☏ Stefan Jakubek, Christoph Hametner, Zhang Peng Du, Lukas Böhler, Johanna Bartlechner

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(Updates are uploaded on Tuesdays and Fridays at 13:30.)

- ▼ SIR-Model
- ▼ \(R_{\text{eff}}\)
- ▼ Forecast
- ▼ SIR-Model for Two Age Groups
- ▼ Scenario Analysis
- ▼ Data Preprocessing

Due to continuing low case numbers, the analysis will be paused. If the number of cases increases considerably, the analysis will be restarted.

Due to non consistend data reporting, the analysis will be paused.

scheduled updates only on Fridays

**Epidemic course associated with non-pharmaceutical interventions: ** The presented analysis of the current epidemic is based on a SIR model which is augmented by an exogenous input. This input is estimated by differential flatness. The figure above shows the two compartments of Infected \(I\) (top) and Susceptible \(S\) (center). The association with intervention measures in form of light (or soft) lockdowns (colored in blue) and hard lockdowns (colored in pink) can be seen in the course of these compartments. By using the ratio \(S/S_{\text{crit}}\) as presented, an interpretation similar to the effective reproduction number \(R_{\text{eff}}\) is possible, which serves as a leading indicator for an epidemic surge for \(S/S_{\text{crit}}>1\).

In the bottom figure, the estimated aggregated exogenous drivers \(u\) are shown, which are the cumulated effects or inputs necessary to describe the observed courses of the epidemic. The aggregated exogenous drivers \(u\) can be interpreted as a flow of individuals into or out of the compartment of susceptibles and thus either fueling up or slowing down the epidemic. If \(u\) is zero, the epidemic behavior is described by a classic SIR model.

On the right side, the state trajectory of \(I\) and \(S/S_{\text{crit}}\) is shown in the phase plane. Critical developments of the epidemic (i.e. an influx into the susceptible compartment due to exogenous driving mechanisms) can be detected easily. The phase evolution can vividly describe the potential and the direction of an epidemic, which is indicated by the trend (orange line) in the phase plane.

**Epidemic course associated with non-pharmaceutical interventions: ** The presented analysis of the current epidemic is based on a SIR model which is augmented by an exogenous input. This input is estimated by differential flatness. The figure above shows the two compartments of Infected \(I\) (top) and Susceptible \(S\) (center). The association with intervention measures in form of light (or soft) lockdowns (colored in blue) and hard lockdowns (colored in pink) can be seen in the course of these compartments. By using the ratio \(S/S_{\text{crit}}\) as presented, an interpretation similar to the effective reproduction number \(R_{\text{eff}}\) is possible, which serves as a leading indicator for an epidemic surge for \(S/S_{\text{crit}}>1\).

In the bottom figure, the estimated aggregated exogenous drivers \(u\) are shown, which are the cumulated effects or inputs necessary to describe the observed courses of the epidemic. The aggregated exogenous drivers \(u\) can be interpreted as a flow of individuals into or out of the compartment of susceptibles and thus either fueling up or slowing down the epidemic. If \(u\) is zero, the epidemic behavior is described by a classic SIR model.

On the right side, the state trajectory of \(I\) and \(S/S_{\text{crit}}\) is shown in the phase plane. Critical developments of the epidemic (i.e. an influx into the susceptible compartment due to exogenous driving mechanisms) can be detected easily. The phase evolution can vividly describe the potential and the direction of an epidemic, which is indicated by the trend (orange line) in the phase plane.

If \(u(t)\) is constant for a prolonged period of time, \(S/S_{\text{crit}}\) approaches one and a state of endemic equilibrium is reached. In the figure above, a range of endemic equilibrium (green area) is displayed. If there are no considerable changes in \(u(t)\), the number of active cases stays within the colored area.

**Epidemic course: ** The presented analysis of the current epidemic is based on a SIR model which is augmented by an exogenous input. This input is estimated by differential flatness. The figure above shows the two compartments of Infected \(I\) (top) and Susceptible \(S\) (center). By using the ratio \(S/S_{\text{crit}}\) as presented, an interpretation similar to the effective reproduction number \(R_{\text{eff}}\) is possible, which serves as a leading indicator for an epidemic surge for \(S/S_{\text{crit}}>1\).

In the bottom figure, the estimated aggregated exogenous drivers \(u\) are shown, which are the cumulated effects or inputs necessary to describe the observed courses of the epidemic. The aggregated exogenous drivers \(u\) can be interpreted as a flow of individuals into or out of the compartment of susceptibles and thus either fueling up or slowing down the epidemic. If \(u\) is zero, the epidemic behavior is described by a classic SIR model.

On the right side, the state trajectory of \(I\) and \(S/S_{\text{crit}}\) is shown in the phase plane. Critical developments of the epidemic (i.e. an influx into the susceptible compartment due to exogenous driving mechanisms) can be detected easily. The phase evolution can vividly describe the potential and the direction of an epidemic, which is indicated by the trend (orange line) in the phase plane.

**Epidemic course: ** The presented analysis of the current epidemic is based on a SIR model which is augmented by an exogenous input. This input is estimated by differential flatness. The figure above shows the two compartments of Infected \(I\) (top) and Susceptible \(S\) (center). By using the ratio \(S/S_{\text{crit}}\) as presented, an interpretation similar to the effective reproduction number \(R_{\text{eff}}\) is possible, which serves as a leading indicator for an epidemic surge for \(S/S_{\text{crit}}>1\).

In the bottom figure, the estimated aggregated exogenous drivers \(u\) are shown, which are the cumulated effects or inputs necessary to describe the observed courses of the epidemic. The aggregated exogenous drivers \(u\) can be interpreted as a flow of individuals into or out of the compartment of susceptibles and thus either fueling up or slowing down the epidemic. If \(u\) is zero, the epidemic behavior is described by a classic SIR model.

On the right side, the state trajectory of \(I\) and \(S/S_{\text{crit}}\) is shown in the phase plane. Critical developments of the epidemic (i.e. an influx into the susceptible compartment due to exogenous driving mechanisms) can be detected easily. The phase evolution can vividly describe the potential and the direction of an epidemic, which is indicated by the trend (orange line) in the phase plane.

If \(u(t)\) is constant for a prolonged period of time, \(S/S_{\text{crit}}\) approaches one and a state of endemic equilibrium is reached. In the figure above, a range of endemic equilibrium (green area) is displayed. If there are no considerable changes in \(u(t)\), the number of active cases stays within the colored area.

**Comparison of effective reproduction numbers with different origins:** For the analysis of the COVID-19 epidemic, the effective reproduction number \(R_{\text{eff}}\) of a country can be utilized as a measure to describe the infection activity over time (e.g. see [3,4,5]). The effective reproduction number \(R_{\text{eff}}\), which is often based on statistical modeling [6], indicates epidemic activity, i.e. if \(R_{\text{eff}}>1\) applies. The reproduction number \(R_{\text{eff}}\) obtained from differential flatness is compared to the officially reported reproduction number \(R_{\text{eff}}\), which is based on governmental data [1] and calculated based on statistical methods [7].

**Forcasting the epidemiological dynamics by using the exogenous drivers: **The aggregated exogenous drivers \(u\) can be utilized to predict or evaluate the epidemic. In this diagram, a potential course of \(u\) is based on previously observed courses of \(u\). This is set to predict a future course (orange line). Note that such a projection of \(u(t)\) critically depends on weather or not governmental interventions are changed in the predicted time interval. The associated responses of \(I\) and \(S\) are visible in the subfigures above together with a prediction interval \(\text{PI}\). Of special interest for such an analysis is, if and when \(S/S_{\text{crit}}\) exceeds one, which would fuel up the epidemic in the observed country.

**Forcasting the epidemiological dynamics by using the exogenous drivers: **The aggregated exogenous drivers \(u\) can be utilized to predict or evaluate the epidemic. In this diagram, a potential course of \(u\) is obtained from a least-squares estimation of the last fourteen days. This is set to predict a future course of three weeks ahead (orange line). Note that such a projection of \(u(t)\) critically depends on weather or not governmental interventions are changed in the predicted time interval. The associated responses of \(I\) and \(S\) are visible in the subfigures above together with a confidence interval. Of special interest for such an analysis is, if and when \(S/S_{\text{crit}}\) exceeds one, which would fuel up the epidemic in the observed country.

**Forcesting the epidemiological dynamics by using the exogenous drivers: **The aggregated exogenous drivers \(u\) can be utilized to predict or evaluate the epidemic. In this diagram, a potential course of \(u\) is obtained from a least-squares estimation of the last four weeks. This is set to predict a future course of three weeks ahead (orange line). Note that such a projection of \(u(t)\) critically depends on weather or not governmental interventions are changed in the predicted time interval. The associated responses of \(I\) and \(S\) are visible in the subfigures above together with a confidence interval. Of special interest for such an analysis is, if and when \(S/S_{\text{crit}}\) exceeds one, which would fuel up the epidemic in the observed country.

scheduled updates only on Tuesdays

scheduled updates only on Fridays

Due to continuing low case numbers, the analysis for Israel will be paused.

**Epidemic course of different age groups with non-pharmaceutical interventions and vaccinations: **The presented analysis of the current epidemic is also based on the SIR model with exogenous drivers, but the model is split into two discrete age groups and their respective compartments. The used datasets (raw data) are based on the data provided by the Austrian Agency for Health and Food Safety [1]. The first group comprises the population that is younger than 65 years, indicated by \({\text{0-64}}\), and the second group those people aged 65 years or above, indicated by \({\text{65+}}\). The course of the epidemic is represented separately for each group by the Infected \(I\) (top) and aggregated exogenous drivers \(u\) (bottom). The Reproduction Number \(R_{\text{eff}}\) from differential flatness is compared to the officially reported \(R_{\text{eff}}\) provided by the Austrian Agency of Health and Food Safety [1], whereas a value greater than one indicates epidemic activity (middle). Additionally, individual state diagrams are presented for each group on the right.

The infection dynamics of the two groups are coupled to each other, meaning that the groups can also infect each other since they are part of the same population and the same epidemic. Due to the size of the first age group (\({\text{0-64}}\)) the obtained results for the exogenous drivers and the phase evolution are similar to the results obtained for the entire population, since they also make up for almost 80% of the Austrian population. The results for the second age group (\({\text{65+}}\)), however, differ visibly. Especially the state diagram indicates, that the \({\text{65+}}\) group has been beneath the critical value \(S_{\text{crit}}\) for an onset of a wave for a longer period than the other group.

The effect of the vaccination becomes visible by comparing the effective and gross exogenous inputs, whereas the gross exogenous inputs \(u_{\text{0-64},gross}\) and \(u_{\text{65+},gross}\) are estimated under the assumption that the same percentage of successfully vaccinated individuals [8] and unvaccinated individuals enter the susceptible compartment. However, the successfully vaccinated individuals are instantly removed from the susceptible compartment as they do not contribute to the epidemic anymore. The parallel course of \(u_{\text{0-64},gross}\) and \(u_{\text{65+},gross}\) indicates analog epidemiological behavior for all age groups as observed before the vaccination launch.

**Epidemic course of different age groups with vaccinations: **The presented analysis of the current epidemic is also based on the SIR model with exogenous drivers, but the model is split into two discrete age groups and their respective compartments. The used datasets (raw data) are based on the data provided by Israel's government services and information website [3]. The first group comprises the population that is younger than 60 years, indicated by \({\text{0-59}}\), and the second group those people aged 60 years or above, indicated by \({\text{60+}}\). The course of the epidemic is represented separately for each group by the Infected \(I\) (top) and aggregated exogenous drivers \(u\) (bottom). The Reproduction Number \(R_{\text{eff}}\) from differential flatness is plotted (middle), whereas a value greater than one indicates epidemic activity. Additionally, individual state diagrams are presented for each group on the right.

The infection dynamics of the two groups are coupled to each other, meaning that the groups can also infect each other since they are part of the same population and the same epidemic. Due to the size of the first age group (\({\text{0-59}}\)) the obtained results for the exogenous drivers and the phase evolution are similar to the results obtained for the entire population, since they also make up for more than 80% of the Israeli population. The results for the second age group (\({\text{60+}}\)), however, begin to differ visibly with the vaccination launch.

The effect of the vaccination becomes visible by comparing the effective and gross exogenous inputs, whereas the gross exogenous inputs \(u_{\text{0-59},gross}\) and \(u_{\text{60+},gross}\) are estimated under the assumption that the same percentage of successfully vaccinated individuals [4] and unvaccinated individuals enter the susceptible compartment. However, the successfully vaccinated individuals are instantly removed from the susceptible compartment as they do not contribute to the epidemic anymore. The parallel course of \(u_{\text{0-59},gross}\) and \(u_{\text{60+},gross}\) indicates analog epidemiological behavior for all age groups as observed before the vaccination launch.

**Epidemic course of different age groups: **The presented analysis of the current epidemic is also based on the SIR model with exogenous drivers, but the model is split into two discrete age groups and their respective compartments. The used datasets (raw data) are based on the data provided by the official UK government website for data and insights on coronavirus [3]. The first group comprises the population that is younger than 60 years, indicated by \({\text{0-59}}\), and the second group those people aged 60 years or above, indicated by \({\text{60+}}\). The course of the epidemic is represented separately for each group by the Infected \(I\) (top) and aggregated exogenous drivers \(u\) (bottom). The Reproduction Number \(R_{\text{eff}}\) from differential flatness is plotted (middle), whereas a value greater than one indicates epidemic activity. Additionally, individual state diagrams are presented for each group on the right.

The infection dynamics of the two groups are coupled to each other, meaning that the groups can also infect each other since they are part of the same population and the same epidemic. Due to the size of the first age group (\({\text{0-59}}\)) the obtained results for the exogenous drivers and the phase evolution are similar to the results obtained for the entire population, since they also make up for more than 75% of the population of the United Kingdom.

**State diagram and alternative turn of events: **The compartments of the model (left) show a fictive scenario for Austria. A fictive lockdown is imposed on Oct. 17^{th}. This can easily be analysed by applying the fictious aggregated input shown in the third subfigure. The fictive course of the of the infection numbers (green) demonstrates that an earlier lockdown of the same duration would have significantly reduced the peak of infections in Austria.

**Data Preprocessing: **The used datasets (raw data) are based on the data provided by the Austrian Agency for Health and Food Safety [1]. They show certain country-specific anomalies, mainly caused by inconsistencies in the reporting of positively tested and recovered persons. In some cases, inconsistencies were also found in the reported data of the electronic reporting system. In addition, a weekly pattern of under-reporting and over-reporting is also present (as fewer tests are usually performed/ entered on weekends). First, in order to reduce the impact of such anomalies, the data were suitably smoothed to create the results. A window length corresponding to a seven-day multiple was chosen, and smoothing is based on local regression using weighted linear least squares and a second-order polynomial model [2].

**Data Preprocessing: **The used datasets (raw data) are based on the data provided by the Johns Hopkins University [1]. They show certain country-specific anomalies, mainly caused by inconsistencies in the reporting of positively tested and recovered persons. In some cases, inconsistencies were also found in the reported data of the electronic reporting system. In addition, a weekly pattern of under-reporting and over-reporting is also present (as fewer tests are usually performed/ entered on weekends). First, in order to reduce the impact of such anomalies, the data were suitably smoothed to create the results. A window length corresponding to a seven-day multiple was chosen, and smoothing is based on local regression using weighted linear least squares and a second-order polynomial model [2].

- [1] BMSGPK. Austrian COVID-19 open data information portal. https://www.data.gv.at/covid-19
- [2] Cleveland, W. S. and Loader, C. Smoothing by local regression: Principles and methods. Statistical Theory and Computational Aspects of Smoothing, pages 10–49. Springer, 1996.
- [3] Santamaría, L. and Hortal, J. COVID-19 effective reproduction number dropped during Spain’s nationwide dropdown, then spiked at lower-incidence regions. Science of the Total Environment, 751, 2021
- [4] Shim, E., Tariq, A., Choi, W., Lee, Y., and Chowell, G. Transmission potential and severity of COVID-19 in South Korea. International Journal of Infectious Diseases, 93:339–344, 2020
- [5] You, C. et al. Estimation of the time-varying reproduction number of COVID-19 outbreak in China. International Journal of Hygiene and Environmental Health, 228, 2020
- [6] Cori, A., Ferguson, N. M., Fraser, C., and Cauchemez, S. A new framework and software to estimate time-varying reproduction numbers during epidemics. American Journal of Epidemiology,178(9):1505–1512, 2013
- [7] Richter, L., Schmid, D., and Stadlober, E. Methodenbeschreibung für die Schätzung von epidemiologischen Parametern des COVID-19 Ausbruchs, Österreich. Technical report, 2020
- [8] Polack, F. P. et al. Safety and Efficacy of the BNT162b2 mRNA Covid-19 Vaccine. New England Journal of Medicine, 383(27):2603-2615, 2020.

- Country Icons made by Freepik from www.flaticon.com

- [1] Dong, E., Du, H., and Gardner, L. An interactive web-based dash-board to track COVID-19 in real time. The Lancet Infectious Diseases, 20(5):533–534, 2020.
- [2] Cleveland, W. S. and Loader, C. Smoothing by local regression: Principles and methods. Statistical Theory and Computational Aspects of Smoothing, pages 10–49. Springer, 1996.
- [3] Israel's government services and information website. https://www.gov.il/
- [4] Polack, F. P. et al. Safety and Efficacy of the BNT162b2 mRNA Covid-19 Vaccine. New England Journal of Medicine, 383(27):2603-2615, 2020.

- Country Icons made by Freepik from www.flaticon.com

- [1] Dong, E., Du, H., and Gardner, L. An interactive web-based dash-board to track COVID-19 in real time. The Lancet Infectious Diseases, 20(5):533–534, 2020.
- [2] Cleveland, W. S. and Loader, C. Smoothing by local regression: Principles and methods. Statistical Theory and Computational Aspects of Smoothing, pages 10–49. Springer, 1996.
- [3] The official UK government website for data and insights on coronavirus. https://coronavirus.data.gov.uk/

- Country Icons made by Freepik from www.flaticon.com

- [1] Dong, E., Du, H., and Gardner, L. An interactive web-based dash-board to track COVID-19 in real time. The Lancet Infectious Diseases, 20(5):533–534, 2020.

- Country Icons made by Freepik from www.flaticon.com