In 2020, distance rules were imposed in many countries to slow down the spread of the corona virus. Obeying such rules leads to a change in crowd behavior: pedestrians keep more distance and, to achieve this, might slow down their walking speed. In summer 2020, at the time of writing of this manuscript, we are noticing an increased use of the Vadere simulation framework for pedestrian dynamics. In particular, we are getting requests from users who want to simulate social distancing with the Optimal Steps Model (OSM), one of several locomotion models implemented in Vadere. The Optimal Steps Model explicitly models psychological personal space needs as introduced by Hall [3] . Default parameters in Vadere's OSM are calibrated to empirical data that was collected without social distancing [4] . In these, agents come into close contact when passing a bottleneck. See Figure 1 .
In this study, we try to find parameter values for the Optimal Steps Model which allow us to simulate social distancing. Our goal is to enable researchers to conduct case studies with default parameters, and without having to understand the mathematical intricacies of the model.
For this, we look at a classic bottleneck scenario. We argue, that this scenario is particularly suitable, because many geometries in the built environment can be interpreted as a sequence of bottlenecks. We also consider different desired social distances, which we define trough the Euclidean distance between the centers of agents. Agents in Vadere are represented by circles. We call contact any violation of the desired social distance, even for the shortest time. Figure 1 : Normal crowd behavior in front of a bottleneck. The agents (circles) try to reach the target on the left hand side. In this example, we consider agents being in contact if they violate a social distance of 1.5m. Red circles represent agents currently in contact. The contact between two agents is represented by a red connection line. Blue circles represent agents out of contact. Vadere's default parameters for the Optimal Steps Model are chosen to reflect such normal behavior. For a simulation with social distancing new parameters must be found.
We investigate two use cases:
• Use case 1: Social distance as lower bound. In this use case, the social distance value must never be violated.
As a consequence, if we find such a parameter combination, there are no contacts at all. In other words, we interpret the social distance as a lower bound of the actual distances kept by the agents. • Use case 2: Social distance as target distance. In this use case, we accept temporal violation of the distance requirement, that is, contacts. In other words, we interpret the desired social distance as an average distance.
For both use cases, agents need to keep larger distances than before. In the Optimal Steps Model, personal distance is modeled by introducing 'repulsion' between agents, or less physically interpreted, the utility of a position decreases when an agent approaches another agent. The repulsion (or utility drop) is captured by a so-called potential function. Hence, we need to find appropriate parameters of the potential function to strengthen the repulsion between agents.
In short, the research question of this study is:
• How can one adjust parameters in the Optimal Steps Model to achieve physical distancing that reflect specific desired 'social' distances?
The basic idea of the Optimal Steps Model is that virtual pedestrians (agents) are attracted by targets and repulsed by obstacles and other virtual pedestrians. Less physically spoken, agents, when moving, maximize the utility of their position. This utility depends (negatively) on the geodesic distance to the target and (negatively) on the close proximity to other agents. Agents move by 'stepping' on the position within a circle around their current spot that optimizes this utility. The circle radius represents each agent's personal maximum stride length, which in turn is linearly correlated to the agent's free-flow speed [5] , that is, an assumed desired speed when the path is free. Thus, agents step towards targets while skirting obstacles and avoiding collisions. In the reminder of the text, we use the physical interpretation, because it is used in the names of parameters in Vadere's implementation of the OSM.
The repulsion between two agents is achieved by a distance-dependent potential function. See Figure 2 . The potential function is based on Hall's theory of interpersonal distances which describes four distance zones around a person [2, 3] . Accordingly, the potential function is defined piece-wise on rings around each agent: a circular core for collision avoidance, a first ring that represents the intimate space, and a second ring that represents personal space. Agents outside the personal zone have no influence on other agents' path choice. This is mathematically modeled by setting the the potential function to zero.
The value of the potential function in the personal space ring is very low. Thus, this area will be kept free only if agents have ample space to avoid each other [2] . As soon as the space becomes more constricted agents will get closer. This The personal space is around the torso of an agent. If the distance between two agents is smaller than the sum of the torso radii, the agents collide (1. Collision area). To prevent this, the potential is set to a high value in the torso area. The personal space (2. Personal space) begins at x = 0.84m and ends at x = 1.59m. The upper bound x = 1.59m is the sum of the personal space width w = 1.2 and the torso radii of the agents.
is typical for normal human behavior. In the intimate space ring the potential function value increases significantly. Again see Figure 2 . In crowds, this area is only kept free if the density is low [2] . Finally, to prevent agents from overlapping, the potential is set to a very high value (when compared to the values in the personal and intimate spaces) in the collision area. The exact definition of the potential function, and default parameters, implemented in Vadere can be found in [1] .
The shape of the potential function is controlled through several parameters that could be adapted to achieve social distancing in the Optimal Steps Model. We try to keep the adaption of the OSM as simple as possible by only changing two parameters: the potential height h (in Vadere: pedPotentialHeight) and the personal space width w (in Vadere: pedPotentialPersonalSpaceWidth). Why do we choose these two parameters? The parameter potential height h controls the strength of repulsion. If h is increased, we expect agents to increase their distance to others. The parameter personal space width w controls how far the repulsion reaches: the larger w the bigger the influence area of an agent.
Note that the personal space width w is related to but not equal to the desired social distance d the user wants to model. When there is ample space it might suffice to set w = d to keep agents at least the desired social distance apart. The true distance among agents is an emergent value and will then be bigger! In a bottleneck scenario, like Figure 1 on the other hand, the true distance will be much smaller than w.
In crowd simulations, we are especially interested in dense crowds which typically occur in front of bottlenecks, such as doors or narrow passages. As a consequence, if we want to observe a virtual crowd that obeys social distancing, we need to choose a value for the personal space width w that is bigger than the desired social distance. In this study, we want to find suitable values for the parameters personal space width and potential height that do just that. Figure 3 : Funnel with balls. The ball size represents the repulsion between agents. If the ball size is small (small repulsion), the balls just fall through the narrowing. If the ball size is too large, balls get stuck. This is why we must not increase the repulsion to an infinite value to achieve social distancing.
There is currently no empirical data available to calibrate the Optimal Steps Model parameters for social distancing. Nonetheless, we need to find a suitable scenario and define plausible criteria which help us to evaluate how well social distancing is captured by the model.
To pick a scenario, we have a closer look at use case 1, where the social distance is a lower bound for permissible distances. In particular, all distances larger than the social distances are accepted.
One might be tempted to achieve such a behavior by increasing the repulsion between agents up to 'infinity', that is, to set the potential height h and the personal space width w to extremely high values.
But there is a problem with this approach: If the potential is too high, agents get stuck, as illustrated in Figure 3 with a funnel analogy. The funnel represents a bottleneck which agents need to pass to reach their target. The ball size represents the repulsion between agents. If the ball size is small (small repulsion), the balls just fall through the narrowing. If the ball size is too large, balls get stuck. This effect can be easily observed in test simulations. This is why we cannot increase the repulsion at will.
We use the bottleneck scenario depicted in Figure 1 with this consideration in mind. The bottleneck scenario covers both aspects, density and repulsion. We can directly observe what happens if the repulsion is too high.
The scenario is set up as follows. The topography of the scenario is 12m wide and 65m long. The width of the corridor is 1.2m and its length is 5.0m. 100 agents are generated in the source on the right hand side (green box) and try to reach the target (orange box) on the left hand side. With these settings, the agents are close to each other in front of the bottleneck. The source is placed sufficiently far away from the corridor to exclude any effects of the spawning process on the observation area. We start to count contacts at simulation time t = 20s to exclude any contacts produced by the spawning process. The distance measure between two agents is the Euclidean distance between their center points.
We do not have any empirical data to calibrate our parameters for social distancing. Hence, we need to define other criteria to decide which parameter values fit our two use cases best. For the choice of the criteria, we consider how the use cases were originally motivated: The goal of social distancing is to decrease the number of contacts which decreases the probability of infection.
The question is: How to count contacts? We decide to use a time measure t m for counting contacts.
The time measure t m is defined as
In our investigation, the number of agents is n = 100. t i,j is the time agent i has contact to another agent j. The time t i,j is zero if the distance x i,j between two agents i and j is above the desired social distance:
where t s,s is the point of time when the social distance is violated for the first time and t s,e is the point of time when the contact ends.
For use case 1, the contact time has to be zero. This means that all agents always keep a distance larger than the desired social distance d. Thus, the acceptable parameter combinations for use case 1 have to fulfill the condition: t m = 0 Condition for use case 1
We expect that this condition is always fulfilled when the personal space width and the potential height are set to high values which lead to high repulsion. Since a high repulsion can lead to clogging, we only want to increase the repulsion as much as necessary. We expect that there are multiple parameter combinations which fulfill condition 1. Hence, the solution to our problem is not unique but forms an indifference curve.
For use case 2, the parameter values are acceptable when the mean value x m of the distribution of the true distance x is equal to the desired social distance d: x m = d. We decide to approach this problem pragmatically.
We accept parameter combinations which fulfill:
x m ≈ d Condition for use case 2 (4)
We make use of the fact that some distances x i,j are below the social distance if t m > 0. Note: The detour with t m saves us from measuring distances between agents, and from deciding when agents are too far apart to count for the measure.
We consider the social distances d: We start with use case 1. For each social distance d i , we mark all parameter combinations which fulfill t m = 0. Then we build the convex hull around these points which serves as an approximation for the indifference curve. Visually this seems a good fit. Then we plot the 5 approximated indifference curves and analyze them visually. Finally, we derive a rule of thumb which relates social distance and parameter values for use case 1.
For use case 2, we need to find a value for the average contact time for which the average distance is approximately the desired social distance. For that purpose, we iteratively increase t m . Then we visually analyze whether the condition is fulfilled for the social distances d: one. We could now pick randomly a parameter combination on each indifference curve and list them in a table. The problem is that these values are true for the five discrete values of the social distance, but not for values in between. In order to provide parameter values for the complete social distance interval [1.25, 2.25], we need to find a relation between parameters and social distance. We anlayze the plot visually and suggest to use
as a rule of thumb. The relation w = 1.75d can be directly seen when analyzing the plot. With Eq. 6 the intersection points of line and indifference curves are almost equally spaced. See Figure 6 . Thus we assume that the behavior is linear in between. Under this assumption the rules of thumb can be used for any social distance between 1.25 and 2.25. One might be tempted to use regression techniques to find more precise equations. We argue that this would not improve the accuracy of the equations, because the points themselves are not accurate due to the discretization error of the grid and the approximation error caused by the convex hull.
Finally, we verify our rule of thumb visually for the social distances d: 1.5, 2.0 , see Figure 7 . We can not observe any contacts. Thus use case 1 is covered by our rule of thumb for at least the social distances d: 1.5, 2.0. We can not guarantee that the rule of thumb holds for any social distance and any topography. Nevertheless, accu:rate [6] has already succesfully tested our rule of thumb for some real-life applications.
In these two examples, social distancing was achieved successfully with the new parameter combinations. On the other hand, we also observe that the increased repulsion produces clogging. See Figure 7 (right). Clogging occurs earlier when the desired social distance d is increased. The screen shots in Figure 7 were all taken at simulation time t s = 40s. For a desired social distance d = 1.5, the agents still move at simulation time t s = 40s. Clogging occurs at a later point. For a social distance d = 2.0, clogging has already occured at t s = 40s. The probability for clogging can be reduced when using a dynamic floor field instead of a static one. But this does not completely resolve the problem of clogging. We will discuss this problem in 4.4. 
For use case 2, we look for an average contact time t m for which Eq. 4 holds. We iteratively increase t m and analyze the crowd behavior visually. For an average contact time t m = 10s, Eq. 4 seems to hold when checked visually. Figure  6 depicts the five indifference curve approximations for use case 2. We repeat the procedure of use case 1 to find a second rule of thumb:
The rules of thumb for the two use cases are summarized in Table 3 .
Again, we verify our results for use case 2 visually. See Figure 8 . In the video analysis, we again observe the problem of clogging which can be improved but not eliminated by using a dynamic floor field.
We are of the opinion, that real pedestrians resolve deadlocks by temporarily adapting their behavior. For example, they might briefly violate social distances to 'squeeze by'. This must be addressed by suitable behavioral models.
Another strategy would be to model a time-dependent agent potential function, where the repulsion decreases when agents experience a certain time of clogging. This would entail social distance violations and, in our eyes, model pedestrians' impatience with deadlocks. Such a behavior is already implemented in the accu:rate's simulator crowd:it [6] . After a certain period of time, agents switch to their default behavior, that is, without keeping any distance rules. This usually resolves the clogging. We plan to implement similar behaviors in the Vadere simulation framework.
For the time being, we recommend to keep this problem in mind when using our parameter values in Vadere. We strongly recommend to use a dynamic floor field and allow agents to switch places in counterflows. Both measures decrease the probability for clogging. See Table 4 Table 4 : Overview of additional settings in Vadere to reduce the probability of clogging.
We have found parameter values for the Optimal Steps Model to simulate social distancing. We considered two different use cases for social distancing. In the first use case contacts have to be completely avoided. The distance between agents must never fall below the required social distance value. In the second use case, the social distance is interpreted as desired distance. The average distance is approximately the social distance, but single distances might be smaller which leads to contact. For both use cases we derived rules of thumb to determine the Optimal Steps Models parameters for any desired social distance d ∈ [1.25m, 2.25m].
In visual analysis the rule of thumb seems to work for use case 2. Despite this, the precision of the equation could be increased by reformulating the problem as an optimization problem where each deviation from the desired social distance is punished within a utility function. For this purpose, several questions need to be answered: In which area around an agent should a distance deviation be punished? How to choose the utility function? The solution to this problem is part of future research.
The rules of thumbs are based on a bottleneck scenario where high densities occur. We expect that our parameter suggestions work for scenarios with bottlenecks. We can not guarantee that the they work for any topography.
When testing the new parameter values, we observed that the probability of clogging increases when increasing the social distance. This is caused by the higher repulsion. The clogging problem can be mitigated but not completely eliminated by using a dynamic floor field and allowing position switches in counterflow. We list the Vadere parameter settings to achieve this. However, we believe that the problem is better addressed by implementing suitable agent behaviors. We plan to do this in the Vadere simulation framework.