Report

Latest update: December 4th, 2021.

Khailaie^1,* Sahamoddin, Mitra^1,* Tanmay, Bandyopadhyay^1,§ Arnab, Schips^1,§ Marta, Mascheroni¹ Pietro, Vanella² Patrizio, Lange² Berit, Binder¹ Sebastian, Meyer-Hermann^1,3,4,# Michael

¹Department of Systems Immunology and Braunschweig Integrated Centre of Systems Biology (BRICS), Helmholtz Centre for Infection Research, Rebenring 56, D-38106 Braunschweig, Germany
²Department of Epidemiology, Helmholtz Centre for Infection Research, Inhoffenstraße 7, D38124 Braunschweig, Germany
³ Institute for Biochemistry, Biotechnology and Bioinformatics, Technische Universität Braunschweig, Braunschweig, Germany
⁴Cluster of Excellence RESIST (EXC 2155), Hannover Medical School, Carl-Neuberg-Strasse 1, 30625 Hannover, Germany
^*shared first author, alphabetic order
^§equal contribution, alphabetic order
^# corresponding author

This report is a frequently updated analysis of the model described in our article.

Time-varying reproduction number for Germany
Evolution of reproduction number for federal states of Germany
- R_t over time for individual federal states
Our model
- Model structure

The basic reproduction number (R_0) is a good measure for the long-term evolution of an epidemic that can be derived from models. However, it assumes constant conditions over the whole period analysed. We opted for a shifting time window in each of which the reproduction number (R_t) is determined (time-window of 10 days and shifting time of 1 day). We developed an automatized algorithm for the fast analysis of the current R_t. Importantly, each time window is not analysed independently but includes the history of the epidemic by starting from the saved state of the simulation at the beginning of each time window. This analysis was developed for the sake of providing a daily updated evaluation of the reproduction number suitable to support political decisions on non-pharmaceutical interventions in the course of the CoV-outbreak and applied to German data.

The estimate of the reproduction number is based on confirmed cases exclusively and ignores any estimate of unreported cases.

Because of dynamic changes in the reported case numbers of the final days as the result of delayed reporting, calculating Rt on those might be misleading. Therefore, we do not show the result for the two last days.

Interpretation of the reproduction number: It is important to note that, while the rate of new infections is decreasing for values of R_t<1, achieving a value of slightly below 1 by itself is not sufficient but means that no uncontrolled outbreak is imminent. In particular, it is not an indication that contact restrictions should be released, as this will most likely increase its value to greater than 1 again. In fact, the value should be substantially lower than 1, as the time to eradication of the virus from the population can still be very long for values of slightly below 1.

Time-varying reproduction number for Germany

Data for Germany were ﬁtted to the cumulative number of reported cases in a sliding time window with a size of 10 days. The transmission rate R_1 was varied to ﬁt to the data. Parameter sets were randomly sampled within the speciﬁed ranges and, upon reﬁtting, this induced a variability of reported R_t values. The box plot shows the 25 and 75 percentiles as well as the min and the max values. Outliers were removed. Red line shows the average of daily Rt median within a week ending at the given date.

Prospective analysis: 7-days incidence

The simulations from the previous section are continued from the final date assuming a stable R_t value equal to the 7-days moving average. The prospective number of weekly new cases (symptomatic and reported) is obtained for the next three weeks and shown as the 7-days incidence per 100K population. The box plot shows the 25 and 75 percentiles as well as the min and the max values. Outliers were removed.

Performance of prospective analysis

The predictive power of the model is analyzed retrospectively by comparing the data with the forecast of the increase in the total registered cases from a given date in two weeks timeframe. The prediction is based on the average of daily median R_t-value in the week ending with the given date. The percentage of the prediction error is shown in the figure below. Positive and negative values indicate overestimation and underestimation, respectively. Changes in the pandemic dynamics can be inferred from a large prediction error (green and red regions).

Current reproduction number (`R_t`) for federal states

Evolution of reproduction number for federal states of Germany

The same analysis as in the previous section was performed for all federal state in Germany separately.

`R_t` over time for individual federal states

Baden-Württemberg

Bayern

Berlin

Brandenburg

Bremen

Hamburg

Hessen

Mecklenburg-Vorpommern

Niedersachsen

Nordrhein-Westfalen

Rheinland-Pfalz

Saarland

Sachsen

Sachsen-Anhalt

Schleswig-Holstein

Thüringen

Our model

Based on a classical model of infection epidemics, we developed a mathematical model particularly adapted to the requirements and specificities of the CoV-outbreak (SECIR model). An in-depth description can be found in our article.

Model structure

*The scheme of the SECIR model. The model distinguishes healthy individuals without immune memory of COVID-19 (S), infected individuals without symptoms but not yet infectious (E), infected individuals without symptoms who are infectious (pre-symptomatic (C_I) and asymptomatic (C_R) carriers), infected symptomatic individuals who are not yet detected (I), and detected (I_{H,R}) and undetected (I_X) symptomatic patients. Further, compartments for hospitalization in non-critical (H_{U,R,S}) and critical/intensive care units (U_{D,R}) were introduced to monitor the load on the healthcare system. Detected patients recover from different states of the disease (R_Z) or die (D). Undetected individuals who went through the infection and recovered are also taken into account (R_X).