithaworn, S., Burke, D. S., 2005. Strategies for containingan emerging influenza pandemic in southeast asia. Nature 437 (7056), 209–214.Frieman, J. A., Turner, M. S., Huterer, D., 2008. Dark energy and the accel-erating universe. Annual Review of Astronomy and Astrophysics 46 (1),385–432.Gabaix, X., Gopikrishnan, P., Plerou, V., Stanley, H., 2003. A theory of power-law distributions in financial market fluctuations. Nature 423 (6937), 267–270.Ganyani, T., Roosa, K., Faes, C., Hens, N., Chowell, G., 10 2018. Assessingthe relationship between epidemic growth scaling and epidemic size: The2014-16 ebola epidemic in west africa. Epidemiology and infection 147, 1–6.Guan, L., Prieur, C., Zhang, L., Prieur, C., Georges, D., Bellemain, P., 2020.Transport e ff ect of covid-19 pandemic in france. In: Journ´ees NationalesAutomtique de la SAGIP - Groupe de Travail Identification et COVID-19.Lille, France.Harman, C. J., Sivapalan, M., Kumar, P., 2009. Power law catchment-scalerecessions arising from heterogeneous linear small-scale dynamics. WaterResources Research 45 (9).He, S., Peng, Y., Sun, K., 2020. Seir modeling of the covid-19 and its dynamics.Nonlinear Dynamics 101 (3), 1667–1680.Huang, J., Qi, G., 2020. E ff ects of control measures on the dynamics of covid-19 and double-peak behavior in spain. Nonlinear Dynamics 101 (3), 1889–1899.Huang, N. E., Qiao, F., Wang, Q., Tung, K.-K., 2020. Herd immunity vs sup-pressed equilibrium in covid-19 pandemic: di ff erent goals require di ff erentmodels for tracking. medRxiv.Iacus, S. M., Santamaria, C., Sermi, F., Spyratos, S., Tarchi, D., Vespe, M.,2020. Human mobility and covid-19 initial dynamics. Nonlinear Dynamics able C.1: Identification parameters for the models in t m of the hospitalized, hospitalized without intensive care, hospitalized in intensive care, and deceased forperiod 1 (convex part) and for period 2 (concave part) Data Dates Part k m A B mean ( E r )hospitalized 17 / − /
03 convex part 1 −
13 1 . .
401 243 . − . − / − /
04 concave part 1 −
14 0 . − .
02 39190 .
75 1 . − hospitalized without 17 / − /
03 convex part 1 −
13 1 . .
690 191 . − . − intensive care 30 / − /
04 concave part 1 −
16 0 . .
149 9504 .
363 4 . − hospitalized in 06 / − /
03 convex part 1 −
21 2 . . . . − intensive care 27 / − /
04 concave part 1 −
12 0 . . .
590 1 . − deceased 01 / − /
03 convex part 1 −
15 3 . . . − . − / − /
04 convex part 1 −
27 1 . . . . −