On this page, we apply the simulation model to several basic scenarios for the spread of the disease and briefly discuss what we can (and cannot) learn from these scenarios for the actual outbreak in Austria.

Scenario 1 | "Best fit"

A canonical use of the simulator is to try and fit the model to the Austrian case data for people with a detected infection, patients in hospital and ICU wards, and the recorded COVID-19 deaths. We provide three such "best fit" scenarios with our simulator, for different assumptions on the effect of seasonality on the spread of the virus. Below, we briefly discuss the main steps of the fitting procedure and the results of the simulations.

We parametrize our model in two steps. In a first step, we use Austrian data (where available) and international data from literature for an initial choice of transition rates between the model compartments. In a second step, we fit the values for the basic reproduction number R0 and the mitigations (reduction of virus transmission due to governmental interventions and behavioral changes) to the Austrian case data for new infections and deaths, as well as patients in hospitals and in ICU wards. The detection ratio δ(t) is also adjusted during this second step. The details of the procedure and our literature sources are described in the comprehensive model documentation.

Assumptions on seasonality

Similar to the influenza (flu) virus, human coronaviruses are generally known to have a marked seasonal transmission pattern with a winter peak in temperate climates of the northern hemisphere. However, the strength of the seasonality (if any) of SARS-CoV-2 transmission is not known. We include three scenarios parametrized by seasonal amplitudes ε=(0,0.1,0.2). The strength ε=0.2 follows a “best estimate” for SARS-CoV-2 in recent literature (Kissler et al., Science 2020).

Reproduction number parameters

The fitting procedure leads to values of the basic reproduction number R0 between 3.2 and 3.33 (depending on seasonality) and reproduces the approx. 30% daily increase in case data that was observed in Austria prior to the lockdown. We include four switch-dates for changes in mitigation. The first switch date t1 corresponds to the lockdown on March 16th and is strongly supported by the fit, the second and third switch dates t2 and t3 parametrize the relaxation of the lockdown starting on May 1st (opening of shops, gradual relaxation of gathering and travel). The fourth switch-date end of July accounts for slightly reinforced measures (such as mask wearing and renewed travel restrictions). Given these dates, we estimate values for R0 and for mitigated values as shown in the following table:

Table 1: The table shows the basic reproduction number R0 and the mitigated reproduction numbers Ri=R0(1-Mi) after the ith switch point. Rows correspond to different strengths of the seasonal effect.

ε R0 R1 R2 R3 R4
0 3.2 0.58 0.67 1.14 1.03
0.1 3.26 0.60 0.73 1.25 1.16
0.2 3.33 0.62 0.82 1.39 1.32

Note that the mitigated reproduction numbers Ri only reflect the effects of governmental interventions and behavioral changes in response to the virus on the effective reproduction number, but not the effects of seasonality and emerging immunity. In all cases, the model assumes that mitigation stays constant after the last switch date on July 22nd.

In our scenarios, we have assumed four switch-points for the effect of mitigation measures and adjusted the date and strength of these changes to the case data accordingly. Note that this does not mean that the reproduction number has changed only at these points in time (see also our R-Nowcasting page). However, if more switch-points are assumed, the signal from the available data is still too weak for an accurate parameter estimate. We invite the user to explore the effects of further switch-points.

Infection fatality ratio (IFR) and detection ratio

In contrast to the reproduction number parameters, the mortality of COVID-19 cannot be estimated from the case data. The reason is that the IFR = deaths / cases has the true number of infections in the denominator, which however depends on the probability that these cases are detected and included into the official records. This detection ratio (δ(t)) is not well known and can only be roughly estimated. From two Austrian PCR surveys (SORA and Statistik Austria), an antibody study at Ischgl and a range of international studies, we arrive at plausible values of 0.7% for the IFR and 15% for the detection ratio during the early phase of the epidemic (until April 1st). With increased testing and contact tracing, a much higher proportion of infections was detected at later stages of the outbreak, as evidenced by a sharp decline in the percentage of positive PCR tests. This is included into the model as a linear increase of the detection ratio until May 18th, where it reaches 47% in the best-fit scenario. However, these values come with a large uncertainty and a “true” IFR between 0.3% and 1.1% (initial δ between 6.4% and 23.6%) appears to be possible. See the documentation and the Scope&Limits section for further arguments and references.


The outbreak dynamics with different levels of seasonality ε are shown in the following figures.

Dynamics: Best fit with epsilon=0 Reff: Best fit with epsilon=0

Figure 1: Outbreak dynamics (top panel) and Reff(t) (bottom panel) with ε=0 for seasonal scenarios with best fit.

Dynamics: Best fit with epsilon=0.1 Reff: Best fit with epsilon=0.1

Figure 2: Outbreak dynamics (top panel) and Reff(t) (bottom panel) with ε=0.1 for seasonal scenarios with best fit.

Dynamics: Best fit with epsilon=0.2 Reff: Best fit with epsilon=0.2

Figure 3: Outbreak dynamics (top panel) and Reff(t) (bottom panel) with ε=0.2 for seasonal scenarios with best fit.

The bottom panels of Figures 1, 2, 3 show how the effective reproduction number Reff(t) changes over the same time scale for the seasonal scenarios. In particular, we see which factor contributes most strongly to the reduction of the maximal level of transmission (corresponding to the fitted R0).

We see that the estimated level of current mitigation (R4) is sufficient to keep the virus largely under control in the absence of seasonality (ε = 0, Figure 1). Although new infections continue to occur, there is no stress on the health system and the total number of COVID-19 deaths in Austria remains moderate. The population-wide level of “herd” immunity due to people who have recovered from the disease remains very low throughout.

A contrasting outcome is obtained under the assumption of moderate seasonality (ε = 2, Figure 3). In this case, the current low level of transmission is partially due to the seasonal effect rather than due to behavioral changes. Assuming that the behavioral pattern remains constant, the total mitigation proves insufficient in winter and a second wave of infections is triggered. Since our model does not include a (likely) response to this wave in terms of reinforced mitigation measures, we see a steep increase in the number of infections and deaths until the further spread is eventually stopped by a combination of emerging immunity, seasonal effects and behavioral mitigation. Finally, the case of weak seasonality (ε = 0.1, Figure 2) produces a “weak second wave” that does not overwhelm the health system, but still leads to more than a million of infections by summer 2021 in the absence of changes in behavior – or a vaccination.

Scenario 2 | Epidemic spread without (or with insufficient) mitigation

The use of the SEIR model is not limited to the attempt to provide best fits to actual case data. It can be equally instructive to explore hypothetical scenarios. A natural reference scenario in this sense addresses the question: What would happen without any mitigation due to either governmental interventions or spontaneous behavioral changes by the population in response to the disease? Or, to rephrase the question: What would happen if we deal with COVID-19 in the same way as we deal with a normal flu season – only without the possibility of vaccination and without acquired (partial) immunity?

To address this question, we maintain all parameters of the "best fit" model described in Scenario 1 (including the initial conditions), but discard all mitigation steps. For simplicity, we focus on the case of no seasonality. In this case, the epidemic continues to grow with its initial rate (corresponding to R0 = 3.2) until it is stopped by emerging herd immunity. As in Scenario 1, we assume a default infection fatality ratio (IFR) of 0.7%. The resulting outbreak dynamic is displayed in the Figure 4.

Dynamics: Unmitigated, best fit with epsilon=0

Figure 4: Outbreak dynamics in Austria in the absence of mitigation and seasonality.

We see that the disease quickly overwhelms the health system and especially the ICU capacities. At the peak of ICU need, more than 28 000 patients simultaneously require intensive care, while only around 1 000 ICU beds are available in Austria.

As a consequence, a large majority of critical cases are not able to receive intensive care. Note that this effect is rather robust with respect to the number of available ICU beds. If instead of 1 000 there were 2 500 ICU beds available for COVID-19 patients (a theoretical upper bound when all Austrian capacities would be used for this purpose), the percentage of patients who can obtain adequate intensive care still remains low.

Our model assumes that this catastrophic overflow increases the death rate for all patients who cannot receive adequate care. Our scenario calculations use an "ICU overflow severity factor" of 2 (corresponding to an increased average IFR of 1.4% for these patients). This way, we obtain more than 112 000 COVID-19 deaths by the end of the epidemic.

As a variation of the above "no mitigation" scenario, we can further ask what happens if governmental interventions and spontaneous behavioral changes do lead to a limited reduction of virus transmission, measured as the mitigated reproduction number R1=R0(1–M1), where M1 is the mitigation strength that was achieved. Figure 5 explores this (in terms of the total number of expected deaths) for an assumed mitigation start date on March 16th for three different baseline IFRs, at 0.3%, 0.7%, and 1%. We assume a severity factor of 2 for ICU overflow in all cases.

Total Deaths R=1

Figure 5: Total number of deaths over R1, for different IFRs in the unmitigated scenario.

As expected, the results show a linear dependence of the total number of deaths on the assumed IFR. In contrast, the dependence on R1 is highly nonlinear with a threshold behavior at R1=1. Once R1 is clearly and consistently larger than 1, the death toll of the disease is unavoidably catastrophic.

For R1 = 1.3 or larger, the death toll is in the order of tens of thousands, even if the infection fatality ratio IFR is as low as 0.3%. The reason is that in all these cases a large part of the Austrian population would get infected within a short period. Herd immunity will eventually stop the pandemic, but this will only happen after a period of extreme stress on the health system.

As an example, consider R1=1.7. In this case, more than 2/3 of the population (more than 6 million Austrians) get infected within half a year. Note that this number is larger than the nominal herd immunity level 1 – 1/R1 (about 41% for R1=1.7) that stops the spread of a new epidemic. An ongoing epidemic stops later since new infections still happen while the total number of infections declines (so-called infection overshoot). This again necessarily leads to severe ICU overflow, where a large majority of cases will not be able to receive intensive care. The death toll therefore reaches almost a level of twice the baseline IFR times 6 million (36 000 to 120 000 deaths in the IFR range considered).

A recent article in Nature by Flaxman et al. (2020) estimated that the lockdown in Austria saved between 40 000 and 85 000 lives by early May. These numbers are entirely in line with our hypothetical “no mitigation” model. Importantly, these model estimates are not meant to predict the real course of the disease, but depend on the assumption of a constant mitigation level. Accumulating deaths and a stressed health system almost necessarily lead to governmental interventions and spontaneous behavioral changes by the population, as seen in many countries with large initial case numbers.

Instead, the scenario refutes claims that COVID-19 is comparable to seasonal flu and shows that significant mitigation measures, driving R1 below 1, are unavoidable. It also reveals that a strategy of insufficient mitigation, based on a hope that herd immunity will bring the pandemic to a halt, leads to a catastrophic death toll. This is expanded in more detail in Scenario 3.

Scenario 3 | Herd immunity strategy

Herd immunity has been considered by some as a potential exit strategy from the epidemic. To achieve herd immunity, a substantial fraction of the population must gain immunity - which, in the absence of immunisation by a vaccine, requires contracting and overcoming the disease. To avoid serious healthcare overload which would increase mortality, this large number of infections must be spread out over a sufficiently long period of time. A natural question to ask is: What is the shortest time in which it might be feasible to achieve herd immunity, without overloading the healthcare system?

Note, however, that lasting immunity has not yet been demonstrated in patients recovered from COVID-19.

Dynamics: Herd immunity stragety

Figure 6: Outbreak dynamics in Austria in a scenario where herd immunity is achieved without overloading the intensive care unit.

In Figure 6, we show a scenario where the mitigation measures are as limited as possible - to allow the development of herd immunity, while still preventing ICU overload. In practice, implementing such measures would be difficult due to stochasticity and uncertainty about the outbreak at any given time. For this reason we require that in our deterministic model, the number of critical COVID-19 patients at any given time is at most 750 - less than the total capacity of 1000. Given the fraction of infected individuals who require critical care, and the assumed average of Tc=6.5 days in ICU, this constraint translates to at most 6600 new infections per day. All other parameters are assumed to be the same as in the "best fit" scenario above, with no seasonal effects.

As shown in Figure 6, herd immunity would under these constraints be achieved in late 2022 - early 2023. By the time that the epidemic subsides, about 6.5 million people have overcome the disease (73% of the population), while about 2.3 million remain susceptible. Around 45 700 people die of COVID-19 - given the IFR of 0.7%. Throughout the process, especially in the early stages when most of the population is still susceptible, strong mitigation measures are necessary to prevent the epidemic from further growth. Early on, mitigation as strong as M=68% is necessary, but this can be gradually relaxed, with complete lifting possible in late 2022.

The healthcare load, as well as the number of deaths, could be reduced if the most vulnerable groups were specifically isolated. This is not included in the scenario.

Scenario 4 | Impact of lockdown timing (sensitivity analysis)

Upon pressing the “Recompute” button, the SEIR simulator derives a model output (curves for number of cases in each category, total number of deaths, etc.) from a set of input parameters, including the reproduction number R0, mitigation strengths and times, and the transition probabilities and times between the model compartments. This points to another key usage of the SEIR simulation app – beyond fitting and the construction of purely hypothetical scenarios: providing a tool for the user to explore how (small) changes in the input parameters change the output.

In mathematical modeling, an analysis of this kind is called “local sensitivity analysis”. It is used (among other things) to assess the relative importance of certain input parameters for the model output. As an example, we will consider here two output quantities: the peak ICU need and the total number of COVID-19 deaths by August 31st, 2020. We study how these quantities depend on two key input parameters, the timing of the lockdown (the first mitigation switch-point) and the timing of the relaxation of lockdown measures (starting with the second mitigation switch-point) that we have used in our “best fit” scenarios. All other parameters are kept constant, including, in particular, the strength of the mitigations.

Figure 7 and 8 show the impact of a shift in the lockdown date (t1) and all following switch-dates (t2 - t4) by a given number of days on the total number of deaths and the peak ICU need for three seasonal scenarios, with no, weak and moderate seasonal effects (ε=0; 0.1; 0.2). For all three scenarios, we obtain large changes for earlier or later lockdown dates, relative to our estimates with lockdown on March 16th , see also Table 2.

What if Lockdown Time_Deaths What if Lockdown Maximum ICU

Figure 7,8: Impact of a change in the lockdown date on the total number of deaths and the peak ICU need for three seasonal scenarios, with no, weak, and moderate seasonal effects.

Table 2: Effect of the lockdown date (no / weak / moderate seasonal effects).

Lockdown start date Deaths by August 31st Peak ICU need
March 16th 773 / 775 / 776 280 / 281 / 282
March 23rd 4 958 / 4 808 / 4 691 1 683 / 1 648 / 1 629
March 9th 128 / 132 / 135 44 / 45 / 45

Delay of the lockdown by one week leads to ~ 6 times larger number of deaths and an overflow of the ICU capacities for all scenarios. Vice-versa, there is also an effect of an earlier lockdown, leading to ~ 6 times lower total death numbers for a shift of one week. Note that the total lockdown period (time from lockdown to relaxation) is not changed in these scenarios.

In Figure 9, we analyze the impact of an earlier or later switch-point to relax the lockdown measures again (where we shift all switch-dates t2 – t4 after the lockdown by a set number of days), see also Table 3. In contrast to the previous case, we thus have a change in the lockdown period.

What if Relaxation Time_Deaths

Figure 9: Impact of an earlier or later switch-point to relax the lockdown measures again.

Table 3: Effect of the date when mitigation measures are relaxed again (no / weak / moderate seasonal effects).

Relaxation start date Deaths by August 31th
May 1st 773 / 775 / 776
April 1st 2 265 / 2 802 / 3 693
June 1st 683 / 682 / 681

The sensitivity of the total number of deaths to a shift in the relaxation date is smaller than for a shift in the lockdown date. This is expected because the absolute change in mitigation at this switch-point is smaller, but note that we also display much larger shifts by up to one month. While an earlier relaxation date on April 1st can lead to a 4-fold increase of total deaths, the effect of a shift of the relaxation switch-point to an even later date is very small.

It is tempting to interpret these findings as predictions for what would have happened if the Austrian government had chosen an earlier or later lockdown date (or relaxation date). Such an interpretation has been made for related studies. However, we want to remind the user that this amounts to an over-interpretation. Neither our model, nor any other model known to us, can predict the behavioral changes of the Austrian population for scenarios that did not happen. Our derivations assume that no behavioral change would have happened other than the assumed temporal shift.

What can be concluded is the fact that the course of the epidemic is highly sensitive to the date of a lockdown of the estimated strength, and to a lesser degree to the relaxation date. It is very likely that both a (relatively) early lockdown and not-too-early relaxation contributed to Austria’s successful course through the epidemic so far, but our sensitivity results should not be read as predictions for the number of saved lives (or extra deaths) for alternative governmental actions.

Further remarks:

  1. Our results on the effect of a shift of the lockdown date on the peak ICU need shows good qualitative agreement with the results from the individual based simulation model by N. Popper. This demonstrates the robustness of our (and Popper‘s) analysis with respect to modeling details.

  2. In our sensitivity analysis above, we have only changed a single input parameter type (mitigation switch-points). For a complex problem with many interacting parameters (as we have it here), this simple view is often not sufficient. As an example, consider the three parameters “infection fatality ratio (IFR)”, “detection ratio” and “initial number of infectious individuals”. If any one or two of these parameters are changed, we observe strong effects on the model output. However, if we change all three in a coordinated way, there are almost no effects for the early phase of the epidemic, where data already exists. This is the reason why we have added the “lock” function to the input panel.

  3. Sensitivity analysis is often also used to quantify the uncertainty of model predictions. If input parameters are known with some uncertainty that can be quantified, sensitivities translate these into uncertainties of the output. Some COVID-19 simulators (e.g., by Basel University) include this functionality and display credible intervals for the output. However, an interpretation of these credible intervals is not straightforward. This is because it is often not clear, which parameters are “known” with which uncertainty. In applications, we often use a best fit to output data (time series of past cases and deaths) to obtain estimates for input parameters, which then produce further estimates of output data (future cases). In addition, the parameter space of the model is large, making a comprehensive statistical analysis prohibitive. For this reason, we encourage the user to build her or his own intuition of model uncertainties by running the simulator in parameter regions of interest. Note that we do provide credible intervals for our R-Nowcasting, which is a much simpler problem. However, even in this case, the interpretation of the uncertainties displayed requires great caution (see our discussion there).