Germany’s next shutdown—Possible scenarios and outcomes

Abstract With the rapid increase of reported COVID‐19 cases, German policymakers announced a 4‐week “shutdown light” starting on November 2, 2020. Applying mathematical models, possible scenarios for the evolution of the outbreak in Germany are simulated. The results indicate that independent of the effectiveness of the current restrictive measures they might not be sufficient to mitigate the outbreak. Repeated shutdown periods or permanently applied measures over the winter could be successful alternatives.


| INTRODUC TI ON
The rapid increase of reported COVID-19 cases in Germany has prompted policymakers to announce on October 28, 2020, a new period of stricter control measures starting on November 2, 2020. 1 As of the first day of this "shutdown light," Germany accounts for 545,027 reported SARS-CoV-2 infections and 10,530 reported deaths. 2 The decision from Oct 28 stipulates softer measures than were applied in spring of 2020. 3 In particular, schools maintain in person teaching, and all shops, not just grocery stores, keep their regular opening hours. The rising case and fatality numbers reported in Europe since early September 4 made people more aware of the risk of infection, but opposing opinions persevered and stimulated protests throughout the country. It is hard to quantify how the measures are going to change the course of the epidemic in Germany, as a match to the March/April shutdown is not exactly possible.
Nevertheless, mathematical models allow simulating different scenarios of how the November shutdown might play out and what is likely to follow under various assumptions on the policies and public behavior adopted.

| ME THODS
The results shown here are based on simulations of a mathematical compartmental model of SEIR (susceptible-exposed-infectedrecovered) type that in addition accounts for undetected and hospitalized cases and partitions the population into age classes.
The compartments are summarized in Figure 1, and transitions between them are described by a system of ordinary differential equations comparable to what was proposed in. 5 Susceptible individuals become exposed by effective contact with an infectious individual. Exposed individuals progress through three stages, E 1 , can be in any of the above states. Age classes evolve in parallel and are coupled to one another by contact rates among individuals. Demographics are neglected, meaning that besides fatalities caused by the disease, no individuals enter or leave the population. The model is calibrated on reported case counts, 10 hospitalizations and ICU occupation as daily reported by the Robert Koch Institute, 2 following methods previously adopted in. 5 All the scenarios described below assume that (i) high incidence numbers lead to automatic contact reduction by making individuals on average more careful, (ii) effective contact rates are higher in winter than in spring/summer due to more frequent indoor activities, 6 (iii) high prevalence leads to higher under-reporting due to limited test and contact tracing capabilities, and that (iv) immunity acquired by having contracted and recovered from the disease does not wane as evidence on loss of immunity is yet debated. 7,8 In what follows possible scenarios for reducing infectious contacts are considered. The simulations shown in Figure

| RE SULTS
Mathematical models similar to the ones used here were employed in the early phase of the pandemic to follow and predict the outcome of initial intervention measures. One of the most salient predictions of these simulations was the inevitability of a second wave of the epidemic if the measures imposed during the first shutdown period in Spring of 2020 were to be relaxed too much. 5 This outcome can be witnessed not only in Germany, but in many countries all over Europe and worldwide where strict interventions have been lifted stepwise during the summer. Now, a similar prediction concerning a third wave can be deduced from the simulations presented here.
As shown in scenario 1, no matter how effective the shutdown in November is in reducing contact rates and hence the incidence of new infections, returning to contact rates close to those prevalent in late summer will most likely lead to a third wave of rising case F I G U R E 1 Model structure and transitions through the model compartments. Susceptible individuals (S) become exposed by effective contact with any infectious (red compartments). Exposed individuals progress through three stages, E 1 , E 2 , and E 3 , before illness onset. Instead of planned wave breaker interventions, fixed for established periods of the year, one might also think of applying and relaxing intervention measures based on the reported case incidence. We have tested (simulations not shown here) also such an intervention strategy with triggered measures, assuming, for example, that an incidence of 50 cases per 7 days and 100,000 inhabitants triggers severe restrictions leading to strongly reduced contact rates, whereas lifting of these restrictions is triggered by the incidence dropping to 8 cases per 7 days and 100,000 inhabitants. One obvious assumption to include in the model is that control measures cannot be put in place instantaneously, but require a few days to be effectively introduced or relaxed. This leads to dynamics similar to that in scenario 2, but with shutdown periods not occurring at predefined intervals.
This holds true even when the threshold values are modified to take into account rising detection ratios due to, for example, improved contact tracing or increasing testing capacity.

| D ISCUSS I ON
We conclude that the 4-week soft to moderate shutdown started on November 2 cannot on its own be expected to prevent a third, pos- The unknown detection ratio, which is most likely fluctuating in dependence on new cases and testing capabilities, might significantly affect the outcome of model simulations. 9 In order to model the occupation rate of ICU beds, we assumed that the same standard for admission to and release from intensive care is uniformly applied over time. Changing this admission policy over time in reaction to higher demand may lead to lower ICU occupation than shown in the model. Increasing the availability of ICU beds by, for example, establishing temporary field hospitals might be a further mitigation strategy for coping with even higher ICU occupancy. If this strategy is considered, the timing of maximal ICU demand may be as relevant as the number of beds required when assessing different scenarios.
Finally, the presented model results cannot-and do not intend tomake any statements about possible economic or social effects of contact restrictions.

ACK N OWLED G EM ENTS
The authors are members of the CoSiMo (COVID-19 Simulation and Modeling) collaboration of FIAS and JSC and would like to thank Thomas Lippert for initiating this collaboration, the other team members, Jan H. Meinke and Stefan Krieg, as well as Daniel Rohe and Anne Nikodemus for their useful inputs on the presentation of the results.

CO N FLI C T O F I NTE R E S T S
The authors declare no conflict of interests.