key: cord-0884254-mhqe3xjz authors: Küchenhoff, H.; Guenther, F.; Höhle, M.; Bender, A. title: Analysis of the early Covid-19 epidemic curve in Germany by regression models with change points date: 2020-10-30 journal: nan DOI: 10.1101/2020.10.29.20222265 sha: 64a78356c2986d6c2f10b2d519aa55967c0d6d0b doc_id: 884254 cord_uid: mhqe3xjz We analyze the Covid-19 epidemic curve from March to end of April 2020 in Germany. We use statistical models to estimate the number of cases with disease onset on a given day and use back-projection techniques to obtain the number of new infections per day. The respective time series are analyzed by a Poisson trend regression model with change points. The change points are estimated directly from the data without further assumptions. We carry out the analysis for the whole of Germany and the federal state of Bavaria, where we have more detailed data. Both analyses show a major change between March 9th and 13th for the time series of infections: from a strong increase to a stagnation or a slight decrease. Another change was found between March 24th and March 31st, where the decline intensified. These two major changes can be related to different governmental measures. On March, 11th, Chancellor Merkel appealed for social distancing in a press conference with the Robert Koch Institute (RKI) and a ban on major events with more than 1000 visitors (March 10th) was issued. The other change point at the end of March could be related to the shutdown in Germany. Our results differ from those by other authors as we take into account the reporting delay, which turned out to be time dependent and therefore changes the structure of the epidemic curve compared to the curve of newly reported cases The first phase of the Covid-19 pandemic in Germany was managed relatively successful in comparison to other countries in Europe. Therefore, it is worth taking a closer look at the course of the pandemic in Germany, which has already led to controversial discussions. This particularly concerns the important question about the effectiveness of various control measures. There are several publications using data from different countries on the effects of control measures, see, e.g., [1] , [2] and [3] . As [4] point out, many of such studies are undermined by unreliable data on incidence. Many papers use data provided from the Johns Hopkins University (JHU) [5] . These data are based on cumulative registered cases in different countries, which induces several problems, particularly the fact that not all cases are reported and that there is delay between the day of infection and the reporting day. Furthermore, the systems of reporting vary between countries, which makes comparisons between countries difficult. Therefore we focus on the analysis of the epidemic curve in Bavaria and Germany. The availability of case based data for Bavaria and detailed data for disease onset for Germany is essential for our analysis. In a recent paper on Germany by [6] , the authors use a complex Bayesian modeling approach based on the daily registrations in the JHU data. An important claim by [6] is that the lock-down-like measures on March 23rd were necessary to stop exponential growth, however, this result contradicts for example results by the German RKI [7] . Furthermore, these approaches were critically questioned by [8] and [9] , where the latter emphasized the importance of taking into account the delay by reporting and incubation time, when analyzing the possible effect of non pharmaceutical interventions. We follow this line of argument and use a statistical model to estimate daily numbers of infected and of persons with disease onset on a certain day. For Bavaria, we use detailed case-based data, while a modeling approach is utilized for German data. We analyze the respective epidemic curves using a segmented regression model with change points. The paper is organized as follows. In Section 2, we present the data and the the strategy of estimating the relevant daily counts. Then the segmented regression model, which is the basis for further analyses, is presented. In Section 3, we present the results followed by a discussion in Section 4. the latter is not always known: partly because it could not be determined and partly because the case did not (yet) have any symptoms at the time of entry into the data base. A procedure for imputation of missing values regarding the disease onset has been developed by [10] , using a flexible generalized additive model for location, scale and shape (GAMLSS; [11] ), assuming a Weibulldistribution for time t d > 0 between disease onset and reporting date. We estimate the delay time distribution from data with disease onset and impute missing disease onsets based on this model. For the German data, no individual case data were available, so instead we used estimated disease onset data provided by the Robert Koch Institute (RKI), see [12] and [7] . The method used by the RKI is similar to our approach applied to Bavarian data [10] . To interpret the course of the epidemic and possible effects of interventions, case based data on time of infection is essential. However, as such data is generally not available, one simple approach is to shift the curve to the past by the average incubation period. The average incubation period for COVID-19 is about five days [13] . A more sophisticated approach is to use the incubation period distribution as part of an inverse convolution, also known as backpropagation, in order to estimate the number of infections per day from the time series of disease onsets [14, 15] . We assume a log-normal distribution for the incubation time with a median of 5.1. days and a 95% percentile at 11.5 days [13] . These are the same values as used by [6] . For our calculation, we use the back-projection procedure implemented in the R package surveillance [16] . To analyze the temporal course of the infection we use the following Poisson regression model with over-dispersion and change points (see [17] , [18] ): where E(Y t ) is the expected number of new reported cases at time t, K is the number of change points, and x + = max(x, 0) is the positive part of x. The change points are used to partition the epidemic curve Y t into K + 1 phases. These are characterized by different growth parameters. In the phase before the first change point CP 1 the growth is characterized by the parameter β 1 , in the 2nd phase between CP 1 and CP 2 by β 2 = β 1 + γ 1 . The next change is then at time CP 2 . In the 3rd phase between CP 2 and CP 3 the growth parameter is given by β 3 = β 1 + γ 1 + γ 2 . This applies accordingly until the last phase after CP K . The quantities exp(β j ), j = 1, . . . , K + 1 can be interpreted as daily growth factors. Since model (1) is a generalized linear model given the change points, the parameters of the model (including the change points) can be estimated by minimizing the quasi likelihood function for the Poisson model. Due to the estimation of the change points the numerical optimization problem is not straight forward. For the estimation of the model we use the R-package segmented, see [19] . The starting values are estimated by discrete optimization using all possible integer suitable combinations of change points. The number of change points K is varied step-wise up to a maximum of K = 4. It is examined whether the increase of the number of break points leads to a relevant improvement of the model fitting (over-dispersion parameter). Models with more than 4 change points since they are hardly interpretable and the danger of overfitting is high. We apply the segmented regression model to time series of the estimated daily numbers of infections for Bavaria and Germany. Since the back propagation algorithm yields an estimate for the expected values of the number of daily infections and does so by inducing a smoothing effect, as a sensitivity analysis for the location of the breakpoints, we also apply the model to the time series of the daily number of disease onsets. When comparing the results, it should be taken into account that the onset of the disease is on average 5 days after infection. In Figure 1 , the three different time series of daily cases (reported, disease onset and estimated infection date) are presented. The delay between the three time series for Bavaria and Germany is evident. Furthermore, the curves do not just differ by a constant delay, but there is some change in structure of the curves. The curve relating to the date of infection is clearly smoothed due to the back projection procedure (cf. Section 2.2) and has a clear maximum both for Bavaria and Germany. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted October 30, 2020. ; For the Bavarian data on disease onset, the model with K = 4 change points gives the best result with an estimate of the over-dispersion parameter of 3.8, i.e., the variance of Y t is 3.8 times higher than the value of Var(Y t ) = E(Y t ) otherwise expected under the assumption of the Poisson regression model. The over-dispersion for a model with K = 3 change points is substantially higher (4.5), which suggests that the model with four change points should be preferred. Table 1 , left panel. The model delivers five phases starting with a steep increase, which is slowed down in the second phase. In the third phase starting at 15th-17th of April, the increase is stopped and there is decrease in number of disease onsets in the fourth phase, which is accelerated in the fifth phase. The Poisson model for the infection date gives a substantially better fit than the disease onset model. The model with four change points has an underdispersion by the the factor 0.39 compared to 0.79 for the model with three change points. Therefore, we use the model with three change points to avoid overfitting. The result can be seen in Figure 2 (right panel) and in Table 1 (right panel). The phases are similar to those for the disease onset, but the change from increasing curve (phase 1 and 2) to a decreasing phase (phase 3) is direct without a plateau in between. Taking the mean incubation period of five days into account, we combine the results for the two models for Bavaria, which implies four phases of the epidemic: 1st phase There is a substantial increase in new infections in both models. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted October 30, 2020. ; https://doi.org/10.1101/2020.10.29.20222265 doi: medRxiv preprint The results for the German data are presented in Figure 3 and in Table 2 . For the disease onset model, the overdispersion is substantially lower for the model with four change points than with three change points. However, the overdipersion of the model with four change points is rather high (20.1) and the confidence intervals for the change points are rather big, especially for the first part of the time series where two change points are estimated. While the distinction in the first three phases is unclear, there is a clear turning point between March 14th and March 18th. The fifth phase with a further slowdown starts at the end of April. As can be seen from Figure 3 There are further estimated change points in both models, which have wide confidence intervals and do not fit well together in the models. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted October 30, 2020. ; https://doi.org/10.1101/2020. 10.29.20222265 doi: medRxiv preprint The present analysis is a retrospective, exploratory analysis of the German and Bavarian COVID-19 reporting data during Mar-Apr 2020. The analysis does not include cases that have not been recorded. If the proportion of undetected cases changes over time, this can distort the curve and thus the determination of the change points. Therefore, additional data on daily deaths and hospital admissions and the number of tests performed should be considered. Furthermore, it is possible to estimate the proportion of undetected cases with the help of representative studies such as the one currently conducted in Munich, see [20] . Our analysis is based to a considerable extent on imputed data, see [10] , which is a results of missing data w.r.t. the disease onset. Since changes in behavior do not occur abruptly, the assumption of change points is also problematic in itself. Therefore, the interpretation of change points should always be done in conjunction with a direct observation of the epidemic curve. Our analysis is based on the onset of the disease (more precisely: the onset of symptoms) and a back projection to the date of infections, and therefore, despite its limitations, is better suited to describe the course of the epidemic than the more common analysis of daily or cumulative reported case numbers. In the analysis of the Bavarian and the German data in different settings, the main result is the change point, where the exponential growth was stopped and is clearly visible between March 9th and 13th. The timing of this change point coincides to the implementation of the first control measures: the partial ban of mass events with more than 1000 people. Furthermore, in a press conference on March 11th chancellor Merkel and the president of the RKI appealed to self-enforced social distancing (https://www.bundesgesund heitsministerium.de/en/coronavirus/chronologie-coronavirus.html. Furthermore, the extended media coverage from Bergamo, Italy, as well as the voluntary transition to home-office work could be related to this essential change in the course of the pandemic. In Bavaria and in Germany, the change point at the end of March of infection date is apparent. This change point is associated with different measures taken in March (closing of schools and stores on March 16th and the shutdown including contact ban on March 21st). Since there were many measures administered simultaneously, it is not possible to attribute individual measures to the development of the epidemic curve. The claim by [6] , that the shutdown on March 21st was necessary to stop the growth of the epidemic is not supported by our analysis. There is a change 10 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted October 30, 2020. point in the epidemic curve after that date, but the major change from an exponential growth to a decrease was before the shutdown. The difference in results can be explained by the different data bases used for the respective analyses. While [6] use data based on daily registered cases, in our analysis, data on disease onset are included. As can be seen from Figure 1 and from the results of our data analysis, the delay distribution of the time between disease onset and reporting day changed over time. Using this information is a crucial difference between our analysis and that of [6] . In a recent technical addendum [21] the authors re-fit their model on more appropriate data. These analysis -in our opinion -clearly show that the effective reproduction number decreased earlier than in their initial analysis, however, they attribute the decrease to a SIR model peculiarity, where a linear decrease in the contact rate can lead to the incidence curve dropping despite R(t) > 1. The above discussions illustrate how complex the interpretation of even simple SIR models is and the question is, if such SIR modeling is not too simple to really allow for questions to be answered model based (no age structure, no time varying reporting delay, no incubation delay). In contrast, our approach is more data driven with a minimum of modeling assumptions and without the need to include strong prior information about the change points. Directly using a segmented curve with exponential growth (decline) is in line with common models of infectious diseases in its early stages, where the limitation of the spread by immune persons plays no role. The problem of using complex models with many parameters for the evaluation of governmental measures has also been highlighted by [22] . Our approach is similar to that of [9] . However, using change point analysis for variables derived from daily new infections appears problematic, since assumptions modeling the reproduction number R(t) or the cumulative numbers are questionable. More specifically, the use of the time-varying reproduction number R(t), a standard measure to describe the course of an epidemic is challenging, as different definitions have been proposed in the literature that also imply different interpretations (see [23, 24] ). However, the analysis of R(t) as a relative measure can be useful, when one wants to analyze data from different countries with non comparable reporting systems, see [3] . We prefer the direct use of a Poisson regression model with a more plausible assumptions about the error terms instead of using OLS for logarithmic case numbers. Furthermore, we apply a direct maximum likelihood estimation of the change points of the segmented regression model. For the interpretation of the model based on disease onsets, we also use a simple difference of five days to take the incubation time into account. Our results are similar to that of [9] . However, the claim that there is no evidence for the effect of governmental measures is not supported by our analysis. Our result is in line with that of [25] , where a stop of exponential growth in Great Britain has been before the shutdown. Furthermore, the effect of governmental measures as whole is clearly documented in the literature, see, e.g., [1] and [2] .Our result on a possible effect of the ban of mass events is also in line with the results of [3] . . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted October 30, 2020. ; The temporal connection between the change points in our analysis and various control measures should be interpreted as an association, rather than a direct causal relationship. In the end many other explanations exists and from a simple time series analysis it is not possible to say to what extent the population already had changed their behavior voluntarily, as for example observed in mobility data [26] , and in what way the measures contributed to this. More speculative alternative explanations would include the possibility of a seasonal effect on coronavirus activity (e.g. related to temperature) or changes in test capacity or the case detection ratio. However, given the re-emergence of the epidemic in the fall of 2020 at high test capacity and at relatively high temperatures shows that contact behavior is the major explanatory factor for virus activity. Nevertheless, any analysis of observational time series data including only a limited amount of explanatory factors has to be interpreted with care and with respect to the many uncertainties which remain regarding COVID-19 [27] . Despite the limitations of the approach, we argue that it is advantageous and important to directly interpret the epidemic curve and the absolute number of cases, rather than indirect measures like the R(t). Furthermore, the reproduction rate does not contain information about how many people are currently affected, or whether the infected persons belong to risk groups. The course of the time-varying reproduction number calculated by us for Bavaria fits well with the change point analysis [10] . A value of R(t) >1 corresponds to a rate of increase >1, noting that the time delays in the interpretation of R(t) must be kept in mind. It should be noted, that the presented analysis is retrospective. Control measures have to be decided based on a completely different level of information than what the retrospectively established epidemic curve suggests. The simple observation of the course of the reported case numbers by reporting date is also problematic because this course is strongly influenced by the reporting behavior and the methods and capacities of the test laboratories. Typically, substantially fewer cases are reported at weekends than during the week. Therefore, the estimation [10] is an important step to estimate the better interpretable curve of new cases, but is limited by assumptions and limitations itself, that need to be considered when interpreting the results. Since the impact of the measures also depends on how they are implemented by the population (compliance), the results cannot be directly transferred to the future. Nevertheless, it remains a remarkable result that the clear turning point of the early COVID-19 infection data in Germany is associated with non drastic measures (no shutdown) and strong appeals by politicians. All data used for the analyses and all code to reproduce the models, figures and tables in the manuscript are openly and freely available from https://gi thub.com/adibender/covid19-changepoint-analysis-germany-bavaria. All 12 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted October 30, 2020. ; analyses were performed using the R programming language [28] . Figures were created using R package ggplot2 [29] . 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R News Protocol of a population-based prospective COVID-19 cohort study Munich Model-based and model-free characterization of epidemic outbreaks -Technical notes on Dehning The limits of estimating COVID-19 intervention effects using Bayesian models A New Framework and Software to Estimate Time-Varying Reproduction Numbers During Epidemics Association of Public Health Interventions With the Epidemiology of the COVID-19 Outbreak in Wuhan, China Did COVID-19 infections decline before UK lockdown ? Estimating the impact of mobility patterns on COVID-19 infection rates in 11 European countries Covid-19's known unknowns R: A Language and Environment for Statistical Computing Elegant Graphics for Data Analysis We would like to thank Katharina Katz and Manfred Wildner from the Bavarian State Office for Health and Food Safety (LGL) for providing the data and for useful discussions. We also thank Nadja Sauter for help with visualizations.