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

In this video, a brief and simplified presentation of the methodology is given. The methodology at full length can be found in the Journal of Nonlinear Dynamics.

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(Updates are uploaded on Wednesdays at 13:30.)

- ▼ Epidemic Analysis
- ▼ \(R_{\text{eff}}\)
- ▼ Forecast
- ▼ Forecast by Age Groups
- ▼ Hospitalization and ICU Forecast
- ▼ Data Preprocessing

- ▼ Epidemic Analysis
- ▼ Epidemic Analysis for Two Age Groups Including Vaccination Effects
- ▼ Data Preprocessing

Due to low case numbers, the analysis is currently paused.

**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 reaches 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].

**Forecasting 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 \(\hat{u}\) is based on previously observed courses of \(u\). This is set to predict a future course (dashed line). Note that such a projection of \(\hat{u}\) critically depends on whether or not governmental interventions are changed in the predicted time interval. The associated responses of \(\hat{I}\) and \(\hat{S}\) 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.

The animated graphic compares previous forecasts based on aggregated exogenous drivers to the actual course of the epidemic.

**Epidemic course and forecast of different age groups with non-pharmaceutical interventions: **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 \({\text{1}}\), and the second group those people aged 65 years or above, indicated by \({\text{2}}\). The course of the epidemic is represented separately for each group by the Infected \(I\) (middle, top) and aggregated exogenous drivers \(u\) (bottom).

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 the obtained results for the exogenous drivers 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, however, differ visibly.

Analog to the non-segregated model above, the exogenous drivers can be utilized to predict the epidemic (dashed lines). A prediction of \(\hat{u}_{\text{1}}\) and \(\hat{u}_{\text{2}}\) is made based on previous observed courses. This is set to predict a future course. Note that such a projection of the exogenous inputs critically depends on whether or not governmental interventions are changed in the predicted time interval. The associated responses of the infection numbers are visible in the subfigures above.

**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.

**Forecasting the hospital and ICU occupancy:** Based on the previous predictions of the exogenous drivers \(\hat{u}_{\text{1}}\) and \(\hat{u}_{\text{2}}\) and the resulting epidemiological behavior, a forecast for the hospital and ICU occupation can be made. By real-time estimation of the case-hospitalization and case-ICU rate, e. g. the share of people tested positive who are admitted to hospital or ICU care, possible changes, such as varying testing strategies or reduced severity of the virus due to vaccination, are accounted for. Based on the predicted course of the exogenous drivers and the estimated case rates, the predictions of the hospital and ICU occupancy are obtained from convolution.

In the figure above, the actual hospital and ICU occupancy (colored in pink and blue), reported by the Austrian Agency for Health and Food Safety [1], are plotted as well as the smoothed occupancys. Furthermore, the predictions obtained by convolution (dashed line) and a prediction interval (grey) are displayed.

In the figure above, previous predictions (dashed line) are compared to the reported and smoothed hospital and ICU occupancy (grey).

In the graphic above, the relative errors of previous predictions after one, two and three weeks are plotted. In orange are the prediction errors based on observed production rates and in blue based on the predicted production rates.

**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