Projecting the spread of COVID-19 for Germany

Hubei

Since the onset of the outbreak in the Hubei province in China, the number of confirmed cases has risen dramatically, peaking at 67,801 (WHO Situation Reports, 2020).^[10] Reporting began in January 2020 and we note that in mid-February, the WHO changed its classification methodology by reporting both clinically and lab confirmed cases rather than only lab confirmed cases. In terms of cumulative numbers, the total reported cases of infected individuals has risen continuously since January, since there has been continuous influx of new cases and very little communication about individuals transitioning back to being healthy.

When reporting began, the rise of new cases seemed to follow an exponential curve, with the number of new cases constantly increasing, however in early February, this trend stopped. Indeed, another trend then took over following a more S-shaped pattern, i. e. that of a sigmoidal function. The total number of reported cases over time is shown in Figure 1, which shows a clear explosive trend at the start of the outbreak before slowing down and approaching what appears to be an asymptote around 70,000.

Figure 1

Total cases in Hubei seem to converge to a constant.

To make this comparable to other countries, we take the population of around 59.17 million individuals in Hubei into account. This implies an infection rate of about 0.11 % or 1 person in 873 being infected in the long-run. While this is only one province, it does provide us with a reference point for what to expect in other regions of the world.

South Korea and Japan

Soon after China, South Korea, Japan, and many other countries all started seeing a rise in their number of confirmed cases. However, these countries have had very different experiences when it comes to infection outbreaks. Indeed, while South Korea has seen the number of confirmed cases rise to 9,241, Japan only has 1,307 confirmed cases (or 14 % that of South Korea). The question remains whether these countries are also experiencing a convergence to a stable number of (cumulative) confirmed cases. Figure 2 shows the cumulative number of confirmed cases in South Korea (left) and Japan (right).

Figure 2

Total cases in South Korea exhibit a similar pattern to the Hubei province.

The data from South Korea shows a reduction in the slope of the curve for total confirmed cases, indicating a significant drop in the number of new infections reported. Assuming the tendency continues, this would point toward South Korea progressively converging to a constant, similarly to the Hubei province in China. Japan on the other hand (as well as all other countries at this point) does not appear to slow down significantly. Relating South Korean infections to population size, we get a long-run infection rate of 1 in 5,337, with a peak number of confirmed cases of ca. 9,606.

Europe

In Europe, however, the situation continues to deteriorate, with new cases being reported by authorities on a daily basis numbering in the thousands (WHO Situation Reports, 2020). However, bearing in mind the lessons learned from the case of the Hubei province, and to a lesser extent from the case of South Korea, this current period of explosive growth should, in theory, be followed by a slow tapering leading to a plateauing of the total number of cases. We report in Table 1 the current infection rates in some European countries. The sample below represents 58 % of the total EU population, including Norway and Switzerland. This sample is by no means intended to be representative and is for illustrative purposes only by considering some of the most affected economies.

As we can see from Table 1, the degree of infection per country varies from 1 in 4,050 to 1 in 784, a fivefold ratio. Other countries fare better, the UK for example has a risk of 1 in 6,914, however we can clearly see that these figures are not yet as low as those in Hubei, with the exception of Italy and Switzerland, which are now below the 1 in 873 rate observed in the Chinese province, and Spain, which is approaching it fast. If this case is to be repeated, then things will likely worsen before they improve. However, with measures taken by governments, and progress being made on the medical front for an effective vaccine, there is still room for governments to curb these rates despite the rapid rise in cases.

3.1 The structure

The model builds on a continuous time Markov chain with 4 states. We look at health of an individual i that can be in four states S={1,2,3,4}. Healthy without infection (s=1), sick (s=2), dead (s=3), healthy with or after infection (s=4). State 1 is the initial state for all individuals, thus si(0)=1 for all i=1...N where N is initial population size.^[11] There are individual transition rates λrs (with r∈S and s∈S) between these states as illustrated in Figure 3.

3.2 Transition rates

Starting with the transition from being healthy to sick, each individual in state 1 has a certain average number of social contacts per day. We capture this by an exogenous arrival or contact rate a. Once a contact takes place, there is a certain probability that the contact is with an individual that can infect the healthy individual. If we assumed random contacts with healthy and sick individuals, this probability would be (N2(t)+ηN4(t))/(N1(t)+N2(t)+N4(t)), where Ns(t) is the (expected) number of individuals in state s at t. This probability assumes that all N2(t) individuals that are sick can infect healthy individuals and a constant and exogenous share η of healthy individuals in state 4 can also infect healthy individuals from state 1. This parameter η captures the idea of “quiet infections” as emphasized e. g. by Wilder-Smith et al. (2020). As opposed to SARS 2003 (or the common flu), individuals infected with CoV-2 do not necessarily display symptoms but can nevertheless infect others. There is also first evidence (Xing et al., 2020) that even recovered individuals (i. e. those that came from state 2) can carry CoV-2, also supporting our use of a positive η.

Figure 3

Transitions between the state of health (initial state), sickness, death and health despite infection or after recovery.

In practice, individuals do not encounter sick individuals with this probability being equal to the share of sick individuals in society as sick individuals might stay at home or behave differently (not shaking hands or similar) than non-sick individuals. We do assume, however, that the probability π(t) to encounter a sick individual rises in N2(t) and ηN4(t) and falls in N1(t), i. e. π(t)=π(N2(t),ηN4(t),N1(t)) with ∂π/∂N2>0, ∂π/∂(ηN4)>0 and ∂π/∂N1<0.^[12] These contacts can lead to infections. Hence, we have an individual sickness rate of λ12(t)≡aπ(t). Finally, when there are N1(t) healthy individuals, the aggregate flow from state 1 to state 2 is ϕ12(t)≡aπ(t)N1(t).^[13]

A further property an infection rate should reflect is the fact emphasized by many virologists and epidemiologists that once an epidemic is over, around two thirds of the entire population will be infected. This includes individuals that had symptoms at some point and asymptomatic cases. We capture this fact by introducing the infection rate which is simply the ratio of the number of infected individuals (sick and in state 2 or without symptoms in state 4) to individuals that are alive,

The infection rate is zero initially at t<0. Illustrated for Germany in a way that is somewhat too simple but perfectly fine for our projection, ρ=0 on 23 February 2020 (and before) when the number of reported sick individuals was zero. On 24 February 2020, a number of N2(0)=16 is introduced into the system, and infections and sickness start occurring. The sickness rate λ12(t) is zero also in the long-run when this infection rate equals the long-run infection rate, which we denote by ρ¯. This captures the above mentioned fact that some individuals will never be infected during the epidemic and will always remain in state 1, limt→∞N1(t)>0.

Summarizing our discussion, an example for the individual sickness rate that captures insights from epidemiology and virology would be

where 0<α, β,γ<1 allows for some non-linearity in the process and a>0. The first term N1(t)−α captures the idea that more healthy individuals reduce the individual sickness rate. The second term (N2(t)+ηN4(t))β increases the sickness rate when there are more infectious individuals. The third term in squared brackets makes sure that the arrival rate is zero when a share ρ of society is sick (state 2) or healthy after infection (state 4). The sickness rate satisfies “no sickness without infected individuals”, λ12(a,N1,0,0,ρ)=0 and “end of spread at sufficiently high level”, λ12(a,N1,N2,ηN4,ρ¯)=0. In between these start- and endpoints, the infection rate will first rise and then fall. This specification makes sure that in the long run a share of around 1−ρ¯ will not have left state 1, i. e. will never have been infected.^[14]

As is widely documented and reported (see e. g. Nishiura et al., 2020), there is also a flow of individuals that are infected but do not display any symptoms. We denote this flow by ϕ14(a,N1(t),N2(t),ηN4(t)) and the corresponding individual rate by λ14. The arguments can be rationalized in the same way as the flows into sickness.

Is it possible to obtain data for λ14? One would need a cohort of inhabitants of a region (and not just those that contact medical services to get tested) to understand the share of individuals that are infected but do not display symptoms. Good data is not available (let alone time series) and might never be available, given the currently still large costs per individual tests (between 80 Euro and 120 Euro). We therefore assume that a constant share r of infected individuals display symptoms. This is a quantity many epidemiologists also work with. Hence, the transition rate λ14 is related to λ12 in the following sense,

In words, the outflow of newly infected, λ14N1+λ12N1 in squared brackets on the left-hand side, times the share r of individuals that show symptoms after infection gives the flow into sickness, λ12N1. We can therefore compute the transition rate from the state of being healthy (s=1) to the state of healthy again or no symptoms (s=4) as

Finally, there is an individual mortality rate and a recovery rate. Both rates depend on many individual characteristics. At the aggregate level, they rise in the stock of sick individuals. We specify them as ϕ23(N2(t)) and ϕ24(N2(t)) with corresponding rates λ23 and λ24, respectively.^[15] For our analysis below, we will assume that both rates are constants. Concerning recovery, we are aware that it strongly varies across age (e. g. Guan et al., 2020) and that recovery is not a state that is identical for each individual. As we abstract from ex-ante heterogeneity, we assume a recovery rate that is identical across individuals and that it takes on average nrec days until recovery. Given the exponential distribution of duration in each state, this means that 1/λ24=nrec or

Death is an absorbing state. We also consider the final healthy state to be absorbing, i. e. we abstract from relapses.^[16]

3.3 Probabilities and means

3.4 The number of individuals who become sick, ever were sick, and die

Because recovery is a process with additional measurement problems, a more reliable data source for our calibration purposes is the number of individuals that have been reported sick since the onset of the epidemic. This is also the time series that is typically reported by countries and regions (such as those reported by the RKI in Germany or the WHO more generally). In our model, the corresponding number follows from the inflows from state 1 to 2 and is given by

When we are interested in the number of individuals that are newly reported sick at some point in time t (incidences at t in short), we should not employ N2(t). The change of the latter, described in (7b), is determined by the newly reported (inflow) minus the outflow, i. e. those that recovered (and those who died). Hence, when data provides incidences on a given day t, we need to compare this with the inflow between yesterday t−1 and t. Hence, our theoretical counterpart to incidences is

Finally, the number N3(t) of deaths from Corona by t is determined by the constant death rate λ23 applied to those that are currently sick,

3.5 Model properties

Before we calibrate our model, we briefly discuss its theoretical predictions. This illustrates the plausibility of our assumptions and also the flexibility of the model. Our model consists of the ODE system (7) where we replace the probabilities ps(t) by the expected number Ns(t) as described in footnote 17. The sickness rate λ12 is from (2), the other arrival rates λ14 and λ24 are from (4) and (5), respectively. The death rate λ23 is a constant. We start the solution of our model with N2ever(0)=Nobservedever(0). We set N3(0)=N4(0)=0 and N1(0)=N−N2ever(0).

One can easily understand the plausible structure of our ODE system (7). The number N1(t) of healthy individuals in state 1 (i. e. the probability or share p1(t)) can only fall. Individuals either become sick with rate λ12 or get infected without symptoms with rate λ14. Both transitions imply an outflow from state 1. The same is true, mutatis mutandis, for state 3 and 4: For state 3, there are only inflows from state 2 implying that N3(t) increases over time. State 4 only experiences inflows from state 1 and 2 implying that N4(t) increases over time. The number N2(t) of sick individuals can rise or fall, depending on the difference between inflows with λ12 and outflows with λ23 or λ24.

In the long-run, i. e. when the epidemic is over, individuals will be distributed across states 1, 3 and 4. Some individuals will never get infected. They therefore remain in state 1 all throughout the epidemic. This finding follows from the fact that the sickness rate λ12 from (2) falls to zero when the infection rate ρ(t) from (1) equals its long-run value ρ¯. Note that λ12=0 implies from (4) that λ14=0 as well, hence outflows from 1 to 4 end as well. The ratio ρ¯ is a widely accepted quantity in epidemiology and will be quantified below. Some individuals that become sick and are in state 2 die and move into state 3 in the long-run. Most individuals will be in state 4 in the long-run, either after having transitioned through state 2 or directly from state 1.

The model also displays a long-run property that is extremely useful to judge its quantitative property in a very simple way. With a population of N individuals and a long-run infection rate ρ¯ discussed after (1), the number of individuals that will have been infected is ρ¯N. When a share r of all infected becomes sick, the long-run number of individuals that were sick at some point, described in (9), is

The long-run number of reported sick individuals therefore only depends on the long-run infection rate ρ¯ and the share r of individuals that become sick and are reported. We will exploit this property when we discuss the effects of policy measures below.

4.1 Data

Our most important data source is the number of individuals that is reported to be sick. We obtain these outflows from state 1 to 2 from the Robert Koch Institute (RKI, 2020). These data fix the left-hand side of our time path for the number N2ever(t) from (9) of individuals that are ever reported as sick. To fix notation, t=0 stands for the 24th February 2020 and T is the day of our final observation.

It seems reasonable to assume that parameters are not constant throughout the entire epidemic. Most centrally, the contact rate a employed in the sickness rate (2) will change as a function of governmental regulations. The first major measures consisted in cancelling sports events (14 March) and closing schools (16 March). Other measures followed.^[18] Given a median incubation of 5.2 days (Linton et al., 2020, Lauer et al., 2020) and a certain delay between feeling symptoms, contacting a doctor and getting reported at RKI (say, 2–3 days), we employ T=21 March 2020 as our last observation to fix the contact rate a.

Data to compute the death rate from Corona are also taken from the RKI (2020). We finally employ various sources, to be discussed in what follows, to fix the share r of infected individuals that become sick and the long-run infection rate ρ¯.

4.2 Targets and parameters

Model for predictions

After some intermediate steps (see app. A.2), and employing the fact that we can replace probabilities in our system (7) by expected numbers of individuals as discussed in footnote 17, we obtain our final theoretical structure,

(15a)N˙1(t)=−arN1(t)1−αN2(t)+ηN4(t)βNβ−αρ¯−N2(t)+N4(t)N1(t)+N2(t)+N4(t)γ,

(15b)N˙2(t)=aN1(t)1−αN2(t)+ηN4(t)βNβ−αρ¯−N2(t)+N4(t)N1(t)+N2(t)+N4(t)γ−λ23+nrec−1N2(t),

(15c)N˙3(t)=λ23N2(t),

(15d)N4(t)=N−N1(t)−N2t−N3(t).

This is the system we work with to compute predictions reported in what follows. Initial conditions for our solution are N2ever(0)=Nobservedever(0)=16 for 24 February 2020 (RKI, 2020), N3(0)=N4(0)=0 and N1(0)=83.100.000−N2ever(0), where 83.1 million is the population size of Germany (Statistisches Bundesamt, 2020).

Figure 4

New incidences (reported) in the model (N2new(t), curve) and in the data (dots) (left) and the number N2ever(t) of sick in model and data (right) for the epidemic without public health measures.

4.3 The goodness of fit of the calibration

We can judge the quality of our calibration in two ways. First, we can check the fit for observed number of individuals that were ever reported as sick, N2ever(t) from (9). Second, we can check the quality of the match of the newly reported sick individuals, i. e. N2new(t) from (10).

The left part of Figure 4 shows the number of individuals that were ever reported as sick. The observations from RKI are depicted as circles. It shows that our parameter choices are sufficiently good to use this model for projections into the future.^[21] This impression is also confirmed from the right part of Figure 4. Obviously, fitting daily values is more demanding due to the greater variance in the data.

Figure 4 also shows where, in the case of no public health measures, Germany would be heading to. Under the assumption that the share of sick and reported individuals out of the infected individuals is r=0.1, the final number of reported sick individuals will be (right part of Figure 4) 5.5 million. This level will be reached towards the end of June.^[22] Hence, towards the end of the epidemic, at stable parameter values, around 7 in 100 inhabitants will have been sick (and reported).

Note that this long run value does not only come out of the solution of our calibrated system (15). It can also be directly read from the long-run prediction in (12), which can be written as limt→∞N2ever(t)/N=rρ¯. When we assume that two thirds of the population will be infected after the epidemic, ρ¯=2/3, and 20 % become sick of which half are reported such that r=.1, we immediately see from (12) that the long run share is 6.7 %, i. e. around 7 in 100.

Figure 5

The number of sick individuals (s=2) by day starting on 24 February 2020 in the absence of public health measures.

5.1 The number of sick individuals and health care

The most important quantity coming out of our analysis is probably information about prevalence, i. e. the number N2(t) of currently sick individuals. This number will decide whether there will be enough hospital services for all individuals.

The number of sick individuals initially rises strongly. This is the consequence of infections. As recovery sets in only after 2 weeks, the initial period sees this strong increase. Once the number of new sick cases reduces (as the number of individuals in state 1 decreases) and recovery sets in, the number of sick individuals falls again. The peak will be reached in mid-May. We note that this is a prediction for the case where individual contacts, as captured by the parameter a in (2), remain unchanged over time. If the policy measures in place as of 14 March have an effect, we would expect a slowdown in the rise of N2(t). This picture therefore shows what would have happened if no interventions had taken place. In this case, the number of sick (and reported) individuals on a given day, taking recovery into account, will rise above 1 million by the end of April.

Figure 5 also provides the maximum number of infected individuals over the epidemic. This allows us to understand whether the number of extreme cases is always below the level that can be handled by the German health system. The peak of the number of sick individuals in mid-April is close to 220,000. According to the Deutsche Krankenhausgesellschaft (2020), there are 28,000 beds available in intensive care units (ICUs) in Germany. A fraction of these are currently available for the treatment of COVID-19 patients. Moreover, an undefined number of additional hospital facilities with ventilation systems might be available in due time through reorganization activities. By contrast, a large fraction of COVID-19 patients might be treatable in regular wards. While it is therefore difficult to pin down the precise number of beds available for COVID-19 patients, a peak of more than 1 million on a given day is definitely beyond any level that can be handled by the health care system. The prevention of such a scenario in the end has been the key issue of all measures currently taken by federal and state governments. We assess these measures in Section 6.

The one ‘advantage’ of a scenario without public health measures: the epidemic is over fast. As the infection rate is very high, the outflow from the state of being healthy is fast. Hence, a large share of individuals, 1−r=90 %, are infected without showing symptoms (or weak symptoms such that they are not reported) and they become immune (in state 4) quickly. Both Figure 4 (left) and Figure 5 show that the epidemic would be over by the end of June.

5.2 Individual risks

One of the concerns of individuals is that they might become infected. How large is this risk in a world without interventions?

Figure 6

The probability per day (dotted) and per week (solid) to become sick in the absence of public health measures.

As Figure 6 shows, the risk to fall sick for each individual from (8) is very low throughout the entire epidemic. At the peak, the probability to fall sick on a given day (blue curve) is around 10−4. In words, 1 person in 10,000 inhabitants falls sick per day. The probability to fall sick within a week (red curve) is of course larger and lies at the peak at around 7 in 10,000 individuals. We emphasize again that this is the number of sick individuals that are actually reported to the RKI.

We compare the daily risk of 1 in 10,000 or the weekly risk of 7 in 10,000 with the long-run risk of 7 in 100 by the end of the epidemic to be or have been sick (as discussed after Figure 4), it seems that at the individual level, the daily or even weekly risk of the epidemic is not really a major health risk.^[23] It is clear, however, that at the level of society, the epidemic is a major issue. This difference between the individual risk and aggregate outcomes is the result of a classic negative externality. Even individuals that take into account the overall risk of 7 in 100 to become sick might not be concerned. If the risk of infecting others was taken into account, individual behaviour would look differently. This is why governmental interventions have a clear justification from the perspective of market failure and public economics.

5.3 Robustness check

The long-run sickness rate r

Given uncertainty about the share r of individuals that become sick from an infection, we undertake robustness checks. Instead of our value of 0.1, we also solve our model for an r ten times lower and up to twice as large – as the left column of Table 3 shows. For each of these new values for r, we recalibrate the values for a and α to match the initial observations of sick individuals.

Table 3

Varying the long-run sickness rate r.

r	a	α	β	γ	max{N2(t)}	tmax	t1000	t100new
0.01	1.03×10−4	.685	.8289	.6952	182,004	23 April	15 July	19 May
0.025	1.282×10−4	.6707	.8397	.6113	440,135	2 May	4 Aug	24 May
0.05	12.3×10−4	.7505	.8513	.7921	775,478	8 May	25 Aug	25 June
0.1	3.02×10−5	.5751	.8662	.6459	1,56×106	14 May	7 Sep	17 June
0.2	2.21×10−5	.5481	.8847	.7511	2,9×106	19 May	27 Sep	16 June

The model predictions are very reasonable from a qualitative perspective. The larger the individual risk r to get sick, the higher the number N2(t) of simultaneously sick individuals and the later the peak tmax of the epidemic.

We also find, as was to be expected, that a larger r implies a longer duration of the epidemic. If we define the end of the epidemic by the day t100new where only 100 new incidences are reported per day, we see that the end of the epidemic lies between 19 May and 25 June. Interestingly, the end of the epidemic is not monotonic in r. When the end of the epidemic is defined by the point in time t1000 where 1,000 sick individuals are left (N(t1000)=1000), the epidemic ends between 15 July and 27 September. Note that the t1000 definition implies much later endpoints (of one and the same epidemic) than the t100new definition.

Figure 7

The number N2(t) of sick individuals as a function of assumptions on the probability r to get sick after an infection.

Why are these numbers important from a public health perspective? The data section shows that Hubei and South Korea seem to have managed to keep the long-run ratio rρ¯ from (12) at 1 in 1,000 (Hubei) or 1 in 5000 (South Korea). If we believe that these values are long-run values, then either the individual probability r to get sick after infection of the long-run infection rate ρ¯ must be much lower than the standard values shown in Table 2 or sickness probabilities r must be lower.

As no information is available at this point about the share of individuals that are already immune to CoV-2 in Germany, one might well hope that Germany is heading towards a Hubei long-run equilibrium. As data quality was often argued to be an issue, our long-run ratio rρ¯ with r=0.01 would correspond to 6.7 in 1,000 individuals that were sick at some point in the long-run (N2ever). This would imply that – in the case of no public intervention – the peak in the number of sick individuals would only be around 180,000 individuals (as the first row of Table 3 shows). The blue time path of N2(t) of Figure 7 would be relevant.

6.1 The experiments

We model policy measures by assuming a time path for a. The first major policy measure, ‘shut down’ for simplicity, took place as of 14 March 2020. Sports events from professional to amateur leagues and schools were closed. This measure is planned to continue until 19 April 2020. As our quantitative analysis starts on 24 February, we employ parameters from Table 2 for the unrestricted epidemic. In particular, we keep the value for the contact intensity a of individuals from Table 2. After 20 days, we reduce a to a lower level alow for the duration of this policy measure. After 37 days, i. e. on 20 April, we return to the original level, assuming that sports events and schooling resumes. Formally,

captures the shut down initiated on 13 March. For all shut down scenarios we assume that alow is 50 % lower compared to a.

To understand whether a later intervention is more effective, we counterfactually assume that the intervention starts 32 days later. Here we set

We call this the ‘delayed shut down’.

Finally, we investigate into the effect of extending the shut down. We extend the period with lower social contact rate alow to last for 74 days, i. e.

This is our ‘extended shut down’.

6.2 The quantitative effects of policy measures

Baseline analysis

For this baseline analysis, we do not recalibrate the model as a in our scenarios changes after T, the day providing the last observation used for calibration. Parameter values are therefore given by the values in Table 2. The effect of various forms of shut downs are captured by a drop of the contact rate a by 50 % to alow, as shown in (16).

The results are shown in Figure 8. Our main variable for judging policy measures is the number N2(t) of individuals that are simultaneously sick.

First, the policy is effective. Reducing contacts reduces the speed of infections and new cases of sickness – but also the transition from healthy in state 1 to healthy (with infection) in state 4. The case without shut down is illustrated by the red curve in Figure 8. It is identical to the curve in Figure 5. The shut down from (16) which reflects the planned closing of schools in Germany until in mid-April is drawn as the green curve. It reduces the rise of the number of sick individuals until mid-April. Afterwards, the number of sick individuals will rise more quickly again. The peak is reached in June and this peak is lower than the one without the shut down. In this respect the shut down is a good policy measure. When the shut down is extended (the black curve), the peak is further shifted to the right and also smaller.^[24]

Figure 8

The effect of a shut down.

Quantitatively speaking, the shut down prolongs the epidemic in terms of its end as defined by t100new also employed in Table 3 (100 or fewer new incidences are reported per day). For the shut down currently in place and for the delayed shut down, the end is beginning of July. For the longer shut down, the end is beginning of August. Concerning the peak demands on the health system, the shut down as currently in place, and also an extended shut down, hardly have any effect. There is a certain delay – which helps the health sector to prepare for higher numbers in the future. But the level of the peak, as Figure 8 shows, hardly changes. This is a fairly bleak outlook.

To be fair, we should stress at this point that the assumption that a shut down reduces the contact rate and thereby the infection rate by 50 % is open to empirical investigation. At this point, this is a model assumption for which empirical evidence is still lacking. Hartl et al. (2020) show that statistically convincing evidence will be available at the earliest beginning of April. It seems fair to say, however, that it can hardly be expected at this point that current measures reduce the infection rate by 50 %. Hence, the above evaluation of the shut down might even be too optimistic.^[25]

The policy effects with lower ρ¯

To return to the uncertainty about the value of the long-run share ρ¯ of infected individuals, we vary r in Table 3 such that we could also study a “data-quality adjusted Hubei-scenario” where rρ¯ implies that there are 6.7 in 1,000 (ever) sick individuals. This was discussed after Figure 7. We now study changes in ρ¯ at unchanged r=0.1 such that we end up at rρ¯=6.7/1000. We therefore reduce ρ¯ by a factor of 10. We do recalibrate our model for this scenario as ρ¯ affects the calibration outcome. The fit is similarly good to the one discussed after Figure 4. Given the new parameter values, we change the contact rate according to the shut down (16) and the extended shut down (18).

Figure 9

The effect of changes in the long-run infection rate ρ¯.

Let us start with the long-run effects. As we know from (12), the long-run effects with ρ¯ reduced by a factor of 10 are 10 times lower. This is visible in the right part of Figure 9 displaying N2ever(t). Once the epidemic is over, there will be 550,000 individuals that were sick at some point, one tenth of the level in Figure 4. Again, this estimate is independent of any policy measure as the latter do not affect rρ¯.

Concerning short-run effects, we see that in the case of no intervention (or interventions that do not have an effect), the peak comes earlier for N2(t). The red curve in Figure 9 peaks in mid-April in contrast to early to mid-May for the higher long-run infection rate of Figure 8. The effect of the shut down currently in place looks qualitatively similar to the effect when ρ¯ is higher. By contrast, the extended shut down in Figure 9 does not display a strong rise in growth rates of N2(t) once it is over. It is directly followed by the decline in the number of sick individuals.

Most importantly, however, theses Figures 9 and 8 display a strong quantitative difference. If the long run infection rate in Germany is in the “data-quality adjusted Hubei range”, the peaks are at a much lower level. In the case without policy measures and in the case where the current shut down (16) is the only policy measure, the peaks of the simultaneously sick number of individuals is just above 200,000. This strongly contrasts to earlier values above 1 million. Another difference consists in the quantitative difference here between the extended shut down and the shut down. In the scenario with a lower long-run infection rate, an extended shut down implies that the peak of N2(t) will be much lower than the peak of the normal shut down. This was not the case in Figure 8. This points to the fact that shut downs need to be sufficiently strong relative to the overall potential number of sick individuals.

When the long-run infection rate is in this Hubei-range, then the epidemic is also over more quickly than in the case of an infection rate of two thirds. In any case, the epidemic will last until June, in the case of extended shut downs (or other measures) it can even last until mid-July.