Agent-based Simulation of Pedestrian Dynamics for Exposure Time Estimation in Epidemic Risk Assessment
AA GENT - BASED S IMULATION OF P EDESTRIAN D YNAMICS FOR E XPOSURE T IME E STIMATION IN E PIDEMIC R ISK A SSESSMENT
Thomas Harweg
Department of Computer ScienceTU Dortmund University [email protected]
Daniel Bachmann
Department of Computer ScienceTU Dortmund University [email protected]
Frank Weichert
Department of Computer ScienceTU Dortmund University [email protected]
July 9, 2020 A BSTRACT
With the Corona Virus Disease 2019 (COVID-19) pandemic spreading across the world, protectivemeasures for containing the virus are essential, especially as long as no vaccine or effective treatmentis available. One important measure is the so-called physical distancing or social distancing . In thispaper, we propose an agent-based numerical simulation of pedestrian dynamics in order to assessbehaviour of pedestrians in public places in the context of contact-transmission of infectious diseaseslike COVID-19, and to gather insights about exposure times and the overall effectiveness of distancingmeasures. To abide the minimum distance of . stipulated by the German government at aninfection rate of 2%, our simulation results suggest that a density of one person per
16 m or below issufficient. The results of this study give insight about how physical distancing as a protective measurecan be carried out more efficiently to help reduce the spread of COVID-19. Keywords
SARS-CoV-2 · COVID-19 · Pedestrian dynamics · Agent-based simulation · Social-force model · Numericalsimulation
Starting at the end of 2019, the Corona Virus Disease 2019 (COVID-19) was first described in Wuhan, China [1], andhas rapidly spread world-wide over the past months causing an unprecedented pandemia with more than 431, 541deaths so far (first wave of disease) [2]. The illness is caused by the Severe Acute Respiratory Syndrome Corona Virus2 (SARS-CoV-2). Despite drastic restrictions in every-day life, numbers of new infections are still on the rise [3].Hence, special attention should be given to the protection of vulnerable patients with high risk of a severe course of thedisease. Diagnostic gold standard to identify SARS-CoV-2 infection is reverse transcription polymerase chain reaction(RT-PCR) of viral ribonucleic acid (RNA) collected by a combined nasopharyngeal swab (NPS) and oropharyngealswab (OPS) [4]. As long as no vaccine or at least no therapy is available, only exit restrictions and social distancing canslow down the spread of COVID-19. Social distancing, also called “physical distancing” means limiting face-to-facecontact with others. A model developed to support pandemic influenza planning [5, 6] was adapted using the data of theCOVID-19 outbreak in Wuhan to explore scenarios for the United States and Great Britain [7] resulting in the adviceof social distancing of the whole population and household quarantine of infected individuals as well as school anduniversity closures. By simulating the COVID-19 outbreak in Wuhan using a deterministic stage-structured SEIR(susceptible, exposed, infectious, recovered) model over a 1 year period,
Prem et al. [8] came to the conclusion that areduction in social mixing can be effective in reducing magnitude and delaying the peak of outbreak. These Britishepidemiologists from the London School of Hygiene & Tropical Medicine suspect that a second wave of COVID-19 a r X i v : . [ phy s i c s . s o c - ph ] J u l gent-based Simulation of Pedestrian Dynamics for Exposure Time Estimation in Epidemic Risk Assessmentdisease could only be prevented if exit and contact restrictions were maintained over the long term or at least resumedintermittently. It is therefore increasingly important to know what distances must be maintained to avoid infection. Asfar as these distances are concerned the study of Bischoff et al. [9] revealed that healthcare professionals within .
829 m of patients with influenza could be exposed to infectious doses of influenza virus, primarily in small-particle aerosols.This led to the advice of keeping a minimum distance of to from the Robert Koch Institut (RKI) [10], which isevaluating available information of the corona virus and estimating the risk for the population in Germany. The RKIand the German Federal Ministry of Health has in response published a handout [11] stating that in public, a minimumdistance of . must be maintained wherever possible. Besides, it is relevant whether people are next to each other,staggered behind each other, or directly behind each other. Especially in indoor areas, e.g. shopping, correspondingconstellations occur in combination. Simulations can help to make the necessary distance measures easier to understand.Different distance scenarios can be simulated and recommendations for the distance can be suggested.In their call to action Squazzoni et al. [12] give an overview of computational models for global pandemic outbreaksimulation and their limitations and
Chang et al. [13] propose a microscopic model for the simulation of the COVID-19outbreak in Australia. The model consists of over 24 million individuals with different characteristics and social contextand was calibrated with the 2016 census data of Australia.In the context of the simulation of pedestrian dynamics there exists a variety of comprehensive surveys [14, 15, 16].Models were differentiated mainly into macroscopic and microscopic. In macroscopic models the crowd is assumed asthe smallest entity. These models allow the representation of high density crowds, but cannot model the interactionbetween pairs or groups of pedestrians. Microscopic models assume one pedestrian as the smallest entity. They areroughly divided into physical force models, cellular automata or queueing models. This work focusses on the so calledsocial force model by
Helbing et al. [17, 18, 19, 20] which can be categorised as a physical force model.Main contributions of this work are • adaptable social force-based model for pedestrian dynamics in realistic environments • exposure time measurement for assessment of the spreading of diseases • discussion of the effects of distancing measures on exposure times.This paper is organised as follows. Firstly, Section 2 gives an overview of existing models for simulations of thespread of diseases. In Section 3 we set forth the agent-based model and the accompanying simulation we used forour experiments. Section 4 describes the experiments concerning the simulation of pedestrian dynamics in realisticenvironments, the effectiveness of distancing measures, and measuring of exposure time in the context of infectiousdiseases in general and COVID-19 in particular. Following, in Section 5 we present the results from the experiment,and in Section 6 we give a short conclusion/summary and outlook. As far as the simulation of the spreading of diseases is concerned most approaches are based on macroscopic models.The so called compartment models [21] divide the population of interest into compartments with different characteristics.The simplest model is the SIR model. It consists of three compartments susceptible , infectious and recovered . Thepopulation is split into these three compartments. Entities in the susceptible group model the entities most likely to beinfected. The entities in the infectious group are the ones already infected and the entities in the recovered group haverecovered from the infection. An entity transfer from for example susceptible to infectious state and from infectious torecovered state could be modelled. Thus, re-infection with the modelled disease would not be depicted by this model.A plethora of different variations of this scheme exist with varying numbers of compartments [22]. The independentvariable in the compartment models is the time t . The transfer ratios of the population from one compartment to anotherare expressed as derivatives with respect to t , thus resulting in differential equations for the compartments of the model.One of the shortcomings of these models is, that each individual in the population is modelled with the same set offeatures. This is overcome by the introduction of metapopulations [23, 24] building sub compartments of, e. g., entitieswith natural immunity or asymptomatic individuals. Still each of the metapopulations share a homogeneous set ofmodel parameters.With rapid growth of available processing power individuals based models (IBM) or agent based models (ABM)are used to model infectious disease outbreaks. Willem et al. [25] and
Nepomuceno et al. [26] give comprehensiveoverviews of IBM/ABM usage in the field of epidemiology. Based on the work by
Brockmann [27],
Frías-Martínez (a) Interaction of agents (infectious andnon-infectious) during simulation. Crit-ical radius is shown in red around infec-tious agents. (b) Visualisation of two agents p i and p j coming below the critical distance d min (c) Underlying model of agent interac-tion between two agents p i and p j Figure 1: Agent interaction in the simulation. Figure (a) shows a visualisation of the running simulation, and (b) showsthe measuring of distances. Figure (c) shows associated quantities of the underlying physical model et al. [28] propose an agent-based model of epidemic spread based on social network information from data of basetransceiver stations (BTC) captured during the 2009 H1N1 outbreak in Mexico. By simulating an outbreak of measlesthat occurred in Schull, Ireland in 2012 based on open data,
Hunter et al. [29] have shown recently, that agent basedmodelling in combination with open data leads to regionally transferable models.
Bobashev et al. [30] discuss thecombination of compartment models and microscopic agent-based models into an so called hybrid multi-scale model.In this paper, we propose a microscopic model for simulating pedestrian dynamics in the context of infectious diseasespread, including monitoring of contacts and exposure times in realistic scenarios. The simulation devised in this workruns in real-time, giving instantaneous feedback about the considered scenario and thus allowing for visual assessment,in addition to resulting statistics.
In the following, we describe in detail the underlying physical model of pedestrian dynamics and the simulationwe implemented. Our simulation is adapted to realistic scenarios in the context of the assessment of infectiousdisease spread, with focus on exposure time measurement. Specifically, we apply it to a supermarket scenario and thecorresponding measures taken with regard to the COVID-19 pandemic.The simulation we present is an agent-based approach based on the model of
Helbing et al. [17, 18, 19, 20]. Thesimulation is carried out on a static scenery with a defined number n ∈ N of agents or particles p i . Each agent has anindividual starting point s i and destination point d i ( t ) (both ∈ R ), the former being static and the latter varying withtime t ∈ R ≥ .Motion of an individual p i is governed by Equation 1, which is composed of the term for self-propelling a self andexternal forces f acting upon p i d x i ( t ) dt = a self i + 1 m i · (cid:88) j ( f soc ij + f phij ) + f wall i ) . (1)Here, x i ∈ R denotes the position of p i , and m i (in kg ) its mass.Self-acceleration of agents, as defined in Equation 2, is the adjustment of the actual velocity v i ( t ) to the desired velocity a self i ( t ) = v i e i ( t ) − v i ( t ) τ . (2)The parameter τ ∈ R > (in s ) determines the amount of delay time for an agent to adapt. The desired velocity of anagent p i is the product of its speed v i and the desired direction of movement e i ( t ) . The latter depends on the destination d i ( t ) and is determined by the pre-computed navigation method described below.The model by Helbing mainly consists of different types of forces acting upon an agent, social and physical forces. Thesocial forces f soc model a pedestrian’s endeavour to avoid contact with other pedestrians, while the physical forces f ph model effects arising when pedestrians are so close that there is actual contact between them. While in the originalmodel of Helbing et al. the social force is only comprised of a normal component (called f norm here), in this paper, weadd an additional tangential term f tang .The social force thus takes the following form: f soc = f norm + f tang . (3)3gent-based Simulation of Pedestrian Dynamics for Exposure Time Estimation in Epidemic Risk Assessment (a) Destination on upper left (b) Destination in the center (c) Destination on lower right (d) Mapping of colours to di-rections Figure 2: Colour-coded direction maps for agent navigation, determining directions of movement for each positionwithin walkable areas (a-c). Destinations are shown as black discs with white dots. Figure (d) shows how colours aremapped to directionsThe accompanying force in normal direction is defined as usual [17, 20]: f norm ij ( t ) = A · exp (cid:18) r ij − d ij ( t ) B (cid:19) · n ij ( t ) · (cid:18) λ + (1 − λ ) 1 + cos φ ij ( t )2 (cid:19) . (4)where A determines the force (in N ) and B the range (in m ) of repulsive interactions, and λ ∈ [0 , the (an-)isotropy.The vector n ij ( t ) denotes the normal from p i to p j of unit length, φ the angle between current direction e i ( t ) ofparticle p i and n ij ( t ) . The scalar value r ij is the sum of the respective radii r i and r j of the considered particles,while d ij ( t ) = (cid:107) x i ( t ) − x j ( t ) (cid:107) is the distance between their centres, both quantities measured in metres. Note that forthe experiments performed, the distance d (cid:48) ij between the perimeters is measured. This is defined as d (cid:48) ij = d ij − r ij .With regard to the measures taken by the German government for prevention of the spread of COVID-19, we setthe critical distance d min for a possible transmission to . . Figure 1 shows different aspects of agent interaction.Figures 1a and 1b show interaction in the simulation with regard to the critical distance, while Figure 1c showsassociated quantities of the mathematical model between two particles.The tangential term f tang (Equation 5) is now defined as a fraction γ ∈ R of the normal term, but in direction t ij ,orthogonal to n ij . Furthermore, this term is added only if the pedestrians involved, p i and p j , are heading into oppositedirections. This is taken care of by the function ψ ij (Equation 6). f tang ( t ) = ψ ij ( t ) · γ · (cid:107) f norm ij ( t ) (cid:107) · t ij ( t ) . (5)The value of γ ∈ [0 , determines the amount of normal social force added in tangential direction. For the experimentsperformed, we set γ = 0 . . The function ψ (Eq. 6) determines the pedestrians’ directions by evaluating the dot productbetween their respective directions of movement e i ( t ) = v i ( t ) (cid:107) v i ( t ) (cid:107) : ψ ij ( t ) = (cid:26) (cid:104) e i ( t ) , e j ( t ) (cid:105) ≤ otherwise. (6)The added tangential term makes for a more realistic movement of the agents, as they evade each other early, if applied.This is even more noticeable in conjunction with the method we employ for pathfinding, as described below.Physical forces acting between pedestrians are also defined in conformance to Helbing et al: f ph ij ( t ) = k · Θ( r ij − d ij ( t )) · n ij ( t ) + κ · Θ( r ij − d ij ( t )) · ∆ v tji ( t ) · t ij ( t ) . (7)Here, constants κ and k determine amounts of friction and the force counteracting body compression. The term ∆ v tji = ( v j − v i ) · t ij describes the tangential speed difference between agents p i and p j . The function Θ ensures thatphysical forces only arise when two particles actually touch, i. e. the distance d ij between them is lower than the sum oftheir radii r ij : Θ( x ) = (cid:26) if x < x otherwise. (8)Finally, wall forces f wall define interaction of agents with obstacles, like walls and stationary objects. The wall forcesare defined in analogy to particle forces: f wall i ( t ) = f wsoc i ( t ) + f wph i ( t )= (cid:18) A wall · exp (cid:18) r i − d ib ( t ) B wall (cid:19) + k · Θ( r i − d ib ( t )) (cid:19) · n ib ( t ) − κ · Θ( r i − d ib ( t )) · ( v i ( t ) · t ib ( t )) · t ib ( t ) . (9)4gent-based Simulation of Pedestrian Dynamics for Exposure Time Estimation in Epidemic Risk Assessment (a) Scene with destinations shown as red circles with crossesinside (b) Scene with 100 agents, infected ones shown in red, othersin green Figure 3: Supermarket scene with walkable areas shown in white and obstacles in blackIn Equation 9, d ib denotes the distance, and n ib the normal and t ib the tangential direction towards the closest obstacle(-point) b to p i , which are all determined directly from the scene representation described below. Parameters A wall and B wall are the analogues to A and B in Equation 4. This is a slight deviation from the model by Helbing et al. , as wedefine separate social distance parameters for obstacles and for other agents.For the experiments conducted in this work, a scene the agents can move around in needs to be defined. This willdefine the supermarket scenario considered in the simulation we performed with regard to the COVID-19 measures,as described in Section 4. The scene is represented as a distance transform [31, 32] of a two-dimensional, binarydiscretised map, as depicted in Figure 3. This map defines the size of the simulated area, as well as the regions withinwhich are walkable (shown as white) or pose an obstacle to the agents (black). The scene in the experiments performedis represented by a discretised map at a resolution of 8 pixels per m . Agent navigation depends on pre-calculated pathsbased on Dijkstra’s algorithm [33]. Accordingly, for each destination, a map containing the directions of movementtowards it for each walkable point in the area is generated. This way the direction of movement e i ( t ) of an agent p i is determined by looking up the given direction of movement at position x i from the map corresponding to p i ’scurrent destination d i (cf. Section 3). Colour-coded renderings of maps for three different destination points are shownin Figure 2. Agent navigation in the context of our COVID-19 simulation, including details on how destinations arechosen, is described in Section 4.The need for the added tangential social force f tang becomes apparent in cases of symmetry, which arise when forcesare at an equilibrium. The pre-calculated paths may impose symmetries, as they minimize the distance towards thedestination. This means, the prescribed way can be a thin line even though there is more space available, resulting inunnatural behaviour and building of queues, especially when agents are approaching others head-on. In the worst case,this can lead to “deadlocks” or clogging of pedestrians, even in cases where the surrounding area provides enough spacefor the agents to evade each other. The simulation was carried out with a basic “supermarket” scenario of size
80 m ×
60 m (cf. Figure 3) with varyingnumbers n of agents, which represent customers. The scene is aimed to mimic a typical German supermarket withshelves, counters and cashiers. Within the simulated area, a set of destinations is defined (Figure 3a), representingpoints of interest within the supermarket. We performed simulations for number of agents (population size) of n ∈ { , , , } . A defined amount of the agents were marked as “infected”.Figure 3b shows the scene with agents. Infectious persons are marked red in the visualisation, all others green.Infected agents carry the virus and can potentially infect others. The amount of infected agents was varied from { . , . , . , . , . } in the experiments, tantamount to ratios of to . The agents’ radii r i were sampleduniformly from [0 . , . . Mass is then determined proportionally to the individual radii as m i = 160 · r i in kg . Thedesired speed v i was sampled uniformly from the range [0 . , . (in m s ).Simulation parameters were chosen as A = 10 000 N , B ∈ { . , . , . , . } m , A wall = 10 000 N , B wall = 0 . , and τ = 0 . . As body contact does hardly ever occur, constants k and κ supposedly have very little impact on the outcomeof the simulation, if at all. For the sake of completeness, we set k = 20 000 kg · m − · s − , and κ = 40 000 kg · s − .5gent-based Simulation of Pedestrian Dynamics for Exposure Time Estimation in Epidemic Risk Assessment Distance in dm E x po s u r e T i m e i n s Infected in % (a) Simulation results for n = 50 . Distance in dm E x po s u r e T i m e i n s Infected in % (b) Simulation results for n = 100 . Distance in dm E x po s u r e T i m e i n s Infected in % (c) Simulation results n = 200 . Distance in dm E x po s u r e T i m e i n s Infected in % (d) Simulation results n = 300 . Figure 4: Box-and-whisker plots of the exposure times for populations of n ∈ { , , , } , desired distances of
50 cm ,
70 cm , and . and infection rates of 2%, 5%, 10%, 15% and 20%Initially, agents were distributed randomly across the area, i. e. their starting points s i are set to random (walkable)positions within the scene. The set of destinations were assigned to the agents in an even split, determining the agents’initial destination points d i . During the simulation, if an agent reaches its destination, a new destination is assignedrandomly from the available set of destinations (excluding the current one). Agent movement is then determined bythe direction map corresponding to the destination, as described in Section 3. Thus, a typical behaviour of customerswalking around the supermarket is simulated. The simulation time was set to fifteen minutes.During the simulation, for each agent, we keep track of exposure time to infected agents. More precisely, the time spanan agent comes below the prescribed safety distance d min of . (cf. Section 1) to an infected agent is accumulatedper individual during the course of the simulation. The box-and-whisker plots [34] in Figure 4 show the average deviation of the exposure time concerning the differentpopulation sizes, distances, and infection rates, as described in Section 4. The boxes are representing the interquartilerange which contains 50% of the values and the whiskers are marking the minimum and maximum values, excludingoutliers (marked as black diamonds).The first scenario depicted in Figure 4a shows the results for a minimal population of agents. With a uniformdistribution of the individuals and if superstructures (cf. Figure 3a) are neglected a density of one individual per
96 m is to be expected. Thus, enough space for avoidance is available. This is supported by the box plots. Even if a veryshort desired distance of
50 cm is considered, the mean exposure time is .
78 s (standard deviation (std) . ) for aninfection rate of 2% of the individuals. Even if very high infection rates of 10% are considered the mean exposure timeis below one minute (
52 s , std .
16 s ). If the requested distance of . (cf. Section 1) is maintained mean exposuretime for an infection rate of 20% is . (std . ). With a growing number of agents exposure times are rising (cf.Figure 4b). Expected density with a population of is one individual per
48 m . For small desired distances andmedium infection rates of 5% the mean exposure time is .
95 s (std . ) which is 61% higher than the results6gent-based Simulation of Pedestrian Dynamics for Exposure Time Estimation in Epidemic Risk Assessmentwith the same parameters and a population of . As the desired distances increase the exposure times are decreasing.Considering the requested . distance the mean exposure time for an infection rate of 20% is .
85 s (std .
79 s ),which is 836% higher than the exposure times for a population of . As far as realistic infection rates of 2% areconcerned, the mean exposure time is .
49 s (std .
84 s ). The results for a population of size is shown in Figure 4c.This population size with an expected density of one agent per
24 m representing the maximum density allowed duringlock down in most of German federal states at the time of writing. For an infection rate of 2% and an desired distanceof . the mean exposure time is .
19 s (std .
02 s ). With an expected density of one individual per
16 m Figure 4dshows the results for a simulation with agents. Mean exposure time is .
31 s (std .
03 s ). The 50% percentile(median) for all simulations with this parametrisation is below ( . , .
64 s , .
37 s , ), showing the effectivenessof distancing tactics in the minimisation of exposure times. We have presented a simulation of pedestrian dynamics in realistic scenarios with focus on the spread of infectiousdiseases by contact transmission. An important measure taken to reduce the spread of COVID-19 is the so-called social distancing or physical distancing , aiming to reduce close contacts between individuals in public places. In theexperiments we conducted, we showed how our simulation can give insights about exposure time to infected individualsand the feasibility and effectiveness of keeping distance in realistic crowded scenarios. Our experiments suggest that, ifwe assume an infection rate of 2%, the prescribed minimum distance of . can be maintained if a density of oneperson per
16 m is not exceeded.In this work, we have presented a method for risk assessment concerning pedestrian dynamics and exposure time inconjunction with COVID-19 in particular and infectious diseases in general. Due to the flexibility of the approach, itcan be applied to a great variety of scenarios prone to transmission of contagious diseases. This especially includespublic places, indoor as well as outdoor.Our simulation can serve as a tool for a better assessment of quantities regarding the number of people to admit, oron guidelines for distances to keep between individual persons. Due to the nature of the simulation, it can also giveinsight about optimisation on the geometry of the surrounding, like identification of bottlenecks and hotspots, in orderto reduce risks for people moving around and meeting in the place in question.With the COVID-19 pandemic affecting countries all over the world at the time of writing, the urgent need of modelsand tools for better assessment of situations in public places is apparent. We are confident that our simulation resultscan serve as a basis for better risk assessment in public places in the context of infectious diseases, and for furtherresearch in this area. References [1] P. Zhou, X. Yang, X. Wang, and et al. A pneumonia outbreak associated with a new coronavirus of probable batorigin.
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