key: cord-0254723-tuhxjzxo authors: Siebler, L.; Rathje, T.; Calandri, M.; Stergiaropoulos, K.; Richter, B.; Nusseck, M.; Spahn, C. title: A Coupled Experimental and Statistical Approach for an Assessment of the Airborne Infection Risk in Event Locations date: 2022-01-11 journal: nan DOI: 10.1101/2022.01.10.22269028 sha: dafd8233a3f9ef5df26c890504f43c3be93305a7 doc_id: 254723 cord_uid: tuhxjzxo Operators of event locations are particularly affected by a pandemic. Resulting restrictions may cause uneconomical business. With previous models, only an incomplete quantitative risk assessments is possible, whereby no suitable restrictions can be derived. Hence, a mathematical and statistical model has been developed in order to link measurement data of substance dispersion in rooms with epidemiological data like incidences, reproduction numbers, vaccination rates and test qualities. This allows a first time overall assessment of airborne infection risks in large event locations. In these venues displacement ventilation concepts are often implemented. In this case simplified theoretical assumptions fail for the prediction of relevant airflows for infection processes. Thus, with locally resolving trace gas measurements and specific data of infection processes, individual risks can be computed more detailed. Via inclusion of many measurement positions, an assessment of entire event locations is possible. Embedding the overall model in a flexible application, daily updated epidemiological data allow latest calculations of expected new infections and individual risks of single visitors for a certain event. With this model, an instrument has been created that can help policymakers and operators to take appropriate measures and to check restrictions for their effect. The pandemic of SARS-CoV-2 forced cultural institutions such as theatres and music halls to restrict or cancel their programs, without knowing the actual infection risk at their location. In order to identify this risk and take appropriate measures scientific evidences and risk estimation models for these venues are required. Airborne transmission of viruses is a complex process involving emission, dispersion in the room, and inhaling [1] . There are many calculation models for an aerosol transmission and infection risk of SARS-CoV-2 in indoor environments [2] [3] [4] [5] [6] [7] . However, these calculators often require idealised assumptions (e.g. ideal mixed ventilation), which lose accuracy with the size of the room. Therefore, they mainly consider small to medium room sizes. In larger event locations, displacement ventilation concepts are often implemented, whose virus transmission towards neighbors is challenging to predict. Unobjectionable vertical buoyancy flows are superimposed by critical horizontal flows due to disturbance effects (e.g. cold walls and leaky doors). The estimation becomes even more critical in large and complex rooms, when relevant boundary conditions are unknown. [8] Moreover, these models neglect the access probabilities of infectious persons, which can be derived from specific epidemiological data. In summary, it has to be stated, that the required overall risk assessment model for large locations is still missing. Within the framework of a project at the Stuttgart State Theatre, locally resolved trace gas measurements were carried out to calculate the airborne infection risk. Based on these investigations, a coupled experimental and statistical model is essential. Depending on the epidemiological data (incidence and reproduction number, vaccination rate and test quality) the access probability of an infectious person to an event location varies. Furthermore vaccinations prevent infections when sitting near an infectious person. All the data are highly relevant for an overall risk assessment, especially for locations with several hundred people. Hence, the research objective is to develop a coupled experimental and statistical model that accounts for these data. The results will provide scientifically based recommendations for actions non-pharmaceutical interventions (NPI) to maintain the operation of cultural institutions under pandemic conditions. The approach of the model can be separated in several sections, which are interacting with each other. This calculation procedure is shown in figure 1 and assists to comprehend the structure of the paper by means of a section assignment. The focus of previous indoor air investigations using trace gas, also conducted at the University of Stuttgart, has mostly been on evaluating the ventilation effectiveness rather than infection risks [9] . For this new purpose, however, such measurements are also well suited, whereby a transfer to viral loads has to be performed. The method even provides reliable results for unknown air flows resulting from partly uncertain boundary conditions. to an infection with a certain probability. As a result the following correlation is announced: with P I and D q as PIRA and dose of inhaled quanta, respectively. Siebler et al. [8] suggests an experimental approach with trace gas for ventilation with outdoor air exchange. It is based on releasing a certain rate of gas with a mass flow controller. Measuring the neighboring concentrations with a gas analyser enables users the quantification of substance dispersion in general. Assuming that trace gases are dispersed in the same way as relevant virus-bearing aerosols [12] , a transfer to infection risk is possible. In order to link trace gas to quanta and to account for mask filtration effects the following approach is introduced: For any other density an alternative positioning might be more suitable. The probability P acc of infectious persons in an event location can be calculated with the binomial distribution of Gauss [13] , if the probability p acc of an individual infectious person gaining access is known. Equation 4 shows the calculation in detail: with n and k as number of persons in the location and number of infectious persons respectively. In order to keep the complexity and the effort of the statistical model in acceptable range it is only practical for maximum one infectious person in the location. For small p (for high vaccination rates and a high quality test procedure) the following assumption and resulting equation is appropriate and above all conservative [13] : For an accurate the identification of p acc , different types and dimensions of classifications are possible in principle, depending on the accessibility of the data. In this case, the status of vaccination is significant. A subdivision in not vaccinated, recovered and vaccinated allows higher accuracy for predicting p acc . An additional subdivision in several different vaccine types due to different effectiveness is specifying the results further. For event locations, the age distribution of visitors might be a relevant parameter. The vaccination rate and the vaccine types are often age dependent. In Table 2 classification of visitors and associated data For each class combination there is a probability of p ij for a person belonging to it. For instance, an arbitrary person of possible visitors, has the probability p 31 to be mRNA-vaccinated and between 0. . . 29 years old with a related vaccine effectiveness η 31 and incidence number I 31 . Incidence numbers in general describe the rate of (new) infections per time and inhabitants. It is assumed that once someone knows that they are infectious they generally stay at quarantine or at least they do not try to get access to an event location. Therefore, predicting incidence rates is relevant to the probability of gaining access being infectious p acc ij because presymptomatic, asymptomatic, and unwittingly infectious persons are critical. This forecast can be estimated by the reproduction number R (R-value), the mean duration between being infected and infecting the next person ∆t inf and the mean critical duration between one becomes infectious and the knowledge about it ∆t crit . R describes how many persons one infectious person infects in average. For even higher accuracies, reproduction numbers could optionally be further classified by age groups and vaccination/recovery status. The following equation shows the correlation between the aforementioned values assuming an exponential course: with I 0 , τ , ∆t inf , R as initial overall incidence (no age and vaccination/recovery status related separation), infection transmission constant in 1/d, mean duration between being infected and infecting the next person and reproduction number (R-value) respectively. The course of the incidence value correlates with the constant τ , which can be derived from equation 6: To estimate the critical ratio of infectious persons, who do not know about their infectiousness at a certain day, it is useful to predict the incidence course up to the duration ∆t crit later. The integral of that course over duration describe persons, who could be infectious over that period (∆t crit ) without knowing it. It is calculated as: The access probability of an infectious person of a certain class combination (e.g. 23) is depending on its certain incidence (e.g. I 23 ). In addition, the test quality with regard to false negative results of different principles (antigen, polymerase chain reaction (PCR)) is considered [14] [15] . with p acc ij , I ij , p fnt as access probability of an infectious person of a certain class combination ij, incidence of a certain class combination ij and probability of a false negative test result (if no test is done p fnt = 1) respectively. The calculation process of an overall probability for one arbitrary person being infectious and getting access is shown in figure (3 Equations (9) and (10) filled in (5) result for the access probability of an infectious person in: In this section the two outputs of the model are derived. Besides the expected value of new infections, which is relevant for operators and policy makers, an infection risk for an individual person is determined. Once an infectious person is in the location, the vaccination status of their neighbors is relevant. Taking the vaccination impact twice into account (access probability and neighbor vaccination status) is noteworthy for this model. Due to the classification, see table 2, equation (3) becomes more complex. Thereby the probability of neighbors belonging to a certain class combination is taken into account. In the following equation the union between measurement and statistical data occurs. with µ vac , η ij as expected value of new infections regarding the vaccination status and their efficiency respectively. Combining equation (12) with the probability of access (equation (11)) results in the overall expected value of new infections: with µ total as expected value of new infections regarding entrance probability and vaccination status. This calculation provides a valuable instrument for operators of event locations and politicians for a total risk assessment of infections. With regard to an individual person the probability of being exposed in a certain neighbor ring of an infectious person has to be taken into account as follows: with P ind , η ind as individual infection risk of one certain person and event date and efficiency of the individual vaccination status respectively. This calculation enables visitors to estimate their individual infection risk regarding the decision of participating an event. The coupled experimental and statistical model assumes that maximum one infectious person is present. Due to superposition of trace gas results risk assessments for more than one infectious person could be calculated. However, the effort for this statistical model adjustment would become laborious. Figure 4 shows the deviation between P acc (k = 1) and k=n k=1 P acc (k) (which are assumed to be approximately equal (equation (5)) in this study) for n = 100 and n = 500 visitors. For n = 500 up to p acc ≈ 1 × 10 4 and n = 100 up to p acc ≈ 1 × 10 3 the deviations are assumed low. Hence even for large locations the model stays conservative and also acceptable for small p acc . With this approach users could estimate whether this model is still accurate enough. Otherwise, operation with such boundary conditions is questionable anyway. It should be noted that included statistical data must be robust and detailed. Given the fact, that infection processes are dynamic, possible virus mutations should be considered when calculating the infection risk. It is assumed that responsible action will be taken and that quarantine measures will be followed in the case of known infection (no attendance at events). Otherwise, the accuracy of the model will be affected. Furthermore, in order to validate the model appearing infections can be taken into account (e.g. p fnt or the classification in general). • Data availability statement: Datasets generated/analysed during the current study are available from the corresponding author on reasonable request. • Funding statement: The authors received a funding from the Stuttgart State Theatre for the development of a simplified infection risk model compared to the presented one. • Conflict of interest disclosure: The authors declare no competing interests. • Ethics approval statement: Not applicable. • Patient consent statement: Not applicable. • Permission to reproduce material from other sources: Not applicable. 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Wiley StatsRef: Statistics Reference Online Comparative sensitivity evaluation for 122 cemarked rapid diagnostic tests for sars-cov-2 antigen, germany Antikörpertests bei COVID-19 -Was uns die Ergebnisse sagen Acknowledgments. We would like to thank Dr. Tjibbe Donker who supported us with his expertise regarding the infection process. Another special thanks goes to Mr. Donagh Hennessy for his scientific editing services.