A phenomenological estimate of the Covid-19 true scale from primary data
AA phenomenological estimate of the Covid-19 true scalefrom primary data
Luigi Palatella a , Fabio Vanni b,c,d , David Lambert d,e a Liceo Scientifico Statale “C. De Giorgi”, Lecce, Italy b Sciences Po, OFCE , France c Institute of Economics, Sant’Anna School of Advanced Studies, Pisa, Italy d Department of Physics, University of North Texas, USA e Department of Mathematics, University of North Texas, USA
Abstract
Estimation of the prevalence of undocumented SARS-CoV-2 infections is critical for understandingthe overall impact of CoViD-19, and for implementing effective public policy intervention strategies. Wediscuss a simple yet effective approach to estimate the true number of people infected by SARS-CoV-2,using raw epidemiological data reported by official health institutions in the largest EU countries andthe USA.
As the coronavirus disease 2019 (CoViD-19) epidemic reached every corner of the world, each country hasadopted different interventions to manage the early and long-term phases of the spread of the epidemic.This epidemic has forced many countries to react by imposing policies aimed at reducing populationmobility together with internal and international border limitations. These policies have been informedby various measures of epidemic risk that all trace back to the number of new cases that the country’shealthcare system has found each day. Unrecorded and unnoticed cases make it difficult to estimate thetrue number of infected persons present at a given time (prevalence).Cases of infection are usually detected through testing, typically occurring when ill people (or theirrecent contacts) seek healthcare. Official data are mainly collected with medical swabs. This favors theexamination of patients showing clear symptoms. However, key features of CoViD-19’s dynamics have todo with asymptomatic or pre-symptomatic transmission [1, 2]. Transmission of severe acute respiratorysyndrome coronavirus 2 (SARS-CoV-2) can occur before symptom onset in the infector, which presentsa stumbling block to efforts to stop the spread of the disease. Infected persons often do not developnoticeable symptoms until after the viral latent period, the time between being infected with the virusand first becoming contagious. (This differs from the incubation period: the time between being infected1 a r X i v : . [ phy s i c s . s o c - ph ] J a n ith the virus and first developing symptoms.) Recent studies [3, 4, 5] have found that SARS-CoV-2 hasthe notable property that the latent period of SARS-CoV-2, is shorter than the incubation period of thevirus. Thus, individuals can become contagious even before they show symptoms. Early in the epidemic,and in various situations afterward, diagnosis has been based on having a certain set of symptoms.More commonly, an infection diagnosis comes from direct detection of SARS-CoV-2 RNA by nucleic acidamplification tests, typically reverse-transcription polymerase chain reaction (RT-PCR) from the upperrespiratory tract. Furthermore, testing capacity depends on the demands a healthcare system is under.So that the confirmed case numbers reported during an outbreak represent only a fraction of the truelevels of infection in a community.Undocumented infections often are not detected due to mildness, limitedness, or absence of symptoms.As they are not quarantined, they can expose far more of the population to the virus, thereby sustainingthe spread of the epidemic. Moreover, the fraction of undocumented (but still infectious) cases is acritical epidemiological feature that modulates the pandemic potential of a virus. The ability to estimatethe scale of an epidemic is of paramount importance medically, socially, and economically, as it affectsviral hot-spot detection, resource allocation, and intervention planning. Restrictive measures have indeedexempted firms producing essential goods and services. In addition, companies able to massively employremote work have succeeded in mitigating the negative effects. Better estimation of the true prevalenceof CoViD-19 cases is crucially important to set the strength and scale of non-medical interventions andimprove the evaluation of the economic and health impacts associated with lockdown and reopeningpolicies.Many modeling methods use the mortality rate to estimate the scale of CoViD-19 [6, 7, 8, 9]. Thesemethods rely on fixed estimates of certain epidemiological properties (such as the onset-to-death intervaldistribution and the generation-time distribution). Further, they assume that the fatality rate is thesame everywhere and that its true value is near the lowest observed value ( ’ . − . R t . We make a quantitative analysis ofthis saturation effect to estimate the true scale of the Covid-19 pandemic, i.e., the order of magnitude ofthe actual number of people that have been infected.In the Methods section, we explain the mathematical background for the estimation of the fraction ofun-diagnosed cases by using both the number of new positive tests and the instantaneous reproductionnumber R t . In the Data and Results section, we show the results for Italy, France, Spain, Germany, andUSA. In the Conclusions section, we summarized our research and conclude the paper. At the very beginning of an epidemic, the fraction of the population that is susceptible to the disease is 1.The basic reproduction number, R , is the average number of people that are infected by a single infectedperson per day when everyone is susceptible. In general, R depends on the capacity of the virus to infect2eople, as well as on aspects that vary from country to country such as social habits, movement patterns,average health status, and sanitation. Moreover, the reproduction number may depend on time due toseasonal conditions, air humidity, UV exposure, and social changes (e.g., an increase in mask wearing orincreased average interpersonal distance). When people become immune or die the susceptible fraction ofthe population decreases. All else being equal, this leads to a decrease in the instantaneous (or apparent,or actual) reproduction number, R t . This phenomenon is described by the following relation: R t = R ( t ) (1 − C ( t )) . (1)Here, C ( t ) is the true fraction of the population that has been infected at time t . Let us suppose that thetesting system of a region is not able to detect every positive case, we define λ ( t ) ≡ detected casesactual cases = c ( t ) C ( t ) , (2)where c ( t ) is the total number of detected cases (the cumulative number of cases) as a fraction of thewhole population.The instantaneous reproduction number, R t , is an important tool for detecting changes in disease trans-mission over time [10, 11]. Policy makers and public health officials use R t to assess the effectiveness ofinterventions and to inform policy. Specifically, the instantaneous reproduction number measures diseasetransmission at a given point in time, t . One can interpret it as the average number of people that eachcontagious individual at time t would infect per day, if conditions remained unchanged. We assume,following [12, 13, 14], that R t depends on the number of new cases per day, j ( t ), only through 1 − C ( t )(as in Eq.(1)) and that j ( t ) obeys the so-called renewal equation: j ( t ) = ddt C ( t ) = Z ∞ A ( t, τ ) j ( t − τ ) dτ + i ( t ) , (3)where i ( t ) is the number of imported cases per day, and τ represents the infection age, the amount oftime since an individual became infected.The function A ( t, τ ) is a product of R t and the generation-time distribution g ( t, τ )[15]. The distribution g ( t, τ ) represents the probability density for a person infected at time t − τ to be infectious at time t (as with any probability density R ∞ g ( t, τ ) dτ = 1). For simplicity, we assume that this distribution isindependent of t , i.e., g ( t, τ ) = g ( τ ), so that j ( t ) = R t Z ∞ g ( τ ) j ( t − τ ) dτ. (4)Typically, the generation distribution is unknown, though it can be approximated by assuming it is thesame as the serial-interval distribution, which refers to the time between successive cases in a chain oftransmission. We follow [16] in using a gamma function given by g ( τ ) = β α τ α − exp( − βτ ) (5)with α = 1 .
87 and β = 0 .
28. We estimate R t by [10] R t = j ( t ) ∞ R g ( τ ) j ( t − τ ) dτ ≈ ˜ j ( t ) ∞ R g ( τ )˜ j ( t − τ ) dτ , (6)3here ˜ j ( t ) ≡ dcdt = dλ ( t ) dt C ( t ) + λ ( t ) j ( t ). We assume that λ ( t ) changes slowly enough that ˜ j ( t ) ≈ λ ( t ) j ( t )and λ ( t − τ ) λ ( t ) ≈ ≤ τ (cid:46) β ), so that the approximation in Eq.(6) holds.Starting from the saturation effect of Eq.(1), we wish to find two unknown variables namely, R and λ .With ideal data, one could immediately apply linear fitting techniques. However, the data reported bymost countries are affected by a strong weekly fluctuation. To reduce this effect, we perform a movingaverage of the R t obtained over 7 or 14 days (the results are almost identical).As Fig.2 shows, R ( t ) abruptly increased at the end of September, then decreased in the first days ofNovember. Since R ( t ) varied significantly, a simple linear fit with R as a parameter would yield poorresults. Instead we make an independent estimate of R ( t ) by doing a weighted average of the R t ’sobtained in all the regions in a country at a given calendar time t . This estimate reads as: h R ( t ) i = 1 N regions X k ∈ regions R ( k ) t − c ( k ) ( t ) , (7)where c ( k ) ( t ) is the fraction of the population recorded as having been infected in region k up to time t ,while R ( k ) t is the value obtained using Eq.(6) at time t in region k . We apply a linear fit to R t / h R ( t ) i as a function of c ( t ). Inserting Eq.(7) into Eq.(1), we obtain R t h R ( t ) i = (1 − C ( t )) = (cid:18) − c ( t ) λ (cid:19) . (8)So we interpret the intercept of the fit as an extrapolation of the initial susceptible population fractionand the slope as λ . In this section, we apply our method to data and obtain satisfactory results. Our data sources are listed inTable 1 below. The analysis was performed at the regional level in all the countries, specifically:
Regione for Italy,
Region for France,
Länder for Germany,
Comunidad autónoma for Spain,
Federal States for theUSA, and
Regions of England plus Wales, Scotland, and Northern Ireland for the UK. For each region,the total population used to calculate c ( t ) was retrieved from the most recent census available online(except for regions of France which report population size directly in the dataset). All the data files offerextended epidemiological reports; however, we only used the number new CoViD-19 cases per day.In some cases, official institutions supply data by date of onset of symptoms, as in the case of Italy, seeFig.1. This type of data is especially nice for finding the instantaneous reproduction number R t . Indeed,Italian Istituto Superiore di Sanità currently uses these values because they are virtually unaffected by One should use instead h R ( t ) i = N regions P k ∈ regions R ( k ) t − c ( k )( t ) /λ but the value of λ at this stage is unknown. We usea recursive approach inserting the value obtained in the linear fit procedure explained below and then repeating the wholeprocedure. In any case we see a negligible difference in the value of λ obtained by using the simplest factor 1 − c ( k ) ( t ) as inEq.(7), so we present the simplest approach. ountry data source Italy Dipartimento della Protezione Civile [17] and Istituto Superiore di Sanità [18]France Santé publique France [19]Spain Ministerio de Sanidad [20] and Datadista [21]Germany Robert Koch Institute [22] and Risklayer GmbH [23]UK The official UK Government website [24]USA The Covid Tracking Project [25]
Table 1:
Summary of primary data sources available both at national and "regional" level. R t da il y sy m p t o m a t i c c a s e s symptoms onset dateITALYR t Symptomatic cases
Figure 1:
Number of symptomatic cases in Italy per day reported by date of onset of symptoms (right axis, log scale) and R t (left axis, linear scale) obtained using Eq.(6) and then averaged over a week. The vertical line indicates the most recentdata reported as consolidated. fluctuations in testing capacity. Unfortunately, these data are only available for the whole country ratherthan particular regions. So, we only used these data to compare our measure of R t with the one basedon the raw data of new infections. We find that our measure is accurate up to reasonable statisticalfluctuations.Now we turn to Fig.2 and Fig.3. For each country we plot the value of R ( k ) t for its regions versus timeand R t / h R ( t ) i versus c ( t ) along with the best fit for λ . The fit was performed neglecting the first 100days of the epidemic when testing was quite irregular and the number of PCR tests extremely low. Theerror on λ was evaluated by changing this threshold from 50 to 150 days.5 R t calendar dateITALY R t(k)
52 1 . . [0 . , . .
45 0 . . [0 . , . .
10 0 . . [0 . , . .
74 1 . . [0 . , . .
70 1 . . [0 . , . .
70 1 . Table 2:
Results of the testing variable estimation as for January 2021.
As the vaccination campaign continues, we must consider the effects of immunization through vac-cination on our analysis. The functional form will change to R t / h R i ∼ (1 − c ( t ) /λ ) (1 − ν ( t )) ’ (1 − c ( t ) /λ − ν ( t )), where new term takes in account the vaccinated fraction of the population ν ( t ).If data on the vaccination campaign are available at the regional level, and ν ( t ) reaches a substantialvalue, one should perform a two-dimensional fit of R t / h R i as a function of c ( t ) and v ( t ). The slope ofthe iso- R t / h R i lines, i.e., the lines in the c ( t ) − v ( t ) plane of constant R t / h R i , will give another estimateof λ . This will allow us to compare and contrast the effects of vaccination and new infectious cases on R t . If λ ’
1, their effects should be the same. If λ <
1, we expect that the effect of a given number ofinfectious individuals on R t to be larger than that of the same number of vaccinated. Robustness check using seroprevalence data
Our findings can be checked by comparing our results withalternative methods used to evaluate the true number of people that have had SARS-CoV-2. One of themost reliable methods is the analysis of seroprevalence of IgG antibodies in blood. We compare our λ values with those obtained by two papers [26, 27] and a dashboard on an official website [28].In particular, [26] reports the fraction of people showing IgG antibodies to SARS-CoV-2 over the numberof officially reported cases as of the end of May 2020. This ratio should be compared with λ − . In Table3 of [26] data for 10 sites are reported. They range from a minimum of 6 . . .
9. In Fig.4 (left panel), we reportthe value of λ using data covering the same period, namely ending May 23 rd , 2020. We report our best fityielding λ = 0 .
06, together with the line corresponding to the λ obtained from serological data describedin [26]. The same approach was followed in Fig.4 (right panel), where we report the typical value retrievedfrom [28] (between 6 and 7 times the official number of cases) with our fit at the date of the last update8 R t / < R ( t ) > c(t)USA until 23th May 2020 λ =0.06Missouri a Connecticut a New York a
80 90 100 110 120 130 da ys s i n c e J an s t, R t / < R ( t ) > c(t)USA until 30th Jul 2020 λ =0.08Dashboard 100 120 140 160 180 200 da ys s i n c e J an s t, Figure 4: (left panel, color online) best fit value for λ through May 23 rd , 2020 compared with λ a obtained from Table 3 of[26] for New York ( λ = 11 .
9) and for the extremal cases of Missouri ( λ = 1 / .
8) and Connecticut ( λ = 1 / . th , 2020 compared with the average value given by[28] ( λ = 1 / of the website (July 2020).Passing from late spring to the end of summer, λ increased considerably. This effect is apparent in thedashboard data [28] and the same pattern holds in all countries analyzed. We presume this increase isdue to increased testing ability and contact tracing efforts made in all countries. Nevertheless, as a finalremark, we stress that if our results are confirmed, in all countries the tracing capacity is not enough, byitself, to mitigate the spread of SARS-CoV-2. Due to the large contribution of asymptomatic or mild-symptomatic cases, we think that reaching a value of λ ’ Knowledge of true prevalence of CoViD-19 is critical for informing policy decisions about how to distributeresources and manage the impacts of CoViD-19 on public health, society and the economy[34, 35]. Thetrue scale of the epidemic can affect economic development since it reduces long-run economic growth bylimiting the size of social networks. On the contrary a low prevalence estimate could lead people to take9ore epidemic-relevant risks, making disease eradication impossible using social distancing policies only.We have proposed a method to estimate the true number of infected people, unveiling the true scale ofCoViD-19 by using PCR test data alone.We the used this method to estimate a more reliable case fatalityrate. Our approach is a phenomenological estimate of the true scale of the epidemic, since it is basedon an empirical relationship between phenomena, in a way which is consistent with fundamental theory,but is not directly derived from that theory. Consequently, our method can be affected by errors whichwe are not able to distinguish from the lack of theoretical understanding. Nevertheless, there are tworemarkable results. The few attempts to assesss the true impact of SARS-CoV-2 via serological testsyield λ -values comparable in order of magnitude with our calculations [36]. Moreover, countries thatperformed better tracing, like the USA, experienced a fatality rate lower than Italy, France, and Spain (inthe first wave). Improved contact tracing performance (higher λ ) leads to a lower fatality rate because therate is calculated using the correct denominator. Good contact tracing can also help keep the epidemicunder control, preventing the virus from reaching fragile and elderly people. Thus leading to a bona fidedecrease in the fatality rate.We have proposed that the study of the graph R t vs c ( t ) and R t / h R ( t ) i vs c ( t ) can provide useful insightinto epidemic dynamics. These methods should be useful in other epidemics with incomplete data dueto the presence of unrecorded contagious individuals, for example, the seasonal flu. Minor modificationsallow this approach to keep its effectiveness even in the presence of active vaccination campaigns. Bycontributing to a better-informed response to epidemics, we believe this work serves to benefit society asa whole. Fabio Vanni acknowledges support from the European Union’s Horizon 2020 research and innovationprogramme under grant agreement No.822781 GROWINPRO - Growth Welfare Innovation Productivity.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
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