Studying the course of Covid-19 by a recursive delay approach
SSTUDYING THE COURSE OF COVID-19 BY A RECURSIVEDELAY APPROACH
MATTHIAS KRECK † , ERHARD SCHOLZ ‡ Abstract.
In an earlier paper we proposed a recursive model for epidemics; in the presentpaper we generalize this model to include the asymptomatic or unrecorded symptomatic people,which we call dark people (dark sector). We call this the SEPAR d -model. A delay differentialequation version of the model is added; it allows a better comparison to other models. We carrythis out by a comparison with the classical SIR model and indicate why we believe that theSEPAR d model may work better for Covid-19 than other approaches.In the second part of the paper we explain how to deal with the data provided by the JHU, inparticular we explain how to derive central model parameters from the data. Other parameters,like the size of the dark sector, are less accessible and have to be estimated more roughly, atbest by results of representative serological studies which are accessible, however, only for a fewcountries. We start our country studies with Switzerland where such data are available. Thenwe apply the model to a collection of other countries, three European ones (Germany, France,Sweden), the three most stricken countries from three other continents (USA, Brazil, India).Finally we show that even the aggregated world data can be well represented by our approach.At the end of the paper we discuss the use of the model. Perhaps the most striking applicationis that it allows a quantitative analysis of the influence of the time until people are sent toquarantine or hospital. This suggests that imposing means to shorten this time is a powerfultool to flatten the curves. Contents
Introduction 2Part I: Theoretical framework 31. The SEPAR model and its comparison with other models 3Part II: Application 152. Determining empirical parameters for the model 153. Selected countries/territories 234. Discussion 59Appendix 63References 65
Date : March 1, 2021. † Mathematisches Institut der Universit¨at Bonn and Mathematisches Institut der Universit¨atFrankfurt, Germany, [email protected] ‡ University of Wuppertal, Faculty of Math./Natural Sciences, and Interdisciplinary Centre forHistory and Philosophy of Science, [email protected] a r X i v : . [ q - b i o . P E ] F e b M. KRECK, E. SCHOLZ
Introduction
There is a flood of papers using the standard S(E)IR models for describing theoutspread of Covid-19 and for forecasts. Part of them is discussed in [19]. Wepropose alternative delay models and explain the differences.In [10] we have proposed a discrete delay model for an epidemic which we callSEPAR-model (in our paper we called it
SEP IR model). In this paper we ex-plained why and under which conditions the model is adequate for an epidemic. Inthe present note we add two new compartments reflecting asymptomatic or symp-tomatic, but not counted, infected which we call the dark sector . We call this modelthe generalized SEPAR-model , abbreviated
SEPAR d , where d stands for dark. Thisis our main new contribution. We will discuss the role of the dark sector in a the-oretical comparison of the SEPAR d -model with the SEPAR model. We will see,that – as expected – as long as the number of susceptibles is nearly constant, thedifference of the two models is small, but in the long run it matters.A second topic in this paper is a comparison with the standard SIR model. Thiscomparison has two aspects, a purely theoretical one by comparing the differentfundaments on which the models are based, and a numerical one. For comparingtwo models it is helpful to derive them from similar inputs. For this we pass fromthe discrete model leading to difference equations to a continuous model, replacingdifference equations by differential equations. These differential equations fit intothe general approach developed by Kermack/McKendrick in [9], as we learnt fromO. Diekmann. The analytic model resulting from our discrete model has been in-troduced independently by J. Mohring and coauthors [12] and, more recently, by B.Shayit and M. Sharma [20]. Also F. Balabdaoui and D. Mohr work with a discretedelay approach with additional compartments and a stratification into different agelayers adapted to the Swiss context [3]. Recently y R. Feßler has written a paper [7]in which different differential equation models are discussed and compared, includ-ing the classical S(E)IR-model and the analytic version of our model. Some hintsto earlier papers on the analytic delay approach can be found there.In the second part of the present paper we apply the SEPAR model to selectedcountries and to the aggregated data of the world. To do so we first lay openhow to pass from the data provided by the Humdata project of the JHU to themodel parameters. The data themselves are obviously not reflecting the actualoutspread correctly, which is most visible by the lower numbers of reported casesduring weekends. But in addition there are aspects of the data which need to becorrected like for example a delay of reporting of recovered cases. All this is discussedcarefully. Reliable data about the size of the dark sector are only available in certaincountries where such studies were carried out. We found such studies for Germanyand Switzerland, for other countries we estimate these numbers as good as we can.The case of Switzerland is particular interesting since the effect of the dark sector
EPAR – COVID-19 3 which started to play a non-negligible role for the overall dynamics of the epidemicin the later part of 2020 for the majority of the countries discussed here (India, USA,Brazil, France) can be studied there particularly well. For that reason we begin withthis country and discuss the role of the dark sector in detail.The paper closes with a discussion about what one can learn from the applicationsof the SEPAR model. We address three topics: The role of the constancy intervals,the role of the dark sector and the the influence of the time between infection andquarantine. The latter is perhaps the most striking application of our model offeringa door for flattening the curves by sending people faster into quarantine, a restric-tion which imposes much less harm to the society than other means.
PART I: Theoretical framework The SEPAR model and its comparison with other models
The SEPAR d model. We begin by pointing out that we have changed ournotation from [10]. The compartment consisting of those who are isolated after sentto quarantine or hospital, which there was called I , is now being denoted by A like actually infected, in some places also described – although a bit misleading – as“active” cases (e.g. in Worldometer). This is why we speak now of the SEPARmodel rather than of SEPIR.Let us first recall the compartments introduced in [10]. We observe 5 compart-ments which we call S , E , P , A , R , which people pass through in this order: Sus-ceptibles in compartment S moving after infection to compartment E , where theyare exposed but not infectious, after they are infected by people from compartment P which comprises the actively infectious people, those which propagate the virus.After e days they move from E to the compartment P , where they stay for p days.After diagnosis they are sent into quarantine or hospital and become members ofthe compartment A , where they no longer contribute to the spread of the virusalthough they are then often counted as the actual cases of the statistics. In ordernot to overload the model with too many details, we pass over the recording delaybetween diagnosis and the day of being recorded in the statistics. After another q days the recorded infected move from compartment A to the compartment R of removed (recovered or dead).We add two more compartments reflecting the role of the dark sector. There aretwo types of infected people, those who will at some moment be tested and counted,and those who are never tested, which we call people in the dark. This suggeststo decompose compartment P into two disjoint sub-compartments: P c of people M. KRECK, E. SCHOLZ who after p c days will be tested and counted and move to compartment A , and thecollection P d of people who after a longer period of p d days get immune and so moveinto a new compartment R d of removed people in the dark. To distinguish theseremoved people in the dark from those who come from compartment P after recoveryor death we introduce another new compartment R c of those removed people whooccur in the statistics. Of course R = R c ∪ R d .The introduction of the dark sector in addition to the sector of counted peopleleads to the picture that for the infected persons leaving compartment E there is abranching process: a certain fraction α ( k ) of people from E moves to compartment P c at day k , whereas the fraction 1 − α ( k ) of people moves to compartment P d .The existence of these compartments is a fundamental assumption which distin-guishes the SEPAR d model from many other models including standard SIR. Theexistence of these compartments is closely related to our picture of an epidemic likeCovid-19. Of course this is a simplification. If one assumes that the passage fromcompartment S to compartment E takes place at a certain moment, the durationof the stay in the next compartments varies from case to case. But it looks naturalto take the average of these durations leading for the different lengths e , p c , p d , and q . All these have to be estimated from available information.Once one has agreed to this there is another fundamental assumption. Thisconcerns the dynamics of the epidemic. Each person in compartment P has acertain average number κ ( k ) of contacts at day k . Depending on the strength of theinfectious power of an individual the contacts will lead to newly exposed people. Itis natural to model this development of the strength of infectiousness by a function A ( τ ), which measures the strength τ days after entry into compartment P . Againwe simplify this very much, by replacing A ( τ ) by a constant γ , the average valueof this assumed function. We will discuss this assumption later on in the lightof information available for Covid-19. Given the parameters κ ( t ) and γ our nextassumption is that, if we ignore the dark sector and set η ( k ) := γκ ( k ), the dynamicsof the infection can be described by the following formula:(1) E new ( k ) = η ( k − S ( k − N P ( k − . Here E new ( k ) is the number of additional members of compartment E at day k infected at day ( k −
1) by people from compartment P and N the total number ofthe population. This is a very plausible formula. We call η ( k ) the daily strength ofinfection . It is an integrated expression for the averaged contact behaviour of thepopulation and the aggressiveness of the virus.This is the dynamics if we ignore the dark sector. But members of compartment P d also infect. We assume that the contacts are equal to those in compartment P c .But the average of the strength of infection of people from P d may be smaller than EPAR – COVID-19 5 for those in compartment P c , since in general they can be expected to stay longer intheir compartment until they are immune and the strength of infection goes furtherdown. Thus we introduce a separate measure γ c for those in compartment P c and γ d = ξγ c , with 0 ≤ ξ ≤
1, for those in compartment P d .Using this the equation (1) has to be replaced by:(2) E new ( k ) = s ( k − η ( k − P c ( k −
1) + ξP d ( k − . Given this infection equation the rest of the model just describes the time shiftingpassage of infected from one compartment into the next and counts their cardinalityat day k . As usual we denote the latter by S ( k ), E ( k ), P c ( k ), P d ( k ), A ( k ), R c ( k )and R d ( k ). How such a translation is justified is explained in [10]. So we can justwrite down the self explaining formulas here: Introduction (definition) of the discrete SEPAR d model: Let e , p c , p d , q be integers standing for the duration of staying in the corresponding compart-ments, ≤ α ( k ) ≤ be branching ratios at day k between later registered in-fected and those which are never counted, η ( k ) be positive real numbers describ-ing the daily strength of infection for k ≥ , while η ( k ) = 0 for k < , and ξ ≤ a non-negative real number. Using s ( k ) = S ( k ) N the quantities S ( k ) , E ( k ) , E new ( k ) , P c ( k ) , P d ( k ) , A ( k ) , R c ( k ) , R d ( k ) of the SEPAR d model are given by a) the start condition:since the model is recursive we need an input for the first e + p d days (whichwe shift to negative values of k ), i.e. start data E start ( k ) for − ( e + p d ) ≤ k ≤ , while E start ( k ) = 0 for all k > , P c ( k ) = P d ( k ) = A ( k ) = R c ( k ) = R d ( k ) = 0 (or some other well definedstart values, cf. sec. 2.3) for k < − ( e + p d ) ; b) the recursion scheme for k ≥ − ( e + p d ) : E new ( k ) = s ( k − η ( k − P c ( k −
1) + ξP d ( k − E start ( k ) E ( k ) = E ( k −
1) + E new ( k ) − E new ( k − e ) P c ( k ) = P c ( k −
1) + α ( k ) E new ( k − e ) − α ( k − p c ) E new ( k − e − p c ) P d ( k ) = P d ( k −
1) + (1 − α ( k )) E new ( k − e ) − (1 − α ( k − p d )) E new ( k − e − p d ) A ( k ) = A ( k −
1) + α ( k − p c ) E new ( k − e − p c ) − α ( k − p c − q ) E new ( k − e − p c − q )] R c ( k ) = R c ( k −
1) + α ( k − p c − q ) E new ( k − e − p c − q ) R d ( k ) = R d ( k −
1) + (1 − α ( k − p d )) E new ( k − e − p d ) S ( k ) = N − E ( k ) − P ( k ) − A ( k ) − R ( k ) with P ( k ) = P c ( k ) + P d ( k ) , R ( k ) = R c ( k ) + R d ( k ) M. KRECK, E. SCHOLZ
The reason for this definition is easy to see. Additional people in, for example,compartment P c at the day k are ( P c ) new ( k ) = α ( k ) E new ( k − e ), while α ( k ) E new ( k − ( e + p )) move to the next compartment. Similar formulas hold for the compartments P d , A , R c and R d .An important parameter in an epidemic is the reproduction number ρ , the numberof people infected by a single infectious person during its life time. If we assumethat κ ( k ) and s ( k ) may be considered as constant during p d days about k , we canderive this number from the equations. It is ρ ( k ) = (1 + δ ) − κ ( k ) s ( k ) (cid:0) γ c p c + δγ d p d (cid:1) , where δ = − αα . For s ( k ) = 1 it is usually called the basic reproduction number, inorder to distinguish it from the effective reproduction number with s ( k ) <
1. In partI of this paper we usually mean the basic reproduction number when we speak ofreproduction number, while in the part II the decreasing s ( k ) hast to be taken intoaccount and we usually speak of the effective reproduction number, also without useof the attribute “effective”.If we set α ( k ) = 1, P d = 0 and R d = 0, we obtain the SEPAR model without darksector as a special case of the SEPAR d model. Effects of vaccination can easily beimplemented by sending the according number of persons directly from S to R .For later use the following observation is useful. The number of people in a givencompartment at day k is the sum of additional entries at previous days, for example E ( k ) = e − (cid:88) j =0 E new ( k − j ) ,P c ( k ) = α ( k ) p c − (cid:88) j =0 E new ( k − e − j ) , and so on for P d ( k ) and A ( k ).We abbreviate(3) H ( k ) = E ( k ) + P ( k ) + A ( k ) + R ( k ) , the number of herd immunized (without vaccination). Then the recursion schemeimplies: H ( k ) − H ( k −
1) = E new ( k )Putting this into the formula above: E ( k ) = (cid:80) e − j =0 E new ( k − j ), we obtain: E ( k ) = H ( k ) − H ( k − e )and similarly P c ( k ) = α ( k ) H ( k − e ) − α ( k − p c ) H ( k − e − p c )) , (4) P d ( k ) = (1 − α ( k )) H ( k − e ) − (1 − α ( k − p d )) H ( k − e − p d )) , EPAR – COVID-19 7 A ( k ) = α ( k − p c ) ( H ( k − e − p c ) − α ( k − p c − q ) H ( k − e − p c − q )) . Using R c ( k ) − R c ( k −
1) = α ( k − p c − q ) E new ( k − e − p c − q ) = α ( k − p c − q ) H ( k − e − p c − q ) − α ( k − p c − q − H ( k − e − p c − q −
1) we conclude: R c ( k ) = α ( k − p c − q ) H ( k − e − p c − q )and similarly R d ( k ) = (1 − α ( k − p d )) H ( k − e − p d ) . This gives a very simple structure of the model in terms of a single recursion equa-tion.
SEPAR d -model: The recursion scheme of the SEPAR d model is given by a singlerecursion equation: H ( k ) − H ( k −
1) =(5) s ( k − η ( k − (cid:16) α ( k − H ( k − − e ) − α ( k − − p c ) H ( k − − e − p c )+ ξ (cid:2) (1 − α ( k − H ( k − − e ) − (1 − α ( k − − p d )) H ( k − − e − p d )) (cid:3)(cid:17) and the functions S ( k ) , E ( k ) , P c ( k ) , P d ( k ) , A ( k ) , R c ( k ) and R d ( k ) are given interms of H ( k ) by the equations above.If we pass from a daily recursion to a infinitesimal recursion, replacing the differ-ence equation by a differential equation, we obtain the continuous recursion scheme,where now all functions are differentiable functions of the time t : H (cid:48) ( t ) = s ( t ) η ( t ) (cid:16) α ( t ) H ( t − e ) − α ( t − p c ) H ( t − e − p c ))+ ξ (cid:2) (1 − α ( t )) H ( t − e ) − (1 − α ( t − p d )) H ( t − e − p d ) (cid:3)(cid:17) (6)In both cases, discrete and continuous, consistent start conditions in an interval oflength e + p have to be added. For the discrete case see sec. 2.3. If we remove thedark sector, the continuous model was independently obtained in [12].The branching ration α and with it the number δ ≈ − αα of unrecorded infected foreach newly recorded one varies drastically in space and time, roughly in the range1 ≤ δ ≤
50. For Switzerland and Germany serological studies in late 2020 conclude δ ≈
2, for the USA a recent study finds δ ≈ M. KRECK, E. SCHOLZ serological study found values indicating δ ≈ For our choice of the modelparameter see below, section 3.Besides the determination of α one needs to know the difference between p c and p d and between η c ( k ) and η d ( k ), if one wants to apply the SEP AR d -model. Asexplained above we estimate p c = 7. The mean time of active infectivity of peoplewho are not quarantined seems to be not much longer, although in some cases it is.According to the study [21, p. 466] “no isolates were obtained from samples takenafter day 8 (after occurrence of symptoms) in spite of ongoing high viral loads”.This allows to work with an estimate p d = 10, and so it is not much larger than p c .A comparison of the SEP AR d model with a simplified version, where we assume p c = p d =: p and η c ( k ) = η d ( k ) =: η ( k ) shows that with these values the differenceis very small (see figure 1). In the following we therefore work with the simplified SEP AR : d model setting p c = p d =: p .Recent studies indicate that the number of asymptomatic infected is often as lowas about 1 in 5 symptomatic unrecorded and is thus much smaller than originallyexpected [13]. Although asymptomatic infected are there reported to be considerablyless infective than the symptomatic ones, their relatively small number among allunreported cases justifies to work in the simplified dark model with the assumption η d ( k ) ≈ η c ( k ) =: η ( k ). If we set P ( k ) = P c ( k ) + P d ( k ) as above, we see that P c ( k ) = α ( k ) P ( k ) and P d ( k ) = (1 − α ( k )) P ( k ). Mar Apr May Jun02 × × × × × Figure 1.
Comparison of SEPAR d model for A ( k ) between dark sec-tor with p c = 7 , p d = 10 , ξ = 0 . p c = p d = p = 7 , ξ = 1 (dashed blue), assuming constant η .In part II we discuss how the time dependent parameter η ( k ) can be derived fromthe data and a rough estimate of the dark factor δ can be arrived at, although itlies in the nature of the dark sector that information is difficult to obtain. For Germany see [8, 18], for Switzerland [11] announcing a forth-coming study of
Corona-Immunitas , for USA [15] and for In-dia a report in ANI retrieved12/21 2020.
EPAR – COVID-19 9
A comparison of the SEPAR model ( δ = 0) with the simplified SEPAR d model isgiven in fig. 2 for a constant parameter α = 0 .
2, respectively dark factor δ = 4 andconstant reproduction coefficient ρ = 3. This illustrates the influence of the darkfactor from the theoretical viewpoint. Mar Apr May Jun01 × × × × × Mar Apr May Jun02 × × × × Figure 2.
Comparison of the course of an epidemic with constantreproduction number ρ = 3 without dark sector (dashed), and withdark sector δ = 4 (solid lines): Left counted number of actual infected A c ( k ) (blue). Right: total number of confirmed infected A tot c ( k ) = A c ( k ) + R c ( k ) (brown).1.2. The S(E)IR models and their assumptions.
Whereas the derivation ofthe SEPAR d -model is based on the idea of disjoint compartments, infected peoplepass through in time, there is a different approach with goes back to the seminalpaper [9]. A special case is the standard SIR-model or SEIR model. It seems thatmost people use this as a black box without observing the assumptions on which itis built. One should keep these assumptions in mind whenever one applies a model.There is a modern and easy to understand paper by Breda, Diekmann, and de Graafwith the title: On the formulation of epidemic models (an appraisal of Kermack andMcKendrick) [4], which explains the general derivation. In the introduction the au-thors state that the Kermack/McKendrick paper was cited innumerable times andcontinue: ”But how often is it actually read? Judging from an incessant miscon-ception of its content one is inclined to conclude: hardly ever! If one observes theprinciples from which the S(E)IR models are derived one should be hesitant to applyit to Covid-19.Following [4] we shortly repeat the assumptions on which the general Kermack/McKendrickapproach is based . The general model considers a function S ( t ) := density (number per unit area) of susceptibles at time tand related to this a function F ( t ) M. KRECK, E. SCHOLZ called the force of infection at time t. By definition, the force of infection is theprobability per unit of time that a susceptible becomes infected. So, if numbers arelarge enough to warrant a deterministic description, we have I new ( t ) = F ( t ) S ( t ) , where I new ( t ) is defined as the number of new cases per unit of time and area. Thefunctions S and I are related by the equation: S (cid:48) ( t ) = − F ( t ) S ( t )Then the central modelling ingredient is introduced: A ( τ ) := expected contribution to the force of infection τ units of time ago.Alone from this ingredient an integral differential equation is derived, which givesthe model equations. For more details we also refer to a recent paper by RobertFeßler who derived the integral equation independently [7].Already here we see a different view of an epidemic. No compartments and theircardinality are mentioned; in their place the authors mention only certain functions.If the function A is assumed to decay exponentially, A ( τ ) = αe − βτ with constants α, β . The model derived from this input is called the standard SIR-model . It leads to two ordinary differential equations in the variable t : I (cid:48) = αsI − βIR (cid:48) = βIS ( t ) = N − I ( t ) − R ( t ) , where N as before is the number of the population and s ( t ) = S ( t ) N .For the standard SEIR-model there is an additional function E ( t ) measuring the exposed and the input function is now A ( τ ) = α ββ − γ ( e − γτ − e − βτ )This leads to 3 ordinary differential equations in tE (cid:48) = αsI − βEI (cid:48) = βE − γIR (cid:48) = γIS ( t ) = N − E ( t ) − I ( t ) − R ( t ) , where N as before is the number of the population. The infection function A ( τ )considered here determines the convolution part of an integral kernel in Feßler’sapproach mentioned above. EPAR – COVID-19 11
If Breda et al. are right, readers should be critical to papers applying the S(E)IRmodels without explaining why the models, given their fundaments, are applicable.As far as we can see, the assumption of an exponential decay A ( t ) is often notmentioned by authors applying it in situations where it would be necessary to discusswhether this assumption can be reasonably made. In a situation like Covid-19 whereinfectious people are isolated as soon as possible, it seems questionable whetherthis assumption holds. We are surprised that in most of the papers we have seen,which apply the S(E)IR model to an analysis of Covid 19, this problem is noteven mentioned. This includes the papers of the group around Viola Priesemannwhich play an important part in the discussion about how to deal with Covid 19 inGermany [6], [5].1.3. Comparing SIR with SEPAR.
When we want to compare the SIR modelswith the delay SEPAR model we have to lower, in a first step, the number of com-partments by removing E , P and A and to replace them by a single compartment,called I , of infected people which are at the same time infectious. For this model weassume that infected susceptibles move right away to compartment I , where theystay for p days. In contrast to the SEPAR model it is assumed that these peopleare counted as actual infected people at the moment they are infected. After p daysthey are counted as recovered or dead. So it is a strong simplification of the SEPARmodel, but it follows the same pattern as the SEPAR model since it is a delaymodel. We call it d-SIR-model (“d-”for delay) to distinguish it from the standardSIR-model. The equations for this model are based on the same principles as theSEPAR model: The continuous delay d-SIR model:
Let p be a positive real number standingfor the duration of staying in the compartment I of infected and infectious people.Let η ( t ) be a differentiable function measuring the strength of infection (includingthe effects of social constraints). The quantities S ( t ) , I ( t ) , R ( t ) of the delay SIRmodel are given by a) the start condition:A differentiable function I ( t ) for ≤ t < p , b) and the delay differential equations: I (cid:48) ( t ) = η ( t ) s ( t ) I ( t ) − η ( t − p ) s ( t − p ) I ( t − p ) R (cid:48) ( t ) = η ( t − p ) s ( t − p ) I ( t − p ) S ( t ) = N − I ( t ) − R ( t )To compare this model with the standard SIR-model above we note that also thed-SIR model (like the continuous SEPAR d model) can be derived from the principlesof Kermack/McKendrick, as explained in [7]. One only has to take the product ofthe characteristic function of the interval [0 , p ] with η = γκ as the function A ( τ ). M. KRECK, E. SCHOLZ
To compare the two models one has to relate the input parameters. In the case ofthe d-SIR model they are η (for the comparison we assume that the contact rate isconstant) and p , whereas for the SIR-model they are α and β . The role of η is that of α in the SIR-model, so we set α = η . There are several ways to relate the paramter β of the SIR model with p occurring in the d-SIR model. One is to assume that thetotal force of infection has to be the same if they describe the same developments,i.e. with the function A ( τ ) which is the product of the characteristic function of theinterval [0 , p ] with η one has the condition: (cid:90) ∞ αe − βt dt = (cid:90) ∞ A ( t ) dt Then the second relation: αβ = p η = p α and thus β = 1 p . In both cases the reproduction number is αβ = p α = p η .If one applies this then there is a problem to find parameters so that at least atthe beginning the two models are approximatively equal. Thus one can relate thetow models in a second way by choosing the parameters so that this is the case. Forthis we fix values for α and β and chose the start conditions of the d-SIR model sothat they agree with the SIR-model during the first days. By construction of theSIR-model the function I is nearly an exponential function as long as the function S is nearly constant. Thus we chose the same exponential function as start valuesfor the d-SIR model.Then the question is whether there are differences of the model curves in the longrun and how large the differences are. One should expect that the assumptions ofan exponential decay regulating the strength of infection of an infectious person inthe case of the SIR-model versus a period of p days, where the strength of infectionis constant and after that goes immediately down to 0 in the case of the delay d-SIR, should result in higher values for the functions I ( t ) and I tot ( t ) = I ( t ) + R ( t ),the total number of infected until time t of the SIR model. The following graphicsin which we assume a constant reproduction rate slightly above 1 show, in fact, adramatic difference supporting the expectation. A similar observation can be foundin [7, fig. 5, 6]This has an interesting consequence for a situation in which a high rate of immu-nity is achieved either by “herd” effects or by vaccination. According to a simple SIRmodel with constant reproduction number ρ = 1 . . ·
80 = 24 million people wouldhave to be infected or vaccinated to achieve herd immunity, whereas according to thedelay model “only” about 0 . ·
80 = 12 million have to be infected (see fig. 3). Thedifference corresponds to the fact that equal initial exponential growth is related to
EPAR – COVID-19 13
100 200 300 4005000001.0 × × ×
100 200 300 4000.10.20.30.40.50.6
Figure 3.
Comparison of SIR (black) and dSIR (blue) for s ( t ) ≈ N = 80 M. Left: Number of infected I ( t ). Right: H ( t ) /N with H ( t ) total number of infected up to time t . Parametervalues: N = 8 M, I = 1 k, α = η = 0 . , β = 0 . p = 8 .
11 for dSIR.different reproduction rates in the two models of the example given: ρ SIR = αβ = 1 . ρ d − SIR = p η ≈ .
2. Such a difference matters because, according to the plausi-bility arguments given above, the delay model may very well be more realistic thanthe SIR model for Covid-19.Next we discuss the differences between the SIR model and the delay SIR modelduring a time when s is still approximately equal to 1, both have constant reproduc-tion numbers, and α = η like above. Moreover we assume that the initial growthsfunctions of both models are approximately identical to the same exponential func-tion (because both are designed to modelling the same growth process).In reality one observes longer periods in the data where the reproduction numberis approximately constant until it changes in a short transition period to a newapproximately constant value. Such changes may be due to containment measures(non-pharmaceutical interventions) imposed by governments, which influence thecontact rate κ ( t ). In the next graphics we show the effect of such a change for bothmodels.In figure 4 we let the dSIR and the SIR curves start with identical exponentialfunctions based on constant reproduction numbers. Then we lower the reproductionrate by 40 percent within three days. As expected the SIR curves have a cusp sinceone exponential function jumps into another, whereas the dSIR equation due tothe delay character shows a slightly smoother transition. The second and moredramatic effect is that a similar phenomenon like in the long term comparison canbe observed: the SIR solution is far above the dSIR solution. The reason seems tobe the same, the different assumptions made by the two approaches about the decayof the strength of infection. M. KRECK, E. SCHOLZ
20 40 60 80 10020004000600080001000012000 20 40 60 80 1002000400060008000100001200020 40 60 80 10020004000600080001000012000
Figure 4.
Comparison of I ( t ) for SIR (top left) and dSIR (top right)for s ( t ) ≈ t ≤
29 and t ≥
31 withidentical exponential increase in the initial upswing. Reduction ofreproduction rate by 40 % in both cases. Bottom: SIR black, dSIRblue. Parameter values: N = 8 M, I = 1 k, α = η = 0 . , β = β = 0 . , α = η = 0 . , β = β ; dSIR p = 8 . d model to data of selected countries. Here the model shows its high quality.Since data about the dark sector are insecure we check how much the dark sectorinfluences the overall dynamics of the epidemic in the discussed countries up to thepresent (until the end of 2020) and choose the dark factor of the model on the basisof the analysis and given estimates for the respective countries. EPAR – COVID-19 15
PART II: Applications of the SEPAR model Determining empirical parameters for the model
JHU data.
The basic data sets (JHU).
The worldwide data provided by the Humdata project(Humanitarian Data Exchange) of the
Johns Hopkins University provides data onthe development of the Covid-19 pandemic for more than 200 countries and territo-ries. The data are compressed into 3 basic data sets for each country/territory
Conf ( k ) , Rec ( k ) , D ( k )where Conf ( k ) denotes the total number of confirmed cases until the day k (startingfrom January 22, 2020), Rec ( k ) the number of reported recovered cases and D ( k )the number of reported deaths until the day k . The last two entries can be combinedto the number of redrawn persons of the epidemic, captured by the statistic,ˆ R ( k ) = Rec ( k ) + D ( k ) . Empirical quantities derived from the JHU data set will be endowed with a hat, likeˆ R , to distinguish them from the corresponding model quantity, here R .The (first) differences of Conf ( k ) encode the daily numbers of newly reported and acknowledged cases:(7) ˆ A new ( k ) = Conf ( k ) − Conf ( k − A tot ( k ) = k (cid:88) j =1 ˆ A new ( j ) = Conf ( k ) , while the difference(9) ˆ A ( k ) = Conf ( k ) − ˆ R ( k ) , is the empirical number of acknowledged , not yet redrawn, actual cases. Some au-thors call it the number of “active cases”; but this is misleading because the phaseof effective infectivity is usually over as soon as an infection is diagnosed and theperson is quarantined.The number Rec ( k ) of recovered people is often reported with much less care thanthe daily new cases and the deaths. By this reason the recorded number of redrawn, https://data.humdata.org/dataset/novel-coronavirus-2019-ncov-cases E.g. in [19, p. 182] . . . A similar identification underlies the numbers for the active case in the
Worldometer . M. KRECK, E. SCHOLZ ˆ R ( k ), may be heavily distorted, with the result that neither itself nor the derivednumbers ˆ A ( k ) can be be taken at face value. The most reliable basic data remaintherefore ˆ A new ( k ) , Conf ( k ) and D ( k ) . Even ˆ A new ( k ) has its peculiarities due to the weekly cycle of reporting activities. Inthis paper we abstain from discussing mortality rates and consider the first two datasets of the mentioned three only. ˆ R and ˆ A play an important role for a completeimage of an epidemic, but they are reliable only for a few countries; for the ma-jority of countries they have to be substituted or complemented by more adequatequantities derived from the basic data (see eq. 12). Smoothing the weekly oscillations of ˆ A new . For all countries the reported numberof daily new infections shows a characteristic 7-day oscillation resulting from thereduction of tests over weekends and the related delay of transmission of data. A3-day sliding average suppresses fluctuations on a day-to-day scale and shows theweekly oscillations even more clearly.
Mar May Jul Sep Nov Jan05000100001500020000250003000035000
Mar May Jul Sep Nov Jan0500010000150002000025000 Mar May Jul Sep Nov Jan0500010000150002000025000
Figure 5.
Acknowledged cases ˆ A new ( k ) for Germany (left), sliding 3-day and 7-day aver-ages ˆ A new, ( k ), ˆ A new, ( k ) (middle) and ˆ A new, ( k ) (right) In some countries these oscillations are corrected for transmission delay by centralinstitutions, but such corrections are not implemented in the JHU data. A simplemethod for smoothing the weekly oscillations consists in using sliding centred A new, ( k ) = 17 (cid:88) j = − ˆ A new ( k + j )and similarly for the centred 3-day average ˆ A new, ( k ). Note that in order to avoida time shift effect which would arise from using a purely backward sliding average,we use a sliding average over 3 days forward and 3 days back. For most countriesthis suffices for carving out the central tendency of the new infections quite clearly(fig. 5). This is done by the
Robert Koch Institut for the German case.
EPAR – COVID-19 17
For some countries already the daily fluctuations of ˆ A new are extreme. The Frenchdata even indicate negative values for ˆ A new for certain days, although this oughtto be excluded by principle. Such effects indicate a highly unreliable system ofdata recording and transmission; they may be due to ex post corrections of earlierexaggeration of transmitted numbers. But even under such extreme conditions thesliding 7-day average leads to reasonable information on the mean motion of the newinfections, so that we don’t have to exclude such countries from further consideration(sec. 3.1).2.2. Data evaluation.
The “actual” cases in the statistical sense.
The difference ˆ A ( k ) = ˆ A tot ( k ) − ˆ R ( k ) (8)can in principle be considered as an expression for the number of actual cases; butit is corrupted by the fact that the number of daily recovering Rec ( k ) is irregularlyreported. For a critical investigation of this number we start from the truism thatany actually infected person recorded at day k has been counted among the ˆ A new ( l )at some earlier day, l ≤ k . The smallest number ˆ q ( k ) of days preceding k (includingthe latter) necessary for supplying sufficient large numbers of infected ˆ A ( k ),(11) ˆ q ( k ) = min l (cid:104) l (cid:88) j =0 ˆ A new ( k − j ) ≥ ˆ A ( k ) (cid:105) , is a good indicator for the mean time of sojourn in the collective of infected whichare recorded as “actual cases”. As long as the number of severely ill among allinfected persons is relatively small and the time of severe illness well constrained,we may expect that the mean time of actual illness does not deviate much from thetime of prescribed minimal time of isolation q min for infected persons. In the case ofCovid-19 q ( k ) surpasses q min ≈
14 only moderately for India, Germany, Switzerlandetc. (fig. 6). For many other countries ˆ q ( k ) behaves differently. It starts near thetime of quarantine or isolation but increases for a long time monotonically withthe development of the epidemic, before often – although not always – it starts todecrease again after the (local) peak of a wave has been surpassed. This is the case,e.g., for Italy and the US; in the last case the deviation is extreme, ˆ q ( k ) surpasses100 and shoots up a little later (see fig. 7)This effect cannot be attributed to medical reasons; the major contribution ratherresults from the unreliability of the statistical book keeping: With the growingoverload of the health system, the time of recovery of registered infected persons isbeing reported with an increasing time delay, sometimes not at all (e.g., Sweden,UK). The difference ˆ A ( k ) = ˆ A tot ( k ) − ˆ R ( k ) gets increasingly confounded by the lackof correctness in the numbers Rec ( k ). In these countries it is an expression of thenumber of “statistically actual” cases only with, at best, an indirect relation to thereal numbers of people in quarantine or hospital. M. KRECK, E. SCHOLZ
May Jul Sep Nov05101520
Mar May Jul Sep Nov0510152025
Figure 6.
Daily values ˆ q ( k ) for the mean time of being statistically counted as an actualcase for India (left) and Germany (right) Mar May Jul Sep020406080
May Jul Sep Nov Jan050100150200250
Figure 7.
Daily values of the mean time ˆ q ( k ) of being statistically counted as actualinfected for Italy (left) and USA (right) The information gathered for Covid-19 proposes the existence of a stable meantime q of isolation of infected persons (including hospital) for long periods in eachcountry. It is usually a few days longer than the official duration of quarantineprescribed by the health authorities. Given q , the sum(12) ˆ A q ( k ) = q − (cid:88) j =0 ˆ A new ( k − j )can be used as an estimate of the number of infected recorded persons who arein isolation or hospital at the day k . Here we do not use 7- or 3- day averages,because the summation compensates the daily oscillations anyhow. The accordinglycorrected number of redrawn ˆ R q is of course given by(13) ˆ R q = ˆ A + ˆ R − ˆ A q . Figure 8 shows ˆ A ( k ) , ˆ A q ( k ) for the USA (with q = 15). It demonstrates thedifference between ˆ A ( k ) (dark blue) and ˆ A q ( k ) (bright blue) and shows that ˆ A q ( k )is a more reliable estimate of actually infected than the numbers ˆ A ( k ) (the “activecases” of the Worldometer).For countries with reliable statistical recording of the recovered we find ˆ q ( k ) ≈ const . In this case we choose this constant as the value for the model q . For other EPAR – COVID-19 19
Mar May Jul Sep Nov Jan02.0 × × × × × × × Figure 8. ˆ A ( k ) (dark blue), ˆ A q ( k ) (bright blue) for the USA ( q = 15) countries one may use a default value, inferred from comparable countries with abetter status of recording the Rec ( k ) data (i.e. ˆ A ( k ) ≈ ˆ A q ( k )). Simplifying assumptions on the duration e of exposition and the duration p of ef-fective infectivity. For Covid-19 it is known that there is a period of duration say e between the exposition to the virus, marking the beginning of an infection, and theonset of active infectivity. Then a period of propagation, i.e. effective infectivity,with duration p follows, before the infection is diagnosed, the person is isolated inquarantine or hospital and can no longer contribute to the further spread of thevirus. Although one might want to represent the mentioned durations by stochasticvariables with their respective distributions and mean values, we use here the meanvalues only and make the simplifying assumption of constant e and p approximatedby the nearest natural numbers.The Robert Koch Institute estimates the mean time from infection to occurrenceof symptoms to about 4 days [16], (5.). This is divided into e plus the time fromgetting infectious to the occurrence of symptoms. According to studies alreadymentioned above the latter is estimated as 2 days, so as a consequence we estimate e = 2. In section 3 we generically use p = 7. We have checked the stability of themodel under a change of the conventions of parameter choice inside the mentionedintervals. Estimate of the daily strength of infection.
As announced in part I we work withthe simplified SEPAR d model. This means that the duration in compartments P c and P d is equal, here denoted by p , and also the strength of infection is assumed tobe equal: η c = η d = η . Furthermore, if P ( k ) = P c ( k ) + P d ( k ) there is a branchingratio α , which has to be estimated for each country, such that P c ( k ) = αP ( k ) and P d ( k ) = (1 − α ) P ( k ). For every counted infected there are then δ = 1 − αα uncounted ones. We call δ the dark factor . M. KRECK, E. SCHOLZ
Once e and p are given (or fixed by convention inside their intervals) one can de-termine the empirical strength of infection η ( k ) using the model equations. Namely η ( k ) = E new ( k + 1) s ( k ) P ( k ) . In terms of the total number of infected H ( k ) (see (4) and with constant α this is η ( k ) = H ( k + 1) − H ( k ) s ( k ) (cid:16) α (cid:0) H ( k − e ) − H ( k − ( e + p c )) (cid:1) + ξ (1 − α ) (cid:0) H ( k − e ) − H ( k − ( e + p d ))) (cid:1)(cid:17) For the simplified SEPAR d model we have αE new ( k ) = A new ( k + e + p ) and αP ( k ) = (cid:80) pj =1 ˆ A new ( k + j ). Thus α cancels and we obtain:(14) η ( k ) = A new ( k + e + p ) s ( k ) (cid:80) pj =1 A new ( k + j )Denoting, as before, the values we obtain from the data by ˆ η ( k ) etc. we obtain(15) ˆ η ( k ) = ˆ A new ( k + e + p )ˆ s ( k ) (cid:80) pj =1 ˆ A new ( k + j ) resp. ˆ η ( k ) = ˆ A new, ( k + e + p )ˆ s ( k ) (cid:80) pj =1 ˆ A new ( k + j ) , where in the second equation we work with the weekly averaged data.An estimation of the total number of new infections induced by an infected personduring the effective propagation time (and thus the whole time of illness) is then(16) ˆ ρ ( k ) = p − (cid:88) j =0 ˆ η ( k + j )ˆ s ( k + j ) ;and similarly for ˆ ρ ( k ). In the following we generally use the latter but write justˆ ρ ( k ). Note also that the determination of the strength of infection ˆ η ( k ) by (15)needs an estimation of the dark factor δ because the latter enters into the ratio ofsusceptibles ˆ s ( k ),while it cancels in the calculation of the empirical reproductionrate ˆ ρ ( k ) (17).This is an empirical estimate for the reproduction number ρ ( k ). In periods ofnearly constant daily strength of infection one may use the approximation(17) ˆ ρ ( k ) ≈ p ˆ η ( k )ˆ s ( k ) = p A new, ( k + e + p ) (cid:80) pj =1 A new ( k + j )Inspection of transition periods between constancy intervals for Covid 19 shows thatthis approximation is also feasible in such phases of change. For p = 7 this variantof the reproduction number stands in close relation to the reproduction numbersused by the Robert Koch Institut , see appendix 4), which gives additional supportto this choice of the parameter.
EPAR – COVID-19 21
SEPAR d parameters. For modelling Covid-19 in the simplified dark ap-proach we use the parameter choice e = 2 , p (= p c = p d ) = 7 as explained in sec.2.2. Where we differentiate between p c and p d we usually use p c = 7 and p d = 10.The value for q depends on the reported mean duration of reported infected beingcounted as “actual (active)” case for each country (sec. 2.1); in the following reportsit usually lies between 10 and 17. For each country we let the recursion start atthe first day k at which the reported new infections become “non-sporadic” in thesense that no zero entries appear at least in the next e + p d days ( ˆ A new ( k ) (cid:54) = 0 for k ≤ k ≤ k +( e + p d )). For k ≥ k − η ( k ) are calculated in the simplifiedcase, p = p c = p d according to (15). Otherwise the formula above (14) has to beused. Start values.
For the numerical calculations we use the recursion (5), with startvalues given by time shifted numbers of the statistically reported confirmed cases,expanded by the dark factor, for k in the interval J − = [ k − , k −
1] where k − = k − − ( e + p d ):ˆ H ( k ) = (1 + δ ) Conf ( k + e + p c ) = (1 + δ ) ˆ A tot ( k + e + p c )Note that the time step parametrization in the introduction/definition of the SEPARmodel in sec. 1.1 works with k = 1.If we set the model parameters η ( k ) = 0 for k < k − η ( k ) = ˆ η ( k ) for k ≥ k −
1, the recursion reproduces the data exactly, due to the definition ofthe coefficients. Then and only then it becomes tautological . Already if we usecoefficients ˆ η ( k ) for k ≥ k from time averaged numbers of daily newly reportedaccording to eq. (15), the model ceases to be tautological. In this case the parameter η = η ( k −
1) may be used for optimizing (root mean square error) the result forthe total number of reported infected A tot in comparison with the empirical data(8). The model acquires conditional predictive ability, if longer intervals of constantcoefficients are chosen.One may prefer to replace A new by the 7-day averages ˆ A new, by introducing Conf ( k ) = k (cid:88) j = k − ˆ A new, ( j ) (+ const )and ˆ H ( k ) = (1 + δ ) Conf ( k + e + p c ). For constant α ( k ) = α the replacement ofthe ˆ A new in the denominator of (15) then boils down to definingˆ η ( k ) = ˆ H ( k + 1) − ˆ H ( k )ˆ s ( k ) (cid:16) α (cid:0) ˆ H ( k − e ) − ˆ H ( k − ( e + p c )) (cid:1) + ξ (1 − α ) (cid:0) ˆ H ( k − e ) − ˆ H ( k − ( e + p d )) (cid:1)(cid:17) M. KRECK, E. SCHOLZ
Then the model becomes tautological also for the use of daily varying ˆ η ( k ) andceases to be so only after introducing constancy intervals as described in the nextsubsection. Main intervals.
The number of empirically determined parameters can be drasticallyreduced by approximating the daily changing empirical infection coefficients ˆ η ( k )( k = k , k + 1 , . . . ) by constant model values η j (1 ≤ j ≤ l ) in appropriately chosenintervals J , . . . J l . We call them the constancy, or main, intervals of the model.Their choice is crucial for arriving at a full-fledged non-tautological model of theepidemic.We thus choose time markers k j (“change points”) for the beginning of such inter-vals and durations ∆ j for the transition between two successive constancy intervals,such that: J j = [ k j , k j +1 − ∆ j +1 ] , j = 1 , . . . , l In the main interval J j the model strength of infection η j (this is the constant dailystrength of infection during this time) are generically chosen as the arithmeticalmean of the empirical values mean { ˆ η ( k ) | k ∈ J j } . Small deviations of the mean,inside the 1 σ range of the ˆ η -fluctuations in the interval, are admitted if in this waythe mean square error of the model A tot can be reduced noticeably. The dates t j ofchange points k j can be read off heuristically from the graph of the ˆ η and may beimproved by an optimization procedure. k is chosen as the first day of a period inwhich the daily strength of infection can reasonably be approximated by a constant.In the initial interval J = [ k , k −
1] the model uses the empirical daily strengthof infection: η ( k ) = ˆ η ( k ) for k ∈ J . In the transition intervals [ k j − ∆ j , k j ] themodel strength of infection is gradually, e.g. linearly, lowered from η j − to η j . For the labelling of the days k there are two natural choices; the JHU day count starting with k = 1 at 01/22/2020, or a country adapted count such that k = 1,where k labels the first day for which the reported new infections become non-sporadic (see above). Both choices have their pros and contras; in the following wemake use of both in different contexts, declaring of course which one is being used. Influence of the dark sector.
Proceeding in this way involves an indirect observanceof the dark sector’s contribution to the infection dynamics of the visible sector. Anunknown part of the counted new infections ˆ E new ( k ) is causally due to contacts withinfectious persons in P d of the dark sector, eqs. (2) or (5)). According to estimatesof epidemiologists there is a wide spectrum of possibilities worldwide, 0 ≤ δ ≤ p d = p c = p = 7 Here we use the smoothing function described in [10]. Also an elementary optimization proce-dure for determining the main intervals is described in this paper.
EPAR – COVID-19 23 and ξ = 1 ( γ c = γ d ) and use estimates for the dark factor δ explained in the respec-tive country section. 3. Selected countries/territories
In our collection we include four small or medium sized European countries(Switzerland, Germany, France, Sweden), three large countries from three differ-ent continents (USA, Brazil, India), and a model for the aggregated data of allworld countries and territories. For the first country analysed here, Switzerland,relatively reliable data on the dark sector are accessible. We take it as an examplefor a discussion of the effects of different assumptions on the branching ration α ,respectively the factor δ , of the dark sector. For the other countries we lay open onwhich considerations our choice of the model the dark factor is based.3.1. Four European countries; Switzerland Germany, France, Sweden.
The four European countries discussed here show different features with regard tothe epidemic: Switzerland and Germany have a relatively well organized health anddata reporting system; the overall course of the epidemic with wave peaks in earlyApril and in early November 2020 and a moderately controlled phase in between is istypical for most other European countries. France, in contrast, shows surprising fea-tures in the documentation of statistically recorded new infections (negative entriesin the first half of the year); and Sweden has been chosen because of a containmentstrategy of its own. In the case of Switzerland and Germany first results of represen-tative serological studies are available. They allow a more reliable estimate of thesize of the dark sector than in most other cases. We therefore start our discussionwith these countries.
Switzerland.
The numbers of reported new infected ceased to be sporadic in Switzer-land at February 29, 2020; we take this as our day t = 1. For the reported dailynew infections (3-day and 7-day sliding averages) see figure 9. At Feb 28 recom-mendations for hygiene etc. were issued by the Swiss government and large eventsprohibited, including Basel Fasnacht (carnival). These regulations were already ac-tive at our day t = 1 (Feb 29) and explain the rapid fall of the empirical strengthof infection (7-day averages) ˆ η ( k ) at the very beginning of our period. During thenext 15 days a series of additional general regulations were taken: Mar 13, ban ofassemblies of more than 100 persons, lockdown of schools; Mar 16 ( t = 12) lockdownof shops, restaurants, cinemas; Mar 20 ( t = 16), gatherings of more than 5 personsprohibited. This sufficed for lowering the growth rate quite effectively. The SEPARreproduction number started from peak values close to 5 and fell down to below thecritical value at March 23, the day t = 24 in the country count (fig. 10, left). Hereit remained with small oscillations until mid May, after which it rose (we choose M. KRECK, E. SCHOLZ t = 85, May 23, as the next time mark), with strong oscillations until late June,before it was brought down to close to 1 in late June ( t = 158, June 23). A longphase of slow growth ( ρ ≈ .
1) followed until mid September. An extremely swiftrise of the reproduction number to values above 2 in late September ( t = 242, Sep.26) brought the number of new infections to heights formerly unseen in Switzerland.In spite of great differences among the differently affected regions ( Kantone ) theepidemic was brought under control at the turn to November ( t = 272, Oct. 27). Mar May Jul Sep Nov Jan02000400060008000
Figure 9.
Daily new reported cases for Switzerland, 3-day slidingaverages ˆ A new, ( k ) and 7-day averages ˆ A new, ( k ). Mar May Jul Sep Nov Jan01234
Mar May Jul Sep Nov Jan0.00.20.40.60.8
Figure 10.
Left: Empirical reproduction rates ˆ ρ ( k ) for Switzerland.Right: Daily strength of infection ˆ η ( k ) (yellow) with model parame-ters η j (black dashed) in the main intervals J j for p = 7 and δ = 2.The empirical values of the infection strengths ˆ η ( k ) determined on the basis ofthe 7-day sliding averages ˆ A new, depend on estimates of the dark factor δ (cf. eq.15). Recent serological studies summarized in [11] conclude δ ≈
2. We choose thisvalue as generic for the simplified SEPAR d model. The values of ˆ η ( k ), assuming adark factor δ = 2, are shown in fig. 10, right. How its values are affected by differentassumptions for the dark sector can be inspected by comparing with the results for EPAR – COVID-19 25 δ = 0 and δ = 4 (fig. 11). The influence of the dark sector on ˆ η becomes visibleonly late in the year 2020. Mar May Jul Sep Nov Jan0.00.20.40.60.8 Mar May Jul Sep Nov Jan0.00.20.40.60.8
Figure 11.
Comparison of empirical values ˆ η ( k ) assuming dark fac-tor δ = 0 (left) and δ = 4 (right); for comparison with generic choice δ = 2 see last figure.With the time markers between different growth phases of the epidemic indicatedabove we choose the following main intervals for our model: J = [1 , , J =[24 , , J = [85 , , J = [119 , , J = [211 , , J = [242 , , J =[270 , t eod = 321) at Jan 14, 2021. The model values η j in theIntervals J j are essentially the mean values of ˆ η ( k ) in the respective interval, wheresmall deviations inside the 1 sigma domain are admitted if the (root mean square)fit to the empirical data ˆ A tot can be improved. They are given in the table below andindicated in fig. 10 (black dashed lines). (Note that η has no realistic meaning; it is afree parameter of the start condition for modelling on the basis of the 7-day averagesˆ η ( k ), see sec. 2.3.) The reproduction rates ρ j in the table refer to the beginningof the intervals; in later times the decrease of s ( k ) can lower the reproduction ratesuntil the end of the intervals considerably. In the case of Switzerland the lattercrosses the critical threshold 1 inside the last constancy interval (see below).Model η j and ρ j in intervals J j for Switzerland J J J J J J J η j -0.127 0.095 0.203 0.156 0.265 0.132 0.165 ρ j — 0.66 1.41 1.08 1.82 0.86 1.01The course of the new infections is well modelled with these values (fig. 12); thesame holds for the total number of counted infected (fig. 15 below).The count of days ˆ q ( k ), necessary for filling up the numbers of reported actualinfected by sums of newly infected during the directly preceding days shows a rel-atively stable value, ˆ q ( k ) ≈
15, until early November. Since then the mean timeof reported sojourn in the department A increases steeply (fig. 13, left). Initiallythis may have been due to a rapid increase of severely ill people during the second M. KRECK, E. SCHOLZ
Mar May Jul Sep Nov Jan02000400060008000
Figure 12.
Model reconstruction for the 7-day averages of the num-ber of new infected A new ( k ) (black dashed) in comparison with theempirical data, 3-day averages, ˆ A new, ( k ) (solid red) for Switzerland.wave of the epidemic; but the number does not fall again with the stabilization inDecember 2021. A comparison of the statistically recorded actual infected ˆ A ( k ) withthe q -corrected one ˆ A q ( k ) (eq. 11) shows an ongoing increase of the first while thesecond one falls again after a sharp peak in early November (same figure, right). Weconclude from this that from December 2020 onward the numbers of recovered areno longer reliably reported even in Switzerland. May Jul Sep Nov Jan05101520253035
Mar May Jul Sep Nov Jan050000100000150000
Figure 13.
Left: Daily values of the mean time of statistically actualinfection ˆ q ( k ) for Switzerland. Right: Comparison of reported infectedˆ A (dark blue) and q -corrected number ( q = 15) of recorded actualinfected ˆ A q (bright blue) from the JHU data in Switzerland.Both quantities cam be well modelled in our approach (fig. 14), although weconsider ˆ A q ( k ) as a more reliable estimate for actually ill persons.On this basis, the SEPAR d model expresses the development of the epidemic inSwitzerland on the basis of only 6 constancy intervals for the parameters η for thewhole year 2020. The three main curves of the total number of recorded infected, A tot ( k ), the number of redrawn R ( k ) and the number of diseased counted as statis-tical “actual” cases A ( k ) are shown in figure 15. EPAR – COVID-19 27
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Mar May Jul Sep Nov Jan020000400006000080000100000120000140000
Figure 14.
Left: Number of actual infected ˆ A ( k ) (blue) for Switzer-land, recorded by the JHU data, and model values calculated withtime dependent q ( k ) = ˆ q ( k ) (black dashed). Right: Empirical values, q -corrected with q = 15, for statistically actual cases ˆ A q ( k ) and thecorresponding model values A ( k ) (black dashed). Mar May Jul Sep Nov Jan0100000200000300000400000500000
Figure 15.
Empirical data (solid coloured lines) and model values(black dashed) in Switzerland for numbers of totally infected ˆ A tot (brown), redrawn ˆ R (bright green), and actual numbers ˆ A , accord-ing to the statistic (blue).This may encourage to look at a conditional 30-day prediction for Switzerlandgiven by the SEPAR d model, the condition being the hypothesis of no considerablechange in the contact behaviour of the population and no increasing influence ofnew virus mutations, i.e., a continuation of the recursion with η , the strength ofinfection in the last constancy interval (fig. 16).The figures show clearly that, under the generic assumption δ = 2 for Switzerland,the ratio of infected s ( k ) starts to suppress the rise of new and actual infectionsalready in January 2021 even for the upper bound of the 1-sigma estimate for theparameter η (fig. 16, top). Of course the question arises what would be changedassuming different values for the dark factor. Figure 17 shows how strongly the ratioof susceptibles is influenced by the choice of δ already at the end of 2020. M. KRECK, E. SCHOLZ
Apr Jul Oct Jan0200040006000800010000
Apr Jul Oct Jan050000100000150000200000
Apr Jul Oct Jan02000004000006000008000001 × Figure 16. A new , A q (top) and A tot (bottom)for Switzerland, assuming dark factor δ = 2; empirical values colouredsolid lines, model black dashed (boundaries of 1-sigma region predic-tion dotted). Apr Jul Oct Jan0.40.50.60.70.80.91.0
Figure 17.
Development of ratio of susceptibles s ( k ) for Switzerland(model values), assuming dark factor δ = 0 (dotted), δ = 2 (solid line)and δ = 4 (dashed).The results for δ = 0 and δ = 4 of the model values of reported new infected, A new ( k ) (black dotted or dashed as above) in comparison with the 3-day averages ofthe JHU data, ˆ A new, ( k ), is shown in figure 18. Remember that the model expressesthe dynamics of 7-day averages of the newly infected. EPAR – COVID-19 29
Apr Jul Oct Jan020004000600080001000012000 Apr Jul Oct Jan020004000600080001000012000
Figure 18. A new for Switzerland, assumingdark factor δ = 0 (left), respectively δ = 4 (right); empirical valuesˆ A new, coloured solid lines, model values black dashed (boundaries of1-sigma region dotted).Assuming a negligible dark sector ( δ = 0) and the upper boundary of the 1-sigmainterval of η , the number of new infected would continue to rise deeply into the firstquarter of 2021. In all other cases the effective reproduction number is suppressedbelow the critical value by the ratio of susceptibles s ( k ) already at the turn to thenew year. We conclude that the role of the dark sector starts to have qualitativeimpact on the development of the epidemic in Switzerland already at this time. Germany.
The epidemic entered Germany (population 83 M) in the second half ofFebruary 2020; the recorded new infections seized to be sporadic at t = Feb. 25,the day k = 1 in our country count. With the health institutions being set in a firstalarm state and public advertising of protective behaviour, the initial reproductionrate (as determined in our model approach) fell swiftly from roughly ρ ≈ t = 30) it dropped below 1 and stayed there,with an exceptional week (dominated by a huge infection cluster in the meet factoryT¨onnies) until late June 2020 (fig. 19 left). The peak of the first wave was reachedby the 7-day averages of new infections ˆ A new, at March 30; 6 days later, i.e. April6, the local maximum for the actual numbers of reported infected ˆ A followed.Serological studies in the region Munich indicate that during the first half of 2020the ratio of counted people was about α = 0 .
25 in Germany, i.e. for each countedperson there were δ ≈ − αα = 3 persons entering the dark sector [14]. Of course this M. KRECK, E. SCHOLZ ratio varies in space and time, for example if the number of available tests increasesor, the other way round, it is too small for a rapidly increasing number of infectedpeople. The rapid expansion of testing,in Germany during early summer seems tohave increased the branching ratio to about α ≈ .
5. In follow up investigations theauthors of the Munich study come to the conclusion that during the next monthsthe ratio of unreported infected has decreased considerably; this brought the factor δ down to ≈ Since the dark segment influences ourmodel mainly through its contribution to lowering of the ratio of susceptibles s ( k ),it is nearly negligible in the first six months of the epidemic. We therefore simplifythe empirical findings by setting δ = 1 for the SEPAR d model of Germany.Contrary to a widely pronounced assessment (including by experts) according towhich the epidemic was well under control until September, the reproduction raterose to ρ ≈ . ρ ≈ .
2, before it accelerated inlate September, brought the reproduction rate to about 1.5, and led straight intothe second wave. Here the weekly oscillations of recorded new infections rose to anamplitude not conceived before (fig. 20). The levels of the mean daily strength ofinfection used in the model (proportional to the corresponding reproduction rates)are well discernible in the next figure 19, right.
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Figure 19.
Left: Empirical reproduction rates ˆ ρ ( k ) for Germany(yellow). Right: Daily strength of infection ˆ η ( k ) for Germany (yellow)with model parameters η j in the main intervals J j (black dashed);critical value 1 /p of η dotted. Short report in [8]. This is consistent with the result in [18].
EPAR – COVID-19 31
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Figure 20.
Daily new reported cases for Germany, 3-day sliding av-erages ˆ A new ( k ) and 7-day averages ˆ A new, ( k ).The dates t j of the time markers k j used for our model in the German case are t = 02/25, 2020, t = 03/24, t = 04/26, t = 06/06, t = 06/16, t = 07/05, t =08/17, t = 08/28, t = 09/27, t = 10/31, t = 11/28, t = 12/14, end of datahere t eod = 01/15, 2021. In the country day count, where t = 1 ( ∼
35 in JHUday count) the main intervals are J = [1 , , J = [29 , , J = [62 , , J =[103 , J = [113 , , J = [132 , , J = [175 , , J = [186 , J =[216 , , J = [250 , , J = [278 , , J = [294 , η and model η j , ρ j in intervals J j for Germany η J J J J J J J J J J J η j ρ j — 0.75 0.83 1.96 0.76 1.21 0.91 1.10 1.49 0.97 1.15 0.93With these parameters the SEPAR model reproduces (pre- or better “post”-dicts)the averaged daily new infections and the total number of reported infected well (fig.21), while one has to be more careful for treating the reported number of actualinfected ˆ A .If one checks the mean duration of being recorded as actual case in the JHUstatistics for Germany by eq. (12) one finds a good approximation q ≈
15 after theearly phase; but the result also indicates that in May/June, and again in the secondhalf of November, the reported duration of the infected state surpassed this value(fig. 22 left). Accordingly the empirical data ˆ A and the corrected ones ˆ A q ( q = 15)drop apart in late November (same figure, right). M. KRECK, E. SCHOLZ
Mar May Jul Sep Nov Jan0500010000150002000025000
Mar May Jul Sep Nov Jan05000001.0 × × × Figure 21.
Left: Daily new reported infected (3-day averages) forGermany; empirical ˆ A new solid red, model A new black dashed.Right: Total number of reported infected; empirical ˆ A tot solid brown,model A tot black dashed. Mar May Jul Sep Nov0510152025
Mar May Jul Sep Nov Jan0100000200000300000400000
Figure 22.
Left: Daily values of the mean time of statistically actualinfection ˆ q ( k ) for Germany. Right: Comparison of reported infectedˆ A (dark blue) and q -corrected number ( q = 15) of recorded actualinfected ˆ A q (bright blue) from the JHU data in Germany.In consequence the model value for the actual infected A ( k ) agree with the JHUdata ˆ A ( k ) only if the model uses time varying values ˆ q ( k ) (fig. 23 left), while the q -corrected numbers ˆ A q ( k ) are well reproduced by the model with constant q = 15(same figure, right).As a result, the 3 model curves representing the total number of (reported) infected A tot , the redrawn R and the actual infected fit the German data well, if the last twoare compared with the q -corrected empirical numbers ˆ A q (fig. 24). EPAR – COVID-19 33
Mar May Jul Sep Nov Jan0100000200000300000400000 Mar May Jul Sep Nov Jan0100000200000300000400000
Figure 23.
Left: Statistically actual cases ˆ A ( k ) for Germany andthe corresponding model values A ( k ) (black dashed) calculated withtime dependent model values for q (see text). Right: Empirical values, q -corrected, for actual cases ˆ A q ( k ) and the corresponding model values A ( k ) (black dashed), q = 15. Mar May Jul Sep Nov Jan05000001.0 × × × Figure 24.
Empirical data (solid coloured lines) and model values(black dashed) for Germany: numbers of totally infected ˆ A tot (brown),redrawn ˆ Rq (bright green), and q -corrected actual numbers ˆ A q (brightblue) .Conditional predictions for A new , A q and A tot , assuming no essential change of thebehaviour, contact rates and the reproduction number from the last main interval J are given in fig. 25. The dotted lines indicate the boundaries of the predictionfor the 1- σ domain for the variations of the values of ˆ η in the last main interval J . M. KRECK, E. SCHOLZ
Apr Jul Oct Jan010000200003000040000
Apr Jul Oct Jan0100000200000300000400000500000
Apr Jul Oct Jan05000001.0 × × × × × Figure 25. A new , A (top) and A tot (bottom)for Germany; empirical values coloured solid lines, model black dashed(boundaries of 1-sigma region prediction dotted). France.
The overall picture of the epidemic in France (population 66 M) is similarto other European countries. But the French JHU data show anomalies which arenot found elsewhere: The differences of two consecutive values of the confirmedcases, which ought to represent the number of newly reported, is sometime negative !This happens in particular in the early phase of the epidemic (until June 2020)where, e.g., ˆ A new (58) = − A new still appear. Sothe French data are a particular challenge to any modelling approach. Even in thisextreme case the smoothing by 7-day sliding averages works well, as shown in fig.26.Another surprising feature of the French (JHU) statistics is an amazing increasein the number of days which infected persons are being counted as “actual cases”.It starts close to 15, but shows a monotonous increase until late October wherea few downward outliers appear, before the curve turns moderately down in earlyNovember 2020 (fig. 27, left).As a consequence the peak of the first wave in early April (clearly visible in thenumber of newly reported) is suppressed in the curve of the actual infected; ˆ A ( k ) EPAR – COVID-19 35
Mar May Jul Sep Nov Jan020000400006000080000100000
Mar May Jul Sep Nov Jan020000400006000080000
Figure 26.
Left: Number of daily newly reported in France ˆ A new ( k ).Right: 7-day sliding averages of new infections ˆ A new, ( k ) for France. Mar May Jul Sep Nov Jan050100150200
Mar May Jul Sep Nov Jan05000001.0 × × × Figure 27.
Left: Daily values of the mean time of statistically actualinfection ˆ q ( k ) for France. Right: Comparison of reported infectedˆ A (dark blue) and q -corrected number ( q = 19) of recorded actualinfected ˆ A q (bright blue) from the JHU data in France.has no local maximum in the whole period of our report. It even continues to rise,although with a reduced slope, after the second peak of the daily newly reported,ˆ A new , in early November. The decrease of the slope of ˆ A starts shortly after thispeak, accompanied by a local maximum of the q -corrected values for the actualinfected reaches ˆ A q (fig. 27). Both effects seem to be due to a downturn of ˆ q ( k ). This extreme behaviour of the data cannot be ascribed to medical reasons; quiteobviously it results from a high degree of uncertainty in data taking and recordingin the French health system.The reproduction numbers are shown in figure 28, left. For the determination ofthe daily strength of infection we have to fix a value for the dark factor. Lackingdata from representative serological studies in France we assume that it is largerthan in Switzerland and choose as a reference value for the model δ = 4. Withthis value the determination of the ˆ η ( k ) are given in figure 28, right, here againwith dashed black markers for the periods modelled by constancy intervals in ourapproach. M. KRECK, E. SCHOLZ
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Figure 28.
Left: Empirical reproduction rates ˆ ρ ( k ) for France.Right: Daily strength of infection are shown in figure 28 for France(yellow) with model parameters η j in the main intervals J j (blackdashed).The markers of change times are here t = 02/25, 2020, t = 05/17, t = 06/15, t = 07/21, t = 08/22, t = 09/29, t =11/03, t =11/27, end of data t eod =12/30, 2020. In the country day count, t = 1 ( ∼
35 in JHU day count), the mainintervals are J = [1 , , J = [83 , , J = [112 , , J = [148 , J =[180 , , J = [218 , , J = [253 , , J = [277 , η j and corresponding reproduction numbers ρ j for the main intervals aregiven in the following table. η and model η j , ρ j in intervals J j for France a J J J J J J J η j ρ j — 1.03 1.09 1.39 1.16 1.40 0.71 1.07An overall picture of the French development of new infections and total numberof recorded cases is given in fig. 29. The so-called “actual” cases are well modelledin our approach (fig.30), if the reference are the q -corrected numbers ˆ A q ( k ) of actualinfected (or if time dependent durations q ( k ) (read off from the JHU data, q ( k ) =ˆ q ( k )) are used). For a combined graph of the 3 curves see fig. 31. EPAR – COVID-19 37
Mar May Jul Sep Nov Jan01000020000300004000050000
Mar May Jul Sep Nov Jan05000001.0 × × × × Figure 29.
Left: Daily new reported number of infected for France(7-day averages); empirical ˆ A new solid red, model A new black dashed.Right: Total number of reported infected (brown); empirical ˆ A tot solid,model A tot black dashed. Mar May Jul Sep Nov Jan0200000400000600000800000
Figure 30.
Empirical values for statistically actual q -corrected casesˆ A q ( k ) and the corresponding model values A ( k ) (black dashed) forFrance. Sweden.
Sweden (population 10 M) has chosen a path of its own for containingCovid-10, significantly different from most other European countries. In the firsthalf year of the epidemic no general lockdown measures were taken; the generalstrategy consisted in advising the population to reduce personal contacts and to gointo self-quarantine, if somebody showed symptoms which indicate an infection withthe SARS-CoV-2 virus. One might assume that the number of undetected infected,the dark sector, could be larger than in other European countries. As we see belowsuch a hypothesis is not supported by the analysis of the data in the framework ofour model.Under the conditions of the country (in particular the relative low populationdensity in Sweden) the first wave of the epidemic was fairly well kept under control, M. KRECK, E. SCHOLZ
Mar May Jul Sep Nov Jan05000001.0 × × × × Figure 31.
Empirical data (solid coloured lines) and model values(black dashed) in France for numbers of totally infected ˆ A tot (brown),redrawn ˆ R (bright green), and q -corrected actual numbers ˆ A q (brightblue).if we abstain from discussing death rates like in the rest of this paper. Once theinitial phase was over (with reproduction numbers already lower than in comparablecountries, but still up to about ρ ≈ ρ ≈ Mar May Jul Sep Nov Jan01000200030004000500060007000
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Figure 32.
Left: Daily new reported cases ˆ A new ( k ) (JHU data) forSweden and 7-day sliding averages ˆ A new, ( k ). Right: empirical repro-duction rates ˆ ρ ( k ) for Sweden.The higher the assumptions for the dark sector, the larger the calculated valuesfor ˆ η ( k ) on the basis of the same data, and vice versa. Figure 33 shows the dif-ferences of ˆ η ( k ) in the case of Sweden under the hypotheses δ = 0 , , , EPAR – COVID-19 39
Until July/August 2020 the four curves show minor differences and indicate relativestable values for the infection strengths leading to reproduction rates close to 1. InSeptember/October the values rose considerably; they went down after the Octoberlockdown only under the assumption of a small dark sector, δ = 0 or 4, while for thelarger dark factors δ = 15 ,
25 the values of the daily strength of infection continuesto increase. This seems implausible. We therefore choose δ = 4 also for Sweden. Mar May Jul Sep Nov0.00.10.20.30.40.5 Mar May Jul Sep Nov0.00.10.20.30.40.5Mar May Jul Sep Nov0.00.10.20.30.40.5 Mar May Jul Sep Nov0.00.10.20.30.40.5
Figure 33.
Daily strength of infection (sliding 7-day averages) ˆ η ( k )for Sweden, assuming different dark factors δ . Top: δ = 0 (left) and δ = 4 (right). Bottom: δ = 15 (left) and δ = 25 (right).In our definition the new infections in Sweden ceased to be sporadic at t = 03/02,2020, the day 41 in the JHU day count. After a month of strong ups and downs of thestrength of infection, the approach of constancy intervals gains traction; with timemarkers of the main intervals t = 03/31, t = 05/17, t = 06/18, t = 07/17, t =08/25, t = 10/10, t = 11/05, t = 12/12, end of data t eod = 12/30, 2020. In thecountry count the main intervals are J = [1 , , J = [30 , , J = [77 , , J =[109 , J = [138 , , J = [177 , , J = [223 , , J = [223 , , J =[286 , Such a hypothesis for the dark sector could be explained only by a drastic and irresponsiblechange of contact behaviour of the Swedish population or an increased infectivity of the virus.Neither of these explanations is supported by available empirical evidence. M. KRECK, E. SCHOLZ
Model η and η j , ρ j in intervals J j for Sweden η J J J J J J J J η j ρ j — 1.01 1.20 0.65 1.00 1.19 1.54 1.13 0.84Although one might want to refine the constancy intervals, already these intervalslead to a fairly good model reconstruction of the mean motion of new infectionsand the total number of infected (fig. 34). Note that since early September thereported numbers of new infections show strong weekly oscillations between null atthe weekends and high peaks in the middle of the week. Mar May Jul Sep Nov Jan01000200030004000500060007000
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Figure 34.
Left: Daily new reported infected for the Sweden, empir-ical 3-day averages ˆ A new, solid red; model A new black dashed. Right:Total number of reported infected (brown); empirical ˆ A tot solid, model A tot dashed.The JHU statistics does not register recovered people for Sweden at all; onlydeaths are reported. In consequence the usual interpretation of (9) as characterizingthe “actual” infected breaks down for Sweden and the estimation (11) for the meanduration of illness becomes meaningless (fig. 35, left). An indication of the extent ofreported actual diseased is given by the q -corrected number ˆ A q (same figure, right).In this sense, the synopsis with a collection of the “3 curves” can be given forSweden like for any other country (fig. 37). Of course the model reproduces theempirical values ˆ A ( k ) even in such an extreme case if the time dependent empiricalvalues of (11) are used for the the model calculation, q ( k ) = ˆ q ( k ), while it recon-structs the q -corrected numbers for the estimate of actually infected ˆ A q ( k ) if therespective constant is used, here q = 15 (fig. 36). The same holds for the UK.
EPAR – COVID-19 41
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Mar May Jul Sep Nov Jan0100000200000300000400000
Figure 35.
Left: Daily values of the mean time of statistically actualinfection ˆ q ( k ) for Sweden. Right: Comparison of reported infectedˆ A (dark blue) and q -corrected number ( q = 15) of recorded actualinfected ˆ A q (bright blue) from the JHU data in Sweden. Mar May Jul Sep Nov Jan0100000200000300000400000
Mar May Jul Sep Nov Jan020000400006000080000100000
Figure 36.
Left: Empirical data ˆ A ( k ) for Sweden (blue) and modelvalues A ( k ) determined with time varying q ( k ) = ˆ q ( k ) (black dashed).Right: Empirical values, q -corrected, for statistically actual casesˆ A q ( k ) and the corresponding model values A q ( k ) (black dashed). Mar May Jul Sep Nov Jan0100000200000300000400000 Mar May Jul Sep Nov Jan0100000200000300000400000
Figure 37.
Left: Numbers of totally infected ˆ A tot (brown), reportedredrawn ˆ R (bright green) – here deaths only – and the difference ˆ A (blue) for Sweden; model values black dashed. Right: the same forˆ A tot (brown), q -corrected ˆ A q (bright blue) and the redrawn ˆ R q (green)as the difference (model values black dashed). M. KRECK, E. SCHOLZ
The three most stricken regions: USA, Brazil, India.
In this sectionwe give a short analysis of the course of the pandemic during 2020 for the threecountries which have to bemoan the largest numbers of deceased and huge numbersof infected (USA, Brazil, India). We expected higher dark factors δ than for the Eu-ropean countries discussed above and checked this expectation by the same heuristicapproach as used for Sweden, i.e. by a comparative judgement of the changes ofthe empirically determined strength of infection ˆ η ( k ), which result from differentassumptions of the values for δ . To our surprise we found no clear evidence for anoverall larger dark factor for the USA than for the European countries and work herewith δ = 4, while for India there are strong indications of a large dark factor whichwe estimate as δ ≈
35 (see below). For Brazil we consider δ ≈ USA.
At the beginning of the pandemics the United States of America (population333 M) suffered a rapid rise of infections with an initial reproduction rate wellabove 5. In early April 2020 this dynamics was broken and a slow decrease startedfor about 2 months. In mid June a second wave with an upswing for about amonth and a reproduction rate shortly below 1 . ρ ( k )determined from the JHU data and the strength of infection ˆ η ( k ) assuming δ = 4 .Figure 39 displays three variants of ˆ η ( k ) for δ = 0 , ,
8. The third one shows animplausible increase for the strength of infection at the end of the year, which wouldseem reasonable only if one of the new, more aggressive mutants of the virus hadstarted to spread in the USA in September 2020 already. Without further evidencewe do not assume such a strong case. As δ = 0 contradicts all evidences collected onunreported infected, we choose δ = 4 for the SEPAR d model of the USA. Also herewe find a moderate increase of the mean level of the ˆ η . To judge whether this maybe due to the inconsiderate behaviour of part of the US population (supporters ofthe outgoing president) or the first influences of a virus mutation and/or still otherfactors is beyond the scope if this paper and our competence.In section 2 it was already noted that the estimation of the time of sojourn inthe “actual” state of infectivity, suggested by the statistics for the USA, leads tosurprising effects. It rises from about 15 in March 2020 to above 100 in earlyNovember, with a moderate platform in between; then it starts falling,before itmakes an abrupt jump (fig 40, left). The jump of ˆ q ( k ) is an artefact of a changein the record keeping: from December 14, 2020 onward the reporting of data of EPAR – COVID-19 43
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Figure 38.
Top: 3-day and 7-day sliding averages of daily new re-ported cases, ˆ A new, ( k ), ˆ A new, ( k ), for the USA. Bottom, left: Em-pirical reproduction rates ˆ ρ ( k ) for the USA. Right: Daily strength ofinfection ˆ η ( k ) for the USA (yellow), with δ = 4; model parameters η j in the main intervals J j black dashed. Mar May Jul Sep Nov0.00.10.20.30.40.5 Mar May Jul Sep Nov0.00.10.20.30.40.5 Mar May Jul Sep Nov0.00.10.20.30.40.5
Figure 39.
Daily strength of infection (sliding 7-day averages) ˆ η ( k )for USA, assuming different dark factors: δ = 0 (left), δ = 4 (middle),and δ = 8 (right).recovered people was given up ( Rec ( k ) = 0 for date of k after 2020/12/14). Ofcourse this jump is also reflected in the numbers of recorded actual infected ˆ A ( k )(same figure, right). M. KRECK, E. SCHOLZ
May Jul Sep Nov Jan050100150200250
Mar May Jul Sep Nov Jan02.0 × × × × × × × Figure 40.
Left: Daily values of the mean time of statistically actual(“active”) infection ˆ q ( k ) for USA. Right: Empirical data (JHU) forstatistically actual cases ˆ A ( k ) (dark blue) versus q -corrected ones, q = 15, ˆ A q ( k ) (bright blue) for the USA.The main (constancy) intervals of the model are visible in the graph of the dailystrength of infection ˆ η ( k ) in fig. 38, bottom right. The dates of the time markerbetween the intervals are: t = 02/29 2020; t = 04/02, t = 06/10, t = 07/19, t =09/04, t = 10/21 t = 11/11, end of data t eod = 12/30. Expressed in terms of thecountry day count with k = 1 ( ∼
39 in the JHU day count) the main intervals for theUSA are: J = [1 , , J = [34 , , J = [103 , , J = [142 , , J = [189 , ,J = [236 , , J = [257 , η and a slightly adapted choiceof parameter values η j inside the 1-sigma domain of the respective interval J j aregiven by the following table.Model η and η j , ρ j in intervals J j for the USA η J J J J J J η j -1.011 0.140 0.189 0.143 0.172 0.215 0.194 ρ j — 0.97 1.28 0.94 1.08 1.30 1.12Here, as for the other countries, the ρ j denote the model reproduction rates at thebeginning of the respective intervals. Assuming the dark factor δ = 4, the effectivereproduction rate ρ ( k ) changes considerably from the beginning of J until the endof the year, s (11 /
11 2020) = 0 .
82 to s (12 /
31) = 0 .
67, (cf.fig. 41).In consequence, the reproduction rate at the end of the year is down to ρ (12 /
31) =0 .
86; and the newly reported infected are expected to reach a peak in late December(fig. 42, right). Apparently this does not agree with the data, while the totalnumber of infections is well reproduced by the SEPAR d model (same figure, left).The difference for the new infected may be an indication that our estimate of thedark factor δ is wrong or the strength of infection at the beginning of the new year2021 is drastically boosted , e.g., by a rapid spread of a new, more aggressive mutantof the virus. EPAR – COVID-19 45
Apr Jul Oct Jan0.40.50.60.70.80.91.0
Figure 41.
Ratio of susceptibles s ( k ) (model values) for the USA, δ = 4. Mar May Jul Sep Nov Jan05.0 × × × × × Mar May Jul Sep Nov Jan050000100000150000200000
Figure 42.
Left: Total number of infected (brown); empirical ˆ A tot solid, model A tot dashed. Right: Daily newly reported for the USA,7-day averages (red); empirical ˆ A new solid red, model new black dashed( δ = 4).As noted above, the recovered people are documented with increasingly largetime delays in the records for the USA. Therefore it seems preferable to comparethe model value of actually infected, A ( k ), with ˆ A q ( k ) rather than with ˆ A ( k ) (fig.43). The empirical determined values of ˆ A q ( k ) are shown in bright blue in the figure.They are marked by three local maxima indicating the peak values of three wavesof the epidemics in the USA. These peaks are blurred in the graph of ˆ A because toomany of the effectively redrawn are dragged along as acute cases in the statistics.Due to under-reporting at the end of the year, the local extremum in December maybe fuzzier than it appears here. But keep in mind that the SEPAR d model with δ = 4 predicts a local maximum inside the interval J , if the contact behaviour andthe resulting strength of infectivity η do not change considerably. M. KRECK, E. SCHOLZ
Mar May Jul Sep Nov Jan05000001.0 × × × × × × Figure 43.
Number of q -corrected actual infected for the USA; em-pirical ˆ A q (bright blue) and model values A q (black dashed).Figure 44, left, shows the 3 curves for the model values (black dashed) of thetotal number of infected A tot , the actually infected A in terms of estimates withconstant q = 15, and the redrawn R , all of them compared with the correspondingvalues ˆ A tot ( k ) , ˆ A q ( k ) , ˆ R q ( k ) determined from the JHU data (coloured solid lines).By using time dependent values q ( k ), like, e.g., in the case of Germany, SEPAR d is able to model the statistically “actual” cases also here (fig. 44, right). Becauseof the growing fictitiousness of the numbers ˆ A ( k ) in the case of the USA we prefer,however, to look at the corrected values ˆ A q ( k ), as stated already. Mar May Jul Sep Nov Jan05.0 × × × × Mar May Jul Sep Nov Jan05.0 × × × × Figure 44.
Left: numbers of totally infected ˆ A tot (brown), redrawnˆ R q (green), and q -corrected actual numbers ˆ A q (bright blue). Right:Numbers of totally infected ˆ A tot (brown), reported redrawn ˆ R (brigthgreen), and actual numbers ˆ A of the statistic (blue) for the USA.Empirical data (solid coloured lines) and model values (black dashed). EPAR – COVID-19 47
Brazil.
The documentation of newly reported became non-sporadic in Brazil (popu-lation 212 M) at t = March 15, 2020. A first peak for the officially recorded numberof actual infected ˆ A ( k ) was surpassed in early August 2020 with a decreasing phaseuntil late October, after which a second wave started (fig. 45 left). In contrast tothe USA we find here a comparatively stable estimate for the mean time ˆ q ( k ) ≈ May Jul Sep Nov Jan0200000400000600000
May Jul Sep Nov Jan05101520
Figure 45.
Left: Acute infected ˆ A ( k ) recorded by the statistics (darkblue) in comparison with q -equalized number ˆ A q ( k ) (bright blue) forBrazil ( q = 14). Right: Empricial estimate ˆ q ( k ) of mean duration ofactive infective according to the statistics for Brazil.The outlier peak of ˆ A ( k ) about October 25 appears also as an exceptional peakin the ˆ q ( k ). Apparently it is due to an interruption of writing-off actual infectedto the redrawn (compare fig. 51). Up to this exceptional phase there is a closeincidence between the ˆ A ( k ) and the q -corrected number ˆ A q ( k ). The minimum ofthe mean square difference is acquired for q = 14. The numbers of newly reportedˆ A new ( k ) show strong daily fluctuations which are smoothed by the 7-day slidingaverage ˆ A new, ( k ) (fig. 46).A comparison of different strengths of infection ˆ η ( k ) indicates a value between δ = 0 and δ = 8 as a plausible choice (fig. 47). For higher values an unnaturalincrease of the infection strength would appear close to the end of the year (if notdue to a new mutant. As we assume that in Brazil the dark factor is higher than inEuropean countries, we use δ = 8 as a plausible model hypothesis.The daily reproduction numbers ˆ ρ ( k ) (independent of δ ) start from a lower levelthan for many other countries, slightly above 2. They show a relatively stabledownward trend, falling below 1 for a few days in early June and for longer periodsafter June 22, 2020 (fig. 48, left). But already at the end of March, when the M. KRECK, E. SCHOLZ
May Jul Sep Nov Jan010000200003000040000500006000070000
May Jul Sep Nov Jan01000020000300004000050000 Figure 46.
Daily varying numbers ˆ A new ( k ) (left)versus 7-day slidingaverages ˆ A new, ( k ) (right) of new infections for Brazil. May Jul Sep Nov0.00.10.20.30.40.5 May Jul Sep Nov0.00.10.20.30.40.5May Jul Sep Nov0.00.10.20.30.40.5 May Jul Sep Nov0.00.10.20.30.40.5
Figure 47.
Daily strength of infection (sliding 7-day averages) ˆ η ( k )for Brazil, assuming different dark factors. Top: δ = 0 (left) and δ = 4(right). Bottom: δ = 8 (left) and δ = 15 (right).reproduction rate was still considerably above 1 ( t = March 30), its downwardtrend was already slow enough to allow for approximation by constancy intervals.In the case of Brazil the initial interval starts at t = 03/15, 2020. The fol-lowing time can be subdivided into main intervals J j in which the averaged dailystrength of infection ˆ η ( k ) can be replaced by their mean values η j , starting with t = 03/30. The main intervals are separated by the days t = 05/14, t = 06/22, t = 07/11, t = 07/19 t = 08/27, t = 08/10, t = 10/29, t = 12/13, 2020. EPAR – COVID-19 49
May Jul Sep Nov0.00.51.01.52.02.53.0
May Jul Sep Nov Jan0.00.10.20.30.40.5
Figure 48.
Left: Empirical reproduction rates ˆ ρ ( k ) for Brazil (or-ange). Right: Daily strength of infection, empirical ˆ η ( k ) (yellow), forBrazil assuming δ = 8, and model parameters η j in the main intervals J j (black dashed).They can well be discerned in fig. 48, right, showing the daily strength of infec-tion ˆ η ( k ) (yellow) and their mean values (black dashed) in these intervals. Inthe country day count k = 1 ( ∼
70 in the JHU count) the main intervals are J = [1 , , J = [16 , , J = [61 , , J = [100 , , J = [119 , , J =[127 , , J = [166 , , J = [206 , , J = [222 , , J = [274 , t eod ], here withthe end of data t eod = 292.The start parameter η and the model reproduction numbers ρ j in the respectiveinterval J j are Model η and η j and ρ j in J j for Brazil η J J J J J J J J J η j ρ j — 1.42 1.14 0.98 1.30 0.96 0.94 0.97 1.17 1.00Also here the ρ j designate reproduction numbers at the beginning of the j -theinterval and the fall of s ( k ) makes the reproduction number cross the critical valuein the last main interval (cf. 49). This does not mean that it will stay there.The resulting model curves and their relationship to the empirical data for newinfections and acual infections are shown in fig. 50. A panel of the three curves A tot , A, R is shown in fig. 51. Here one sees clearly that the outlier bump of ˆ A ( k )is accompanied by an inverse outlier in ˆ R ( k ). M. KRECK, E. SCHOLZ
May Jul Sep Nov Jan0.40.50.60.70.80.91.0
Figure 49.
Model values of the ratio of susceptibles s ( k ) for Brazil( δ = 8). May Jul Sep Nov Jan01000020000300004000050000
May Jul Sep Nov Jan02000004000006000008000001 × Figure 50.
Left: Empirical values (3-day average) for daily newlyreported for Brazil ˆ A new, (solid red line)) and model values blackdashed). Right: Actual cases ˆ A (blue), model values (black dashed). May Jul Sep Nov Jan02 × × × × Figure 51.
Empirical data (coloured solid lines) for the numbers oftotally infected ˆ A tot (brown), redrawn ˆ R (bright green), actual infectedˆ A (blue) and the respective model values (black dashed) for Brazil( δ = 8). EPAR – COVID-19 51
India.
The recorded data on ˆ A new ( k ) for India (population 1387 M) start to benon-sporadic at t = March 4, 2020. From this time on we find a steady growthof the number of reported actual infected ˆ A ( k ) until early September. Because thesize of the country and the life conditions in large parts of it a comparatively highnumber of unrecorded infected may be assumed, with a dark factor at the order ofmagnitude δ ∼
10 probably 20 ≤ δ ≤ In early September the tide changed anda nearly monotonous decline of actual infected started. With the exception of a shortintermediate dodge the decline continues at the end of 2020 (fig. 52, left). Althoughin late December 2020 there were only about 0.7 % recorded infected in India, thehigh quota of unreported infected poses the question whether the downturn in latesummer may already be due to a the decrease of the fraction of susceptibles s ( k ).Before we discuss this point let us remark that the time of being statisticallyrecorded as actual case is relatively stable in the Indian data, with a good approx-imative constant value q ≈
11 (fig. 52, right). In consequence ˆ A ( k ) does not differmuch from ˆ A q ( k ) (same fig. left). This allows to use the recorded data ˆ A in thefollowing without the proviso to be made in the case of the USA. Mar May Jul Sep Nov02000004000006000008000001 × May Jul Sep Nov05101520
Figure 52.
Left: Actual infected ˆ A ( k ) recorded by the statistics(dark blue) in comparison with q -equalized number ˆ A q ( k ) (bright blue)for India ( q = 11). Right: Empiricial estimate ˆ q ( k ) of mean durationof actual infectived according to the statistics, i.e. in ˆ A , for India. A serological investigation of over 4000 inhabitants found 24 % infected(from which over 90 % were asymptomatic). With about 140 k reported in-fected in a population of roughly 31 M this amounts to a dark factor δ ≈ ). M. KRECK, E. SCHOLZ
The reproduction numbers and the corresponding daily strength of infection ofthe model are derived from the 7-day sliding averages of newly reported. Figure 53shows both the daily varying ˆ A new and the averaged ˆ A new, . Mar May Jul Sep Nov020000400006000080000100000
Mar May Jul Sep Nov020000400006000080000
Figure 53.
Top: Empirical number of daily new infections ˆ A new and7-day sliding average ˆ A new, for India.The reproduction number fell rapidly from roughly 3.5 at the beginning to below1.5 in early April, and 1.2 in late May, after which it continued to decrease withminor fluctuations. In early September it dropped below the critical value 1, whereit stayed with few exceptional fluctuations until December (fig. 54). It runs, ofcourse, parallel to the daily strength of infection ˆ η ( k ) calculated from the 7-dayaverages if abstraction is made from the dark sector, δ = 0 (fig. 55, left). Here weconfront it with the more realistic graph of ˆ η calculated under the assumption of adark sector. Mar May Jul Sep Nov01234
Mar May Jul Sep Nov0.00.10.20.30.40.50.6
Figure 54.
Left: Empirically determined reproduction rates ˆ ρ ( k )for India (iorange). Right: Empirical infections strength ˆ η (yellow)assuming a dark sector with δ = 35; model values for η black dotted.Such an idealized scenario with δ = 0 is shown in fig 55, left. If, on the other hand,the empirical daily strength of infection are determined under the more realisticassumption of a non-negligible dark sector, e.g. δ = 35, the picture is different (same EPAR – COVID-19 53
Mar May Jul Sep Nov0.00.10.20.30.40.50.6 Mar May Jul Sep Nov0.00.10.20.30.40.50.6
Figure 55.
Left: Empirical daily strength of infection ˆ η ( k ) for India(yellow) with no dark sector, i.e. assuming δ = 0. Right: Empiricalstrength of infection ˆ η ( k ) for India (yellow), assuming a dark sectorwith factor δ = 35figure, right). Here one finds a daily strength of infection moderately fluctuatingin a narrow band between 10 and 20 % above the critical value η crit = ≈ . δ ≈
0) looks highly unrealistic. In both cases the empirically determinedreproduction rates ˆ ρ ( k ) are the same (fig. 54). In the second case the fall of ˆ ρ ( k )below 1 in early September is due to the lowering of ˆ s ( k ), i.e. as an effect of anincipient herd immunization.But how can that be with a herd immunization quota of (1 + δ ) ˆ A tot ( k ) N ≈
12 %,even with δ = 35, in September 2020? The reason lies in the comparatively lowoverall daily strength of infection. Between May and December 2020 it fluctuatedbetween 10 and 20 % above the critical level (fig. 55, right). Even if part of the lowlevel had to be ascribed to an intentional under-reporting of the ˆ A new ( k ), this wouldmean an increasing size of the dark sector; the overall effect would be the same. For modelling the epidemic in India we can do with 7 constancy intervals. Afterthe initial interval J = [1 ,
30] in the day count of the country ( k ∼
43 in the JHUcount) the main intervals are J = [31 , , J = [74 , , J = [142 , , J =[169 , , J = [191 , , J = [234 , , J = [260 , k eod ], with end of data k eod =297. The date of the interval separators are t = 03/04, t = 04/03, t = 05/16, t = 07/23, t = 08/19, t = 09/10, t = 10/23, t = 11/18, end of data t eod = 12/262020. In 12/2020 it was already twice as much. Only a permanently increasing amount of under-reporting could emulate a fake picture of anon-existing downswing of the epidemic for several months. We exclude such a hypothesis. M. KRECK, E. SCHOLZ
Similar to the Brazilian case, the reproduction rate surpasses 1 in the first fourmain intervals; only in mid September a downswing of the epidemic started, inter-rupted by an intermediate dodge at the beginning of November. In mid Septemberthe total number of acknowledged infected was roughly A tot (240) ≈ M , about 3.8per mill of the total population; but with a dark quota of δ = 35 the total numberof infected had probably already risen above the 10 % margin (see above).The start parameter of the model η is chosen according to the best adaptation tothe 7-day averaged data (without a claim for a directly realistic interpretation) andthe parameter values η j essentially as the mean values of the ˆ η ( k ) in the respectiveinterval J j . They are given in the table.Model η and η j , ρ j in J j for India η J J J J J J J η j -0.138 0.192 0.175 0.164 0.177 0.157 0.182 0.166 ρ j — 1.34 1.2 1.07 1.12 0.90 0.99 0.87With these parameters the SEPAR model leads to a convincing reconstructionof the epidemic in India. This is shown by the graph showing the three curves A tot , A, R (fig. 56). Mar May Jul Sep Nov02 × × × × Figure 56.
Empirical data (colored solid lines) for the numbers oftotally infected ˆ A tot (brown), redrawn ˆ R (bright green), actual infectedˆ A (blue), and the respective model values (black dashed) for India.All with dark sector, δ = 35 and constancy intevals (see main text). EPAR – COVID-19 55
The numbers of newly reported ˆ A new ( k ) and the number of actual cases ˆ A ( k ),including a conditional prediction for the next 30 days on the basis of the lastinfection strength η (fig. 57). Black dotted the boundaries of the 1 σ domain forthe ˆ η -variation in J . In the case of India the data show exceptional low variabilityinside the constancy intervals. Thus the width of the 1 σ domain is smaller than inany of the other countries considered. Apr Jul Oct Jan020000400006000080000100000120000
Apr Jul Oct Jan02000004000006000008000001.0 × × × Figure 57.
Left: Daily newly reported for India, empirical ˆ A new (solid red line) and model A new (black dashed). Right: Reported ac-tual cases for India, empirical ˆ A (blue) versus model A (black dashed).Both with dark sector, δ = 35 and 30-day conditional prediction as-suming no large vchange of the infection strenght in J , the last maininterval.With a dark sector roughly as large as assumed in the model ( δ = 35) the ratio ofsusceptibles went down in late 2020 to s ( k ) ≈ . Apr Jul Oct Jan Apr Jul0.40.50.60.70.80.91.0
Figure 58.
Ratio of susceptibles s ( k ) for India.. M. KRECK, E. SCHOLZ
Aggregated data of the World.
Let us now see how the aggregated data ofall countries and territories documented in the JHU data resource can be analysedin our framework, and how they are reproduced by the SEPAR d model. For thesake of simplicity we speak simply of the World . The number of daily new infectionsˆ A new shows clearly three or four steps, expressed by phases of accelerated growth ofˆ A new between February and December 2020 (fig. 59): March (European countries),May to July (two waves in the US, bridged by rising numbers in Brazil and India),October (second wave in Europe and Brazil, third wave in US), and less visible theJanuary/February wave in China and South-Korea. Mar May Jul Sep Nov0100000200000300000400000500000600000700000
Figure 59.
Daily new reported cases ˆ A new ( k ) for the World and 7-day sliding averages ˆ A new, ( k ).These steps of steeper increase of the daily new infections correspond to localpeaks or elevated levels of the mean strength of infection and reproduction numbers.The first two peaks of the mean reproduction numbers with ρ peak − ≈ ρ peak − ≈ . Mar May Jul Sep Nov01234
Mar May Jul Sep Nov0.00.10.20.30.40.5
Figure 60.
Bottom, left: Empirical reproduction rates ˆ ρ ( k ) for theWorld (orange). Right: Daily strength of infection ˆ η ( k ) for the World(yellow) with model parameters η j in the main intervals J j (blackdashed). EPAR – COVID-19 57
We let the model start at t = 01/25, 2020, the fourth day of the JHU day count,and use the following time separators for the main (constancy) intervals t = 03/29, t = 05/06, t = 07/19, t = 10/02, t = 11/10, t = 11 /
26, end of data t eod =12/11, 2020. In the count adapted to the t chosen here the main intervals are J = [1 , , J = [65 , , J = [103 , , J = [177 , J = [252 , , J =[291 , , J = [307 , , ]. The parameters η j and the corresponding mean repro-duction numbers in these intervals are give by the following table.Model η and η j , ρ j in intervals J j for the World η J J J J J J η j ρ j — 1.01 1.09 1.02 1.12 1.01 1.11 Mar May Jul Sep Nov02000004000006000008000001 × Mar May Jul Sep Nov01 × × × × × × × Figure 61.
Left: 3-day averages of daily new reported infected forthe the World (red); empirical ˆ A new solid red, model A new blackdashed. Right: Total number of reported infected (brown); empir-ical ˆ A tot solid, model A tot dashed.Because of the lack of reliable reporting for recovering dates in several countries,among them some large ones like the USA, we cannot expect a balanced value forthe sojourn in the state of actual disease, documented in the statistics. Fig. 62,left shows that the estimated values ˆ q keeps close to 15 or even 20 until late March.Later on the weight of the countries with reliable documentation of recovering datesis large enough to keep the mean number of ˆ q ( k ) between 30 and 40, even withthe rise of the pandemic after May 2020 (fig. 62, left). Accordingly the q -correctednumber of actually infected ˆ A q separate from the the ones given directly by thestatistics ˆ A ( k ) only in mid April. Since October 2020 they difference between thetwo is rising progressively (same figure, right). M. KRECK, E. SCHOLZ
Mar May Jul Sep Nov010203040
Mar May Jul Sep Nov05.0 × × × × Figure 62.
Left: Daily values of the mean time of statistically actualinfection ˆ q ( k ) for the World. Right: Comparison of reported infectedˆ A (dark blue) and q -corrected number ( q = 15) of recorded actualinfected ˆ A q (bright blue) from the JHU data in the World.Like in the case of those countries which have an unreliable documentation ofthe actual state of infected (e.g. US, Sweden, . . . ) we can here reconstruct thestatistically given number ˆ A ( k ) by the model value A ( k ) by using time variabledurations q ( k ) = ˆ q ( k ). This is being displayed in the graph of the 3 curves of theWorld (fig. 63). Mar May Jul Sep Nov02 × × × × Figure 63.
Empirical data (solid coloured lines) and model values(black dashed) for the World: numbers of totally infected ˆ A tot (brown),redrawn ˆ R (bright green), and numbers of those which are statisticallydisplayed as actually infected ˆ A (blue). EPAR – COVID-19 59 Discussion
The data evaluation in sec. 3 shows clearly that the SEPAR d model works well forcountries or territories with widely differing conditions and courses of the epidemic.For the “tautological” application of the model with daily changing coefficients ofinfection η ( k ) this is self-evident, while it is not so for the use of a restricted numberof constancy intervals. The examples studied in this paper show that in this modeof application the model is well-behaved, able to characterize the mean motion of anepidemic and to analyse its central dynamic. In the country studies we have shownthat this is the case not only for the number of acknowledged daily new reported,our A new ( k ) but also for data which, in the standard SIR approach, are not easilyinterpretable like the number of actual infected persons, A ( k )or the q -normalizednumber A q ( k ).What is the SEP AR d model good for? It is clear that it cannot predict thefuture. The main reason for this is that nobody knows how the contact rates arechanging in the future. It allows – though – a prediction under assumptions. In thedifferent countries we carried this out with different scenarios.The main value of the model is as a tool for analysing the development, and tolearn from such an analysis. We will discuss three such topics:– the role of constancy intervals– the role of the dark sector– the influence of the time between infection and quarantine The role of constancy intervals.
The empirical values of the infection strength ˆ η ( k )are calculated from data on reported new infected and are therefore subject to ir-regularities in data taking and reporting. The most drastic consequences of thisare the obvious weekly fluctuations. Different methods can be applied to smooththese weekly fluctuation, sliding 7-day averages (used here), stochastic estimateused by the RKI (see appendix), band filter etc. Independent of the applied methodthere remain effects (e.g. non- weekly reporting delays) which distort the calculatednumbers away from being correct empirical values for the intended quantities (e.g. η ( k ) = γκ ( k )). Even if they were, one would encounter day to day fluctuationsresulting from the variation of intensities of contacts and of the strengths of infec-tiousness involved, which one is not really interested in if one wants to gain insightinto the dynamics of the epidemic. For this one needs to distil a cross-sectionalpicture of the process. In our approach this is achieved by constructing constancyintervals (main intervals) J j and model strengths of infection η j , read off from thedata, and to apply the infection recursion (5). The role of the dark sector.
With increasing numbers of herd immunized, the in-fluence of the dark sector on the ratio s ( k ) of susceptibles in the total population M. KRECK, E. SCHOLZ gains increasing weight, in particular for countries in which a high dark ratio δ maybe expected. In most of the European countries studied here we find the ratio ofrecorded infected at the order of magnitude of 1 % all over the year 2020. Withthe non-reported ones added it can easily rise to the order of magnitude 10 % andstart to have visible effects. If our estimated values of the dark factor δ are notutterly wrong, our model calculation shows that in nearly all countries of the study,Germany being the only exception, the development of the epidemic is already no-ticeably influenced by the dark sector at the beginning of the year 2021. The lattercontributes essentially to turning the tide of the reported new infected, if one as-sumes constant contact ratios κ ( k ) and mean infection strength γ of the virus. Ofcourse the appearance of new mutants may change γ , and counteract the decrease ofthe numbers of infected predicted by the model. This seems to be the main problemfor the early months of 2021. Apr Jul Oct Jan0.40.50.60.70.80.91.0
Figure 64.
Ratio of susceptibles s ( k ) = S ( k ) /N for Germany(dashed) and Switzerland (solid line) at the end of the year 2020,assuming a dark ratio δ = 2 for Switzerland and δ = 1 for Germany.This becomes particularly succinct by a comparing the Swiss situation with Ger-many at the end of the year 2020 (fig. 64). In both countries containment measureswere taken after a rise of the reproduction rate to 1.4 to 1.5 in late September /earlyOctober, although with different degrees of resoluteness and results (figs. 10, 19).The weight of the dark sector is much stronger for the non-European countries ofour study. In the case of the USA and Brazil it has started to suppress the effectivereproduction number below the critical value 1, according to our model assumptionson the dark factor. But even if one would set it dowm to δ = 1 or 2 the effect wouldalready occur, although a bit later and weaker. That this is not yet reflected in thenumbers of newly infected may have different reasons; one of it would, of course be,that the model can no longer be trusted in this region. Others have been mentionedin the country section. And finally it could be that persons infected some monthsago need not necessarily be immune against a second attack. If virologists come to EPAR – COVID-19 61 this conclusion, the whole model structure would need a revision. At the momentit is too early to envisage such a drastic step.
The influence of the time between infectivity and quarantine.
A central input intothe
SEP AR d model is the assumption that there is a rather short period of length p c , where people, who later are positively tested, are infectious. This is closelyrelated to the fact that people with positive test results are sent to quarantine orhospital. One can wonder what would happen, if the time between infectivity andquarantine or hospital is changed.It is a bit confusing, but there are two answers to this question. To explain thedifference we recall the role of p c in our model. We usually derive the η parame-ters from the data (eq. 15). For the reproduction number (17) in the simplified SEP AR d -model with p c = p d = p and a constant coefficient η this means: ρ ( k ) = p η s ( k ) = ( p c η p d η s ( k )Here we assume that p c is given. This number is only a rough estimate and may bechosen slightly differently. So, for each choice of the estimated number p c one getsmodel curves and one might ask, how much these model curves differ, in particularhow much the reproduction rates would differ. The answer is: not very much. Thereason is that p c enters implicitly also in the formula (14) for η , since the denominatoris a sum over p c values of the daily newly infected. If we assume that this number isconstant (which often is approximately the case) then in the denominator we wouldhave the factor p c and in the formula for ρ it cancels out. Thus in this situation thereconstruction of ρ from p c and η is independent of the choice of p c . If the valuesof the newly infected changes more drastically this is not the case and one has touse the general formula (16), but the difference is not dramatic. So the first answerto the question is: A different estimation for p c does not have a noticeable influencefor the model curves.For understanding the second very different answer we have to recall that η maybe interpreted as the product of the contact rate κ (as measured in the model)and the strength of the infection γ . If we assume that γ is constant, the change of p c discussed above amounts to a change of the model- κ , which does not express achanging contact behaviour. This means that our measure for the contact rate isrelated to our choice of p c .Now we come to the second answer. Here we assume that the contact rate remainsthe same, the contact behaviour of the society is not changed. But suppose that bysome new regulations the value of p c is changed. Then, as expected, if the contactbehaviour is unchanged the reproduction number changes proportionally and so thecurves are different. This second answer is what we are interested in here. Let usassume that one finds means by which the time until the people go to quarantine or M. KRECK, E. SCHOLZ hospital is reduced. Then less contacts take place and so the curves are flattened.This fact is well known, e.g., [2, appendix]. But how much?For answering this question we have taken the model description for Germany,lowering the value of p c from 7 to 6 days from a certain moment on. Here we haveto discuss an important point. One can only influence the time until quarantine orhospital for those who are registered, while the infected people who end up in thedark sector behave as before. At this moment we have to give up our assumptionthat p c = p d . So, from a certain moment on we assume that p d is still 7 but p c is 6.We have carried this out in two different scenarios for the expected numbers ofdaily new recorded infected A new ( k ) and the numbers of actual infected A ( k ). In thefirst one we compare the past development in Germany during the year 2020 with afictitious reduction of p c from 7 to 6 during May 2020, keeping p d = 7 fixed (fig. 65).In a second one we take a look into the future, perpetuating the contact rate of thelast constancy interval, i.e., assuming that the contact behaviour of the populationis unchanged for a while and assume the same fictitious reduction as above in thesecond half of January 2021 (fig. 66). This doesn’t mean that we make a predictionof the future, our only aim here is to demonstrate what would happen if we couldlower p c from 7 to 6. The lowering of the numbers of infected, newly recorded andactual ones, are very impressive. Mar May Jul Sep Nov Jan0500010000150002000025000
Mar May Jul Sep Nov Jan050000100000150000200000250000300000350000
Figure 65.
Model calculations for reported new infected A new ( k )(left) and reported actual infected A ( k ) (right) for Germany. Solidlines with parameter values given in sec. 3( p c = 7 all over the year2020). Dashed p c = 7 from March to May, p c = 6 from Augustonward, smooth transition in June.In the past none of the regulations imposed by the German federal authoritiesmade an attempt to reduce the time until people got to quarantine or hospital asidefrom raising the number of tests. Our considerations suggest to make a serious at-tempt in this direction. It has the big advantage that it does not require additional We thank S. Anderl for the hint.
EPAR – COVID-19 63
Apr Jul Oct Jan0500010000150002000025000
Apr Jul Oct Jan050000100000150000200000250000300000350000
Figure 66.
Model calculations for reported new infected A new ( k )(left) and reported actual infected A ( k ) (right) for Germany (30 daysprediction on the basis of data available 14 Jan 2021). Solid lines withparameter values given in the sec. 3, in particuilar p c = 7. Dashed p c = 7 from March 2020 to 15 Jan 2021, p c = 6 from February 2021onward, smooth transition in between.restrictions of the majority of the population and can be expected to be very effec-tive at the same time. Appendix
Comparison with RKI reproduction numbers .
The estimates of the reproductionnumbers for Germany by the
Robert Koch Institut (RKI), Berlin, are based on anapproach using the generation time as crucial delay time. The generation time τ g ofan epidemic is defined as the mean time interval between a primary infection andthe secondary infections induced by the first one; similarly the length τ s of the serialinterval as the mean time between the onset of symptoms of a primary infected andthe symptom onset of secondary cases. There are various methods to determinetime dependent effective reproduction numbers on the basis of stochastic models forinfections using both intervals. In our simplified approach with constant e and p these numbers correspond to τ g = τ s = e + p − .The RKI calculation uses a method of its own for a stochastic estimation of thenumbers of newly infected, called E ( t ), from the raw data of newly reported cases,described in [1]. The calculation of the reproduction numbers works with these E ( t ) and assumes constant generation time and serial intervals of equal lengths M. KRECK, E. SCHOLZ τ g = τ s = 4 [17]. Two versions of reproduction numbers are being used, a day-sharp and therefore “sensitive” one ρ rki, ( t ) = E ( t ) E ( t − , and a weekly averaged one, ρ rki, ( t ) = (cid:80) j =0 E ( t − j ) (cid:80) j =0 E ( t − − j ) , which we refer to in the following simply as ρ rki ( t ).The paper remarks that the RKI reproduction numbers (“ R -values”) ρ rki ( u ) in-dexed by the date u of calculation refer to a period of infection which, after taking theincubation period ι between 4 and 6 days into account, lies between u − , . . . , u − u −
12 in the interval). We reformulate this redating by setting(18) ˆ ρ rki ( t −
12) = (cid:80) j =0 E ( t − j ) (cid:80) j =0 E ( t − − j ) , , For a comparison with the SEPAR reproduction numbers we write (17) asˆ ρ ( k − ( e + p + 3))) = p (cid:80) j =0 ˆ A new ( k − j ) (cid:80) p − j =0 ˆ A new ( k − ( e + 2) − j ) , which for e = 2 , p = 7 boils down toˆ ρ ( k − (cid:80) j =0 ˆ A new ( k − j ) (cid:80) j =0 ˆ A new ( k − − j ) . This is very close to (18). The main differences lie in the usage of different raw databases (RKI versus JHU) and the adjustment of the raw data (stochastic redistribu-tion E ( t ) versus sliding 7-day averages ˆ A new, ). This may explain the differences inthe level of low or high plateaus shown in fig. 67 (with 1 day additional time shift).In this sense, our model supports the claim of the RKI that their reproductionnumbers can be used as indicators of “a trend analysis of the epidemic curve” [17,p.1]. Acknowledgements:
We thank Odo Diekmann for discussing our thoughts asnon-experts at an early stage of this work; he helped us to understand compartmentmodels better. Moroever, we appreciate the exchange with Stephan Luckhaus, andthank Robert Schaback, Robert Feßler, Jan Mohring, and Matthias Ehrhardt forhints and discussions. Calculations and graphics were made with
Mathematica For e = 2 this would correspond to p = 5, while for τ g = τ s = 5 we arrive at our p = 7. EPAR – COVID-19 65
Mar May Jul Sep Nov01234
Figure 67.
Empirical reproduction numbers ˆ ρ ( k ) of the SEPAR q model for Germany (orange) and reproduction numbers ρ rki ( k − References [1] an der Heiden, Matthias and Osamah Hamouda. 2020. “Sch¨atzung der aktuellen En-twicklung der SARS-CoV-2-Epidemie in Deutschland – Nowcasting 23.”
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