How detection ranges and usage stops impact digital contact tracing effectiveness for COVID-19

To combat the COVID-19 pandemic, many countries around the globe have adopted digital contact tracing apps. Various technologies exist to trace contacts that are potentially prone to different types of tracing errors. Here, we study the impact of different proximity detection ranges on the effectiveness and efficiency of digital contact tracing apps. Furthermore, we study a usage stop effect induced by a false positive quarantine. Our results reveal that policy makers should adjust digital contact tracing apps to the behavioral characteristics of a society. Based on this, the proximity detection range should at least cover the range of a disease spread, and be much wider in certain cases. The widely used Bluetooth Low Energy protocol may not necessarily be the most effective technology for contact tracing.


General simulation model
Our simulation model consists of a population of 10,000 individuals living in households. The household size distribution follows the German household size distribution (1) and is illustrated in Table S1. We assume that individuals return to their household every day. Following prior work on the spread of infectious diseases (2), individuals can meet at four different types of places. In our simulation, these places are workplaces, schools, supermarkets, and households. Similar to the German society, we assumed that 70.1% of the population regularly visit a workplace or school (including universities and kindergartens). This estimated share is based on ca. 44.7 million individuals working (3), ca. 8.3 million pupils in school (4), ca. 2.8 million university students (5), and 2.4 million kindergarteners (6), out of a German population of 83 million individuals. In our simulation, these workplaces or schools have a quadratic shape (as illustrated in Figure S1 B) with a row capacity between 2 and 8, thus, the total capacity of the specific location is between 4 and 64. When visiting a workplace or school, an individual randomly selects one seat, which ensures a mixing of different contacts in our simulation. The distance between each row and column for a workplace or school are randomly sampled between 1.5 and 2-meters. Thereby, individuals can move along aisles in order to go to their seats with a constant speed of 1 . Individuals arrive between 8 AM and 8:30 AM and stay there for 8 hours.  S1. Illustration of the spatial simulation locations. Besides households, infectious individuals can infect other individuals at supermarkets, workplaces, and schools. This figure shows individuals at these places, some of the shown individuals are moving, illustrated by a speed component in x and y direction. Specifically, A shows a supermarket scenario, where individuals move along an aisle. B shows a workplace or school scenario in quadratic shape, where an individual moves to a randomly assigned seat when arriving, and may get infected while doing so.
If an individual is not at their workplace or school, they can visit a supermarket once a day with a daily probability of 20%. These are open Monday through Saturday from 8 AM till 8 PM. For the 10,000 individuals in the population, there are three supermarkets, resulting in a comparable distribution of supermarkets per capita to Germany (7). A supermarket is simulated as one long aisle, as illustrated in Figure S1 A. Individuals can go in both directions, forward, and backward, which increases the mixture of contacts. To do so, each individual has a constant base speed in y direction, which is determined such that the planned duration of the visit is a random number between 15 and 45 minutes. To allow individuals walking backwards, an individual walks with an acceleration in y direction which takes a sinusoidal form. The period duration is three minutes, the amplitude is determined such that the maximum additional speed gain from this sinusoidal function is the difference to the maximum speed of 1 . However, this maximum additional speed gain is divided by a random integer between 1 and 10. As a result, some individuals occasionally walk backward (meaning they walk with a negative speed), whereas others walk almost with a straight speed. The x acceleration is determined randomly, but capped such that the maximum total speed (consisting of a speed in x direction and a speed in y direction) is 1 m/s. Furthermore, the individuals always stay within the physical boundaries.
In general, the next position of a person as a vector is determined by its current speed ⃗, its acceleration ⃗, and the simulation time step : The speed vector of the person after the next time step is defined as follows: The time step in our simulation is 100 ms, as this allows us to simulate walking speeds in the order of about 1 and measure distances in the order of decimeters at the same time.

Epidemic simulation
Another essential part of our simulation model is the infectious disease. We aim to model this infectious disease realistic to COVID-19 with a SEIR model, where individuals are either in a susceptible, an exposed, an infectious, or a recovered state. This flow of states is also illustrated in Figure 4 of the main article.
At the start of the simulation, ten individuals of the population are exposed and the rest of the population is susceptible. The transition of a susceptible state to an exposed state occurs when an individual gets infected. Afterwards, it takes a certain duration until the individual develops symptoms. In accordance with extant knowledge on COVID-19 (8), we model this duration as a random number sampled from a triangular distribution with a minimum of 1 day, a mode of 5.5 days, and a maximum of 14 days. Similarly, two days before an individual develops symptoms, its disease status transits from the exposed state to the infectious state. However, the duration in the exposed state cannot be negative. The duration in the infectious state is 9 days, effectively meaning 2 days before the symptoms and 7 days after the symptoms are noticed (9). We implemented the spatial simulation for the workplaces, schools, and supermarkets, but did not implement it for the households. Instead, we assume that household members typically have a close contact between each other, therefore, in the simulation model an individual infects other household members once they are infectious. Furthermore, an individual and their household members go into quarantine the day after an individual showed symptoms.
A further important set of parameters determines the probability of an infection event. For such an infection event, an infectious and a susceptible individual need to be within a certain distance and the closer they are, the more likely the infection event occurs. Specifically, an infection event can occur only if the two individuals are in proximity of 2 m or closer. In our simulation, each infectious individual is assigned a fixed random value from a triangular distribution with a minimum of 0, a mode of 0.5 m, and a maximum of 1 m. This value accounts for a varying infectiousness between individuals which is observed for COVID-19 (10). The higher it is, the more infectious an individual is, given it is in the infectious state of the disease. Once a susceptible individual is in proximity of an infectious individual, a random number from a half normal distribution is drawn. This half normal distribution has a standard deviation of from the infectious individual. An infection event can then occur if the distance between the two individuals is smaller than this drawn random number. This mechanism alone, however, would lead many more infections if the simulation time step is small (it is 100 ms in our simulations). Therefore, another condition has to be met: a random event of 5% probability needs to be drawn to be true. The first condition ensures that an infection event is more likely, the closer two individuals are. A half normal distribution was chosen instead of a normal distribution because the distance between individuals is always a non-negative value. The second condition is necessary in order to adapt the likelihood of an infection to the simulation time step. Based on 30 simulations without any CT, this setting results in in an average base reproduction number of R0 = 2.799, a median of R0 = 2.792, and a 90% confidence interval between 2.787 and 2.811. The median value in our simulation was close to the median value of 2.79 that was identified for the SARS-CoV-2 pandemic by extant research (11).

Measurement of the base reproduction rate
As discussed above, we set a few parameters such that we obtain a reasonable value for the base reproduction number R0. This affects parameters that define the likelihood of an infection, given an infectious individual is in proximity of a susceptible individual. This section describes how we calculated the base reproduction rate 0 out of our simulation results.
In a simplified SEIR model (12,13) with a natural death rate of µ = 0, the base reproduction rate is given by 0 = . Thereby, is the reciprocal of the infectious period and is the effective contact rate. The SEIR model is described by a system of differential equations, which themselves describe the change of four quantities. These four quantities are the number of susceptible individuals at a given time ( ), as well as the number of exposed individuals ( ), the number of infectious individuals ( ), and the number of recovered individuals at a given time ( ). The system of differential equations is as follows ( is the size of the population, the average incubation period is

Simulation process
We simulated on a high-performance computing cluster with nodes consisting of an Intel Xeon Gold 6230 CPU at 2.1 GHz frequency. In total, we simulated 46 different types of scenarios (composed of different CT adoption rates, different PDRs, and varying probabilities of a usage stop) Each scenario type was simulated 30 times (with different random seeds), in order to obtain smoothed means and confidence intervals. Each simulation ran on one CPU core, therefore, we computed with 1,380 CPU cores in total. Depending on the simulation parameters and random influences, the simulation time varied, most of the simulations finished within 1 and 2 days, the longest one took ca. 7.8 days. In order to decrease the required simulation time, one could increase the simulation time step (e.g., to 1 s instead of 100 ms), and one could reduce the size of population. However, this can lead to less accurate results, and a higher variance; this is the reason we set the simulation time step to 100 ms. The source code to simulate and visualize the results is available at (14). Table S2 shows the main results of our study with the mean value and 90 % confidence intervals based on 30 simulations. Most of these results are also visualized in Figure 2, 3, and 4 within the main article.