3.1. Activity model

COVID-Town is a discrete-time model, in which each week is split into seven days ranging from Monday until Sunday, and each day is split into three phases (morning -evening- night). Each human agent spends one of these phases at home to rest, which is typically the night phase¹.

On each workday, one phase is devoted to going to work. Most workers work during the morning phase of Monday until Friday. Service workers (who work at commercial leisure facilities) and health care workers (who work at hospitals) are exceptions to this rule. They work in shifts to keep leisure facilities and hospitals staffed during each day including Saturday and Sunday. While hospitals are active during every shift, commercial leisure facilities are only open during morning and evening phases.

Finally, each phase which is not dedicated to work or rest is a leisure phase. During leisure time, agents decide via a boundedly rational stochastic process, how they want to spend their phase: They either want to stay at home, meet a friend, go to a non-commercial leisure facility (representing, e.g., parks) or a commercial leisure facility (representing, e.g., restaurants).

Needless to say, not every plan will work out. If an agent wants to meet a friend, but this wish is not reciprocal, the plan will fail. Leisure plans will also fail if e.g. agents lack the money to go to a commercial leisure facility, the agent or its friend is isolated, the location is under quarantine, etc.

The preferences of each agent are drawn from a random distribution, where the mean is given by age-dependent data from the German time use study. The number of friends is also drawn from a random distribution, where the mean is given from survey data. Thus, some agents do not have many friends and like to stay at home, whereas others are extroverted and potential “superspreaders”. On average, however, they represent the empirical characteristics of their age group. Thus, the activity model is empirically micro-founded.

3.2. Epidemiological model

During each of the phases described in the previous subsection, agents move to explicitly modeled locations, where they meet other agents and may become infected. Each infectious agent can meet up to a certain number of agents per phase, which can be altered in policy scenarios to reflect the fact that people who are aware of the virus typically try to keep their distance. At home they meet their family or cohabitors, at work their colleagues or customers and during leisure time their friends or other agents who happen to be at the same place coincidentally (e.g. cohabitors of a friend that they are meeting at their friend’s house or strangers who happen to go to the same restaurant).

The progression of the disease is modeled as in a state-of-the-art sir-type model (^{Imperial College, 2020}). After an agent is infected, it becomes exposed, i.e., it does not show any symptoms and is -initially- non-infectious. Later on, it becomes first infectious, and then moves out of the exposed category to either become infected with mild symptoms (including a certain fraction of permanently asymptomatic individuals) or become a subsequent hospital case. Hospital cases are either severe (who require a standard hospital bed) or critical (requiring first an intensive care unit and then -if the treatment is successful- a standard hospital bed during their recovery phase). All critical and a majority of severe cases die, if they do not receive the treatment that they need because the capacities are exhausted.

Source: ^{Mellacher (2020)}.

Figure 1 Progressionof the disease, based on ^{Imperial College (2020)}  

The probability to become a critical or severe case or die is age-dependent and based on empirical data.

3.3. Economic model

In the economic model, eight types of agents receive income, pay taxes, and spend their funds at two different markets. Sources of income are wages, profits, subsidies (unemployment benefits, family benefits, sickness benefits, quarantine benefits). Schools and hospitals are operated and paid for by the government. Factories and offices are private enterprises owned by firm owners. The types of agents and their economic relations are depicted in Figure 2.

Source: ^{Mellacher (2020)}.

Figure 2 Overviewof the economy. Solid lines represent work, dashed lines production, dotted lines consumption relations 

The consumption goods market in this first version of COVID-Town is highly simplified and aggregated. Factories (using blue-collar workers) and offices (using white-collar workers) produce consumption goods. Human agents plan to spend their average weekly income over the course of a week, where a fraction is dedicated to expenses at the leisure market. This fraction is age-specific in order to ensure that each age group is able to carry out its empirically observed leisure behavior. Both consumption and production happen on each day from Monday until Friday and the market is always cleared.

The leisure market, on the other hand, behaves completely differently, as the interactions on this market are disaggregated, firms are heterogeneous, and a state of disequilibrium is the exception rather than the rule. Agents visit and spend money at commercial leisure facilities during their leisure time as described in the activity model. On a weekly basis, commercial leisure facilities adapt their prices based on their utilization in the past week (i.e., the number of visitors divided by a maximum capacity). Leisure facilities are heterogeneous with regard to their attractivity, which in turn co-determines the leisure preferences of the human agents (together with individual tastes). Thus, even if the simulation is initialized with an equal number of employees per leisure facility, a pattern in which some facilities attract a lot of customers and employ a lot of workers emerges endogenously.

Like the consumption goods market, the labor market is also highly simplified. On a weekly basis, all firms (factories, offices, and commercial leisure facilities) decide on their labor force by comparing their average rate of profit with a target rate of profit. If the former exceeds (falls behind) the latter by a certain threshold, the firm will hire (fire) one worker. Unemployed workers always accept job offers. The target rate of profit is calibrated with the empirical counterpart in order to ensure that the artificial economy hovers around the empirical economy in the medium run in absence of any virus or containment efforts.

3.4. Calibration

The baseline calibration of the model with empirical data is described in detail in ^{Mellacher (2020)}, but that paper only includes a calibration to the first German wave of the pandemic and only the lockdown, but not the “opening up” phase. Since the first British wave was considerably longer and possibly started at a different period of time than the German wave, I adapt the calibration for the present research question. In order to do so, three steps have to be taken:

3.5. Infection detection rate

A key parameter in any epidemiological model that involves the isolation of detected infected, but does not model testing explicitly, is the infection detection rate, i.e., the probability of detecting one infected individual. This detection rate is set to be constant in ^{Mellacher (2020)}, which is a useful simplification for the short time horizon covered in that paper. If we assume, however, that testing is somehow targeted at people showing symptoms or identified contacts of infected individuals, the detection rate cannot be independent of the number of infected. This means that a successful lockdown both reduces the number of infections and increases the detection rate (thus further reducing the number of new infections via isolation and contact tracing). This effect is furthermore increased by the fact that governments massively expanded their testing capacities. In order to approximate the infection detection rate, one may thus look at the number of tests per capita and the test positivity rate, i.e., the fraction of tests which came back positive.

Up until 23rd of April 2020, Germany conducted 629,085 tests on 537,155 people (^{Robert Koch Institute, RKI, 2020}). In calendar week 11, i.e., the start of the simulation, Germany conducted 127,457 tests. No information about the number of people tested in Germany is available for this period. If we assume, however, that the same relation between the number of tests and the number of people tested as in mid-April holds, we can estimate the number of people tested to be approximately 108,000.

In the UK, testing data is available on a weekly basis, but these weeks range from Thursday until Wednesday (in contrast to the week from Monday until Sunday covered in the German data). In the two weeks encompassing the 11^th calendar week, on average 18,000 people were tested per week in England (^{UK Government, 2020a}). If we account for the difference in population size, this amounts to about 27,000 people tested in Germany. If we assume a direct relationship between the number of people tested and the detection rate and take the detection rate as estimated in ^{Mellacher (2020)} as given, this corresponds to a detection rate of 8.2% in the beginning of the simulation.

British (as well as German) testing capacity was increased drastically, however. If a) the number of tests increases more than the true number of infected, and b) tests are directed equally well towards actually infected (or at least not much worse), the detection rate goes up. A strong indication for a) is given in the daily positive rate, i.e., the share of COVID-19 tests which are positive as given by ^{Roser et al. (2020)}, which is about 30% at the beginning of April, 10% at the end of April and 2% at the end of May. In Germany, the positive rate plateaued at about 10% in the beginning of April, 3% at the end of April and under 1% at the end of May.

If we translate this data into detection rates and take the detection rate of 1/3 for the whole first wave (until the end of April) as estimated by ^{Mellacher (2020)} and the average German positivity rate of 6.3% in this period as given, the average positivity rate of 1.5% (Germany) and 4.2% (UK) in May translates to an increased detection rate, as in the height of the pandemic, only severe cases will be detected, whereas broader testing is possible if the testing capacities are not flooded with severe cases. The test positivity rates quickly fell in June to 0.7% (Germany) and 1.2% (UK), and were 0.6% in July (both in Germany and in the UK).

A simple approach to operationalize the assumption of a direct relationship between the test positivity rate and the detection rate is to calibrate the model to a certain detection rate in July and set the detection rates in May and June as linear combinations based on the detection rates until April and July. The goal of doing so is to find an evolution of the detection rate that produces the empirical German epidemiological dynamics in order to obtain a baseline calibration (in-sample-fitting, i.e., descriptive output validation). Once the model is calibrated, we can test whether it is able to produce the empirical British epidemiological dynamics using the same approach and a British policy scenario (out of sample fitting, i.e., predictive output validation).

3.6. Timeline

In a second step, we have to make the British and German policy timeline comparable. This is necessary, because the true number of infections per capita likely differed at the beginning of the German anti-COVID policies. Since British testing (based on the positivity rate) was much lazier in the beginning of the pandemic, we cannot rely on the reported number of infections. Thus, we must look at deaths in order to get a rough estimate of whether the virus had a “head start” in one of the countries.

According to data from the ^{ONS (2020)}, 115 COVID-related deaths were recorded in England and Wales from the 10^th to the 16^th of March, which corresponds (after accounting for the difference in population size) to about 162 German deaths, which corresponds to the number of German deaths between the 16^th and the 22^nd of March (^{RKI, 2021}). As a rough estimate, we can thus conclude that the virus had a 6-day head start in the UK with respect to Germany.

3.7. Policy scenarios

Table 2 shows the German and the uncorrected British containment policy scenarios. In the corrected British scenario, each policy takes place six days later to account for the six-day head start of the virus in the UK described in the previous subsection. While these scenarios are relatively detailed, some level of stylization is necessary, because of regional policy variation. This is especially true for Germany, where the “opening up” phase was largely left to the 16 states (^{Merkur, 2020}), but also for the UK, where we can observe variation in policies between the four nations comprising the UK (England, Wales, Scotland, and Northern Ireland), as well as variation in policy adaption (^{The Guardian, 2020}).

If family isolation and workplace isolation is activated, household members or colleagues of a detected infected are isolated. Minor social distancing ( SD ) increases the preference of agents to stay at home during leisure phases by 10%. Major SD increases it by 100% and cuts the maximum number of agents met during each phase by 50%. If schools close, children and teachers will stay at home during work phases. A household member will stay at home as a caregiver if there is a young child in the household. If commercial leisure facilities close, agents may not enter them anymore and the respective leisure plans will fail. Service workers will get paid by the government. If there is a contact ban, agents may not visit their friends anymore (but still go to non-commercial leisure facilities). Under curfew, agents may only leave their households in order to work. If non-essential industries close, a share of factories is closed down. In absence of any data on this, I assume the share to be 10%. If sanitary standards are increased, the infection chance goes down to 0% (hospitals), 25% (schools and commercial leisure facilities) and 50% (factories and offices).

4.1. Validation

I now turn to model validation, i.e., investigate the relationship between my model and the empirics (^{Gräbner, 2018}). Following ^{Tesfatsion (2017)}, ^{Mellacher (2020)} conducts extensive input validation, process validation, descriptive output validation and predictive output validation on the COVID-Town model. While some inputs, all processes and most outputs are stylized facts of the real world (i.e., validated qualitatively), most inputs are directly calibrated using empirical data and the deaths curve, as well as an estimated infection curve using nowcasting data are fitted to empirical data, i.e., are validated quantitatively.

However, the data derived from the scenario presented in ^{Mellacher (2020)} only captures the lockdown phase, but not the “opening up” phase. This is reasonable, since the first German wave lasted only about 50 days in terms of new infections from the start of the scenario, whereas the restrictions were only lifted after about 100 days. Since the first wave of infections took considerably longer in the UK than in Germany, the model has to be calibrated to fit the German data for a longer timeframe, also covering the “opening up” phase after the lockdown.

Figure 3 shows that the model can -building on the baseline calibration by ^{Mellacher (2020)}- be calibrated to reproduce the empirical German death pattern of the first five months of the pandemic (descriptive output validation). Furthermore, building on this calibration, a counterfactual policy scenario in which the German government would have adopted the British containment policy yields to a death curve that is very similar to the empirical British one (the “corrected” scenario), which suggests the model fares very well at predictive output validation, at least in the UK case. Figure 3 finally also shows that the evolution of the pandemic before the usual start of the simulation (i.e., the 11^th calendar week of 2020) played a crucial role, as the simulated death toll is considerably smaller, if we assume that the British situation was the same as the German one at period zero (i.e., the “uncorrected” scenario).

The solid black line shows local regression (LOESS), the dotted line quantile regression results

Figure 3 Simulatedcumulative total dead for Germany (left), the UK scenario which is corrected for the evolution of the pandemic until the beginning of the containment measures (middle) and the uncorrected UK scenario (right)  

4.2. Counterfactual policy analysis

In contrast to purely epidemiological models, COVID-Town is also able to evaluate economic, as well as social outcomes, albeit in a more or less stylized way. The model produces many economic outputs -such as profits, production and unemployment rates in different sectors, average income of various household or worker types, the income Gini etcetera. However, in order to keep the analysis simple and not go beyond the scope of this paper, I will only compare the cumulative Gross Domestic Product (GDP) in the various scenarios with regard to economic outcomes.

It is important to note that this simulated GDP is not an accurate estimate of what the German GDP would have been, if it had enabled Johnson’s anti-COVID-19 policy, because the model lacks important features of the real world that would make such an analysis possible (e.g., an outside world and consumption / investment behavior that depends on sentiment)².

However, the model is calibrated with empirical data and captures other important economic features of the crisis, such as heterogeneous industries which are differently affected by the containment measures, and Keynesian intersectoral spillovers of demand shortages. I also explicitly model a variety of reasons for productivity losses that are caused by the virus and the containment measures, namely a) caregivers working with lower productivity (or missing work), because schools are closed, b) workers being too sick to work, c) workers being unable to come to work, because they are under quarantine, d) workers dying. Most economic-epidemiological models do not account for any of the features a), b) or c), but COVID-Town is to my knowledge the only model that accounts for all of them. Thus, a comparative economic analysis seems to be fruitful, even though we know that it will not yield quantitatively correct predictions.

Due to the presence of heterogeneous services with a variable price in our economy, calculating real GDP is not straightforward. In order to approximate daily GDP, GDP _t , I calculate the following for each day, where l _i,t denotes the number of workers of type i (service workers, blue-collar workers or factory workers) employed in the private sector in the period t, l _j,t the number of workers of type j (health care workers or teachers) employed in the public sector, ( the target rate of profit, and w _i/j the gross wage rate of a worker. This calculation would correctly predict the GDP, if the economy were in equilibrium.

Finally, to get a quick impression of the social impact of the crisis and the containment policies, I present data on the cumulative leisure plans thwarted, i.e., the number of leisure activities planned by the agents that failed because a) the agents did not have enough money to pursue them, b) they were under quarantine or too sick, c) they were not allowed to visit a certain location, e.g., due to closures or due to a curfew, d) they wanted to meet a friend, but this friend did not want to meet them, e) they wanted to meet a friend, but this friend was too sick or under quarantine. While d) cannot be related to the pandemic in a straightforward way, all other options are more or less directly influenced by the crisis and governmental response. Thus, if a scenario causes relatively many leisure plans to fail, its negative social impact within the model is higher.

COVID-Town is highly flexible and offers, in principle, an infinite number of containment policy scenarios. In order to confine myself to the research question at hand, I study the public health, economic and social outcomes for only six scenarios: a) The German baseline scenario, b) the corrected UK scenario, which aims to mimic the empirical UK containment policy timeline, c) the corrected UK scenario with the exception that contact tracing is never interrupted, d) the corrected UK scenario, but with German-level testing, e) the corrected UK scenario with both contact tracing and German-level testing, and f) the uncorrected UK scenario, which does not account for the different evolution of the pandemic in the two countries until the beginning of March 2020.

Table 3 shows the cumulative outcomes of the first 300 periods, on the 10^th of June 2020 (for the German and the uncorrected UK scenario) or the 4^th of June (for the other scenarios). This time horizon was chosen, since the virus is eliminated in almost all simulation runs of scenarios involving German-level testing after 150 days and the remaining simulation runs may not be representative anymore, even though the German and the UK corrected scenario largely match the empirics for the first 120 days, as can be seen from Figure 3. The simulated GDP and the simulated thwarted leisure plans are normalized to the German baseline scenario. Table 3 reads as follows: In the UK corrected scenario, on average 48.6 agents out of 82,000 died within the first 100 days of the simulation. Adjusting for population size, the UK recorded 48.5 COVID-19 deaths within the same time horizon in the real world. Simulated GDP was 0.61% lower than in the German baseline scenario, whereas the number of leisure plans which failed during this timeframe was 32.9% higher.

We can derive five main insights from these results: