Disease contagion models coupled to crowd motion and mesh-free simulation
Parveena Samim Abdul Salam, Wolfgang Bock, Axel Klar, Sudarshan Tiwari
aa r X i v : . [ m a t h . D S ] J a n Disease contagion models coupled to crowdmotion and mesh-free simulation
Parveena Shamim Abdul Salam
Department of Mathematics, TU KaiserslauternKaiserslautern, 67663, [email protected]
Wolfgang Bock
Department of Mathematics, TU KaiserslauternKaiserslautern, 67663, [email protected] of Mathematics, TU KaiserslauternKaiserslautern, 67663, [email protected] of Mathematics, TU KaiserslauternKaiserslautern, 67663, [email protected]
January 6, 2021
Abstract
Modeling and simulation of disease spreading in pedestrian crowdshas been recently become a topic of increasing relevance. In this pa-per, we consider the influence of the crowd motion in a complex dy-namical environment on the course of infection of the pedestrians.To model the pedestrian dynamics we consider a kinetic equation formulti-group pedestrian flow based on a social force model coupled with n Eikonal equation. This model is coupled with a non-local SEIS con-tagion model for disease spread, where besides the description of localcontacts also the influence of contact times has been modelled. Hydro-dynamic approximations of the coupled system are derived. Finally,simulations of the hydrodynamic model are carried out using a mesh-free particle method. Different numerical test cases are investigatedincluding uni- and bi-directional flow in a passage with and withoutobstacles. keywords: pedestrian flow models; disease spread models; multi-groupmacroscopic equations; particle methods AMS Subject Classification:
The COVID-19 pandemic struck the everyday life worldwide. To avoid fur-ther spread in absence of a vaccine or a well-established medical therapy,most countries in the world invoked non-pharmaceutical intervention mea-sures as extensive backtracking, testing and quarantine [21, 12]. One keypoint is the contact reduction, which is often based on the minimization ofthe number of contacts and contact time [1, 21, 47]. In many cases, such asin schools or large working facilities social distancing is not always possible,especially at the end of classes or shifts, crowds are formed to leave the fa-cilities. Simulations of disease spread in a moving crowd could give valuableinformation about how risky these mass events are and support the designof such intervention measures.Modelling of crowd motion has been investigated in many works on dif-ferent levels of description. Microscopic (individual-based) models based onNewton type equations as well as vision-based models or cellular automatamodels and agent-based models have been developed, see Refs. [24, 25, 11,16, 38, 41]. Associated macroscopic pedestrian flow equations involving equa-tions for density and mean velocity of the flow have been derived as well andinvestigated thoroughly, see Refs. [6, 38, 23, 20, 2, 13, 14]. An elegant way toinclude geometrical information and goal of the pedestrians into these mod-els via the additional solution of an eikonal equation has been developed byHughes, see Refs. [2, 18, 28, 29, 35]. For the derivation and relations of the2boves approaches to each other we refer to Refs. [19, 15, 20]. More com-plex geometries and obstacles have been included into the models by manyauthors, see, for example, Refs. [41, 45, 46]. Pros and Cons of these modelshave been discussed in various reviews, we refer to Refs. [6, 26, 3, 7] for adetailed discussion of the different approaches.On the other hand, there is a vast literature on disease spread models,see Ref. [5] for an overview on multiscale models and connection to crowddynamics. Moreover, we refer to Refs. [22, 34, 39, 17] for a small selection ofpapers on mathematical models based on dynamical systems and to Refs. [27,40, 9] for agent-based and network models. Models coupling crowd motionand contagion dynamics are far less investigated. We refer to Ref. [31] for arecent investigation coupling a crowd motion model with a contagion modelin a one-dimensional situation.One main objective of the present paper is to include a kinetic diseasespread model in the form of a multi-group equation into a kinetic pedestriandynamics model, derive hydrodynamic approximations and provide an effi-cient numerical simulation of the coupled model for complex two-dimensionalgeometries. For the pedestrian flow model we consider a kinetic equation formulti-group pedestrian flow based on a social force model coupled with anEikonal equation to model geometry and goals of the pedestrians. This modelis coupled with a non-local contagion model for disease spread, where localcontacts as well as the influence of contact times is included. A second ob-jective is the extension of the methodology to situations with more complexgeometries and moving objects in the computational domain. This is a wayto model, for example, the detailed interaction of pedestrians with largergeometrically extended objects like cars in a shared space environment. See[10] for another approach to the interaction of pedestrians and vehicles. Thenumerical simulation is, as in Refs. [20, 33] , based on mesh-free particlemethods [43] for the solution of the Lagrangian form of the hydrodynamicequations. Such a methodology gives an efficient and elegant way to solvethe full coupled problem in a complex environment.The paper is organized in the following way: in section 2 a kinetic modelfor pedestrian dynamics with disease spread is presented. The section con-tains also the associated hydrodynamic equations derived from a momentclosure approach. The meshfree particle methods used in the simulations isshortly described in Section 3. The section contains also the results of thenumerical simulations. The solutions of the macroscopic equations are pre-sented for different physical situations and parameter values including uni-3nd bi-directional flow in a two-dimensional passage without obstacles andwith fixed and moving obstacles.
We use a kinetic model for the evolution of the distribution functions ofsusceptible, exposed and infected pedestrians as a starting point and deriveassociated hydrodynamic equations.
We consider an equation for the evolution of pedestrian distribution functions f ( k ) = f ( k ) ( x, v, t ) , k = S, E, I . Here f S stands for the distribution of sus-ceptible pedestrians, f E for the exposed pedestrians and f I for the infected.The evolution equations are given by ∂ t f k + v · ∇ x f k + Rf k = T k (1)with k = S, E, I . The operators R and T k are given by the following defini-tions. Rf k ( x, v ) = ∇ v · (cid:0) [ G ( x, v ; Φ , ρ ) − ∇ x U ⋆ ρ ( x )] f k (cid:1) . with ρ = ρ S + ρ E + ρ I , where ρ k ( x ) = Z f k ( x, v ) dv. Here, U is an interaction potential describing the local interaction of thepedestrians and ⋆ denotes the convolution. Common choices for the inter-action potential are purely repelling potentials like spring-damper potentialsor attractive-repulsive potentials like the Morse potential. In this paper wehave, for simplicity, considered a Morse potential without attraction givenby U = C r exp (cid:18) − | x − y | l r (cid:19) , (2)where C r is the repulsive strength and l r is the length scale. The part of theforcing term involving G describes the influence of the geometry on the pedes-trian’s motion and a potential long range interaction between the pedestrians,4ee below for a detailed description. The operators T k are defined using aSEIS-type kinetic disease spread model leading to T S = νf I − β I f S T E = β I f S − θf E (3) T I = θf E − νf I with constants ν, θ , see Remark 3 below, and the non-local infection rate β I = β I ( x, v ; f S ( · ) , f E ( · ) , f I ( · )) depending in a non-local way on the rate ofinfected persons, compare Refs. [31, 9] for similiar approaches. We define β I = Z ρ ( y ) Z φ ( x − y, v − w ) f I ( y, w ) dwdy. (4)The kernel φ in the infection rate is chosen as φ = φ ( x, v ) = i o φ X ( x ) φ V ( v ) . (5)with R φ X ( x ) dx = 1 = R φ V ( v ) dv . Here, φ X is determined as a decayingfunction of | x | to take into account the effect that infections between pedes-trians are more probable the closer pedestrians are approaching each other. φ V is chosen in a similar way depending on | v | to take into account the fact,that infections are more probable the longer the interacting pedestrians stayclose to each other, that means the smaller their relative velocities are. Theparameter i o is determined by the infectivity. We refer to the section onnumerical results for the exact definition of these kernels.Finally, G is given by G ( x, v ; Φ , ρ ) = 1 T (cid:18) − V ( ρ ( x )) ∇ Φ( x ) k∇ Φ( x ) k − v (cid:19) , (6)where Φ is determined by the coupled solution of the eikonal equation V ( ρ ) k∇ Φ k = 1 . (7)This describes the tendency of the pedestrians to move with a velocity givenby a speed V ( ρ ) and a direction given by the solution of the eikonal equa-tion. The eikonal equation essentially includes all information about theboundaries and the desired direction of the pedestrians via the boundaryconditions. These boundary conditions for the eikonal equation are chosen5n the following way. For walls or for the boundaries of obstacles in the do-main we set the value of Φ at the boundary to a numerically large value. Foringoing boundaries, free boundary conditions for Φ are chosen, whereas foroutgoing boundaries, where the pedestrians aim to go, we set Φ = 0.We note that on the one hand, the eikonal equation includes the geomet-rical information via boundary conditions. On the other hand, it models aglobal reaction of the pedestrians to avoid regions of dense crowds via theterm V ( ρ ) in (7). Remark 1.
The parameters in the above formulas, in particular in the def-inition of (7) and (2) have to be chosen consistent with empirical data, see[8, 30].
Remark 2.
Instead of the social force model used here, one could as welluse more sophisticated interaction models, see for example [16, 4, 36]. Wenote that the differences between these models in the present hydrodynamiccontext are small. The behaviour of the solutions is rather dependent on thechoice of the parameters.
Remark 3.
The dynamical system called the SEIS-model with constant in-fection rate is given by dSdt = − βIS + νIdEdt = βIS − θEdIdt = θE − νI where β is the infection rate, ν the recovery rate and θ the rate with whichexposed persons are becoming infected. Pedestrians are potentially becomingexposed, when they are in contact with infected pedestrians. However, ex-posed pedestrians are only becoming infectious with a certain rate θ . Exposedpedestrians do not infect other pedestrians. Usually, in the situations and onthe time scales under consideration here, ν and θ are very small and set tozero in numerical simulations such that the number of infected pedestriansremains constant during the simulation. Remark 4.
A key difference between other agent-based models as e.g. [9] isthe dependence of the infection on the relative velocity of the agents. Thishas direct implications on the process of infection during the dynamics, ascan be seen the the case studies. .2 The multi-group hydrodynamic model Integrating the kinetic equation against dv and vdv and using a mono-kineticdistribution function to close the resulting balance equations, i.e. approxi-mate f k ∼ ρ k ( x ) δ u ( x ) ( v )one obtains a continuity equation for group k∂ t ρ k + ∇ x · ( ρ k u ) = Z T k dv (8)and the momentum equation ∂ t u + ( u · ∇ x ) u = G ( x, u ; Φ , ρ ) − ∇ x U ⋆ ρ (9)with the total density ρ given as ρ = ρ S + ρ E + ρ I . Moreover, G ( x, u ; Φ , ρ ) = 1 T (cid:18) − V ( ρ ( x )) ∇ Φ( x ) k∇ Φ( x ) k − u (cid:19) , where Φ is determined by solving the eikonal equation V ( ρ ) k∇ Φ k = 1 . The continuity equations are explicitly written as ∂ t ρ S + ∇ x · ( uρ S ) = νρ I − β I ρ S ∂ t ρ E + ∇ x · ( uρ E ) = β I ρ S − θρ E (10) ∂ t ρ I + ∇ x · ( uρ I ) = θρ E − νρ I . with β I = β I ( x ; ρ S ( · ) , ρ E ( · ) , ρ I ( · ) , u ( · )), where now β I = Z φ ( x − y, u ( x ) − u ( y )) ρ I ( y ) ρ ( y ) dy. (11) Remark 5.
Multi-group models for pedestrian flows have been also used indifferent contexts. For example in [37] a hydrodynamic multi-group modelfor pedestrian dynamics with groups of different sizes has been developed andanalysed in [37]. .3 The hydrodynamic model using volume fractions For numerical computations this is rewritten using volume fractions, thatmeans we solve the continuity equation ∂ t ρ + ∇ x · ( ρu ) = 0 (12)and the momentum equation ∂ t u + ( u · ∇ x ) u = G ( x, u ; Φ , ρ ) − ∇ x U ⋆ ρ. (13)Then we compute the volume fractions α S , α E , α I with α S + α E + α I = 1as ∂ t α S + u · ∇ x α S = να I − β I α S ∂ t α E + u · ∇ x α E = β I α S − θα E (14) ∂ t α I + u · ∇ x α I = θα E − να I with β I = β I ( x ; α I ( · ) , u ( · )) defined by β I = Z φ ( x − y, u ( x ) − u ( y )) α I ( y ) dy. (15)Finally, one computes ρ I = α I ρ, ρ S = α S ρ, ρ E = α E ρ. We allow the domain on which the above equations are defined to depend ontime. In particular, we consider moving obstacles, which change their pathand speed in order to avoid collisions with the pedestrians, while movingtowards a specific target. The interaction between the pedestrians and themoving obstacle is additionally modeled by kinematic equations using a re-pulsive potential similar to the pedestrian-pedestrian interactions. A secondEikonal equation is integrated for modelling the path of the moving obstaclein the geometry and the desired destination of the obstacle. The velocityupdate equation for the obstacle is given as dx O dt = v O , v O dt = −∇ x U O ∗ ρ ( x O ) + G O ( x O , v O ; Φ O , ρ ) , (16)where x O and v O are the position and velocity of the obstacle’s centre ofmass, ρ is the density of pedestrians at x O . U O is an interaction potentialdescribing the interaction of the pedestrians on the obstacle. G O is obtained from the gradient of the Eikonal solution φ O for the ob-stacle as G O ( x O , v O ; Φ O , ρ ) = − T O (cid:18) − V O ( ρ ( x O )) ∇ φ O ( x O ) ||∇ φ O ( x O ) || − v O (cid:19) , ||∇ φ O || = 1 . That means, the eikonal equation is for the obstacle only used to includethe geometrical informations and the goal of the obstacle. We note that theaction of the obstacle on the pedestrians is given by the solution of equation(7) via the boundary conditions at the obstacle’s boundaries. In contrast,the action of the pedestrians on the obstacle is given via U O . For the numerical simulation we use a meshfree particle method, which isbased on least square approximations. [44, 33] A Lagrangian formulation ofthe hydrodynamic equations is used and coupled to the SEIS model and theobstacle’s kinematic equations Eq. (16).
The spatially discretizted system in Lagrangian form is given by dx i dt = u i ,dρ i dt = − ρ i ∇ x · u i ,du i dt = G ( x i , u i ; Φ , ρ ) − X j ∇ U ( x i − x j ) ρ j dV j , (17)and dα Si dt = να Ii − β Ii α Si (18)9 α Ei dt = β Ii α Si − θα Ei (19) dα Ii dt = θα Ei − να Ii (20)with β Ii = X j φ ( x i − x j , u i − u j ) α Ij dV j , and dx O dt = v O ,dv O dt = G O ( x O , v O ; Φ O , ρ ) − X j ∇ U O ( x O − x j ) ρ j dV j . (21)Here dV j is the local area around a particle. Remark 6.
Although the equations in Lagrangian form look similiar to amicroscopic problem, there are important differences. In particular, the effi-cient solution of the continuity equation (compared to a determination of thedensity in a purely microscopic simulation) and the consequent availablity of ρ i on each grid-point allows the efficient use of the density, used at variousplaces in the model. For a description of the mesh-free method for the pedestrian flow equationsin a fixed rectangular domain, we refer to [43, 33]. There, the eikonal equa-tion has been solved on a separate regular mesh on the entire domain usingthe fast-marching method. The information from the irregular point cloud,used for solving the fluid equations has been interpolated to the eikonal gridand vice versa. Such an approach requires for a moving obstacle to takespecial care of the grid points being overlapped by the obstacles. Using animmersed boundary method, they are activated and deacitvated dependingon the location of the obstacle.Here, we have followed a different approach. The eikonal equation isdirectly solved on the irregular point cloud used for the hydrodynamic equa-tions. Thus, in each time step an eikonal equation on an unstructured gridhas to be solved, see [42] for the Fast-marching method in this case. The10pproximation of the spatial derivatives in the eikonal equation is obtainedas for the hydrodynamic equations: the spatial derivatives at an arbitrarygrid point are computed from the values at its surrounding neighboring gridpoints using a weighted least squares method. We refer to [32] for a numer-ical study of the accuracy and complexity of such a method for the eikonalequation.Finally, we note that for a uni-directional flow of pedestrians, we haveto solve only one eikonal equation. If there is bi-directional flow, we haveto solve an eikonal equation for each direction. The same would be true forseveral obstacles with different goals.
We have performed numerical simulations of equations (Eq. (12) to Eq. (15)and (16) for different scenarios. In all our simulations, we consider a com-putational domain given by a platform or corridor of size 100 m × m .The top and bottom boundaries are rigid walls without any entry or exit.Right and left boundary are exits for pedestrians and obstacles depending onthe situation under consideration. We consider uni- as well as bi-directionalflow of the pedestrians. Initially, the pedestrians are distributed as shown inFigure 1 with a distance of 1 . m from each other. We consider the caseswith and without obstacles, which are either fixed or moving. We initializethe infected pedestrians (colored in red) with α I = 1 , α S = 0 , α E = 0, thesusceptibles (in green) by α I = 0 , α S = 1 , α E = 0.We have used for the infection rate β I the functions φ X = exp( −| x − y | )and φ V = exp( −| u − v | ). Moreover, we choose V ( ρ ) = V max (cid:16) − ρρ max (cid:17) and V O ( ρ ) in the same way with V max substitued by V Omax . For the parameterswe have used the values given in Table 1.Variable Value Variable Value V max m/s ρ max
10 ped/ m V Omax m/s T 0.001 sC r = C Or l r l Or i o m /s Table 1: Numerical parameters.11
Figure 1: Initial situation at t = 0. Top row: Uni-directional (left) andbi-directional (right) flow. Bottom row: Fixed obstacle (left) and movingobstacle (right). Red indicate infected, green indicate susceptible pedestri-ans. 12uring the evolution, infected pedestrians are colored in red, susceptiblesin green and exposed pedestrians in blue according to the values of α k . If α E > .
05 the colour is switching from green to blue, meaning that the proba-bility of being exposed has exceeded a certain threshold. The red pedestriansremain red throughout the simulations, since the recovery rate ν is set equalto 0 in the simulations. Moreover, since θ is also set to 0 exposed patientsare not becoming infected and cannot infect others in the simulations.The fixed and the moving obstacle considered are rectangular in shapeand initially located as shown in Figure 1.Explicit time integration of the equations in Lagrangian form is done witha fixed time step size of 0 .
001 in our simulations.
In this first test case we have considered a fixed obstacle and comparedthe results to a situation without obstacle. We consider bi-directional flowwithout an obstacle and uni- and bi-directional flow with an obstacle. Thisis done for the case φ v = 0 (no influence of contact time) and the case where φ V is chosen as defined above, that means for a situation where the influenceof the contact time is included. The present initial configuration is chosenin such a way, that there is no increase of the number of probably exposedpedestrians, if a uni-directional flow without obstacle is considered with orwithout influence of the contact time.Figures 2 to 5 show the time evolution of the moving grid points and theassociated infection labels with influence (left column) and without influence(right column) of contact time. Row 1 shows bi-directional flow withoutan obstacle, row 2 shows uni-directional flow around an obstacle and row 3shows bi-directional flow around an obstacle. Red indicate infected, greenindicate susceptibles and blue indicate exposed pedestrians.Here one observes, e.g. in Figure 4 or 5, that in situations with bi-directional flow (top- and bottom row), the number of exposed patients isstrongly reduced, if the contact time is taken into account. For uni-directionalflow, the differences are much smaller as expected, since pedestrians stay nearto each other for a longer time during the evolution.Comparing row 2 (uni-dirctional with obstacle) with the uni-directionalcase without obstacle (no exposed pedestrians), one observes that the numberof exposed patients is considerably increased due to the denser pedestriancrowd surrounding the obstacle. Similar observations can be made comparing13ow 1 and row 3.In Figure 6 we have plotted the number of pedestrians with a higherprobability of being exposed versus time. One can observe, that the numberof these pedestrians is much higher if the influence of the contact time isneglected. This happens mainly in bi-directional flows, which is as expected,since, even if pedestrians are coming close to each other, they pass eachother quickly and the contact time is short such that a contagion is lessprobable. On the other hand if pedestrians are walking in the same direction,the effect of neglecting the contact time is comparably small. We mention,that the contagion model is based on very simple assumptions and obviouslythe parameters of the contagion model have to be adapted to experimentalfindings. We refer again to [31] for similiar investigations.Finally, in Figure 7 we show the macroscopic density of the pedestriansat times t = 10 , , ,
40 for the situation with influence of contact time foruni-directional flow with fixed obstacle.
In this subsection we consider the interaction of pedestrians with a movingobstacle, e.g. a vehicle in a shared space. We consider the same computa-tional domain as in test-case 1 with pedestrians, which are initially located asin Figure 1 (right bottom). Their destination is the right exit. The movingobstacle of size 4 m × m is located on the right with the left exit as destina-tion. Typically in a restricted traffic area the vehicle has a low speed limit.We have chosen 10 km/h ∼ m/s . The maximum speed of the pedestrians ischosen as 2 m/s .In Figure 8 we have plotted the positions of pedestrians and obstacleat different times t = 8 s, s, s and 22 s . We observe an interaction ofpedestrians and obstacle between times t = 14 s and t = 22 s . Moreover,one observes a slight increase of the number of exposed people which is lesspronounced than in the case of the big fixed obstacle in test-case 1.Moreover, in Figure 9 we have plotted the x -velocity component of theobstacle along its center of mass. One observes that the obstacle (comingfrom the right) accelerates and maintains almost its maximum speed. Whenit encounters the pedestrian crowd, it reduces its speed. Finally, it acceleratesagain, when there are no pedestrians anymore in the surroundings.14 Figure 2: Pedestrian dynamics at time t = 10 with influence (left) andwithout influence (right) of contact time. Row 1: Bi-directional flow. Row2: Uni-directional flow around obstacle. Row 3: Bi-directional flow aroundobstacle. Red indicate infected, green indicate susceptibles and blue indicateprobably exposed pedestrians. 15 Figure 3: Pedestrian dynamics at time t = 20 with influence (left) andwithout influence (right) of contact time. Row 1: Bi-directional flow. Row2: Uni-directional flow around obstacle. Row 3: Bi-directional flow aroundobstacle. Red indicate infected, green indicate susceptibles and blue indicateprobably exposed pedestrians. 16 Figure 4: Pedestrian dynamics at time t = 30 with influence (left) andwithout influence (right) of contact time. Row 1: Bi-directional flow. Row2: Uni-directional flow around obstacle. Row 3: Bi-directional flow aroundobstacle. Red indicate infected, green indicate susceptibles and blue indicateprobably exposed pedestrians. 17 Figure 5: Pedestrian dynamics at time t = 40 with influence (left) andwithout influence (right) of contact time. Row 1: Bi-directional flow. Row2: Uni-directional flow around obstacle. Row 3: Bi-directional flow aroundobstacle. Red indicate infected, green indicate susceptibles and blue indicateprobably exposed pedestrians. 18 T i m e ( t ) No. of exposed pedestrian (in %) V = V = e x p (- | u - v | ) T i m e ( t ) No. of exposed pedestrian (in %) V = V = e x p (- | u - v | ) T i m e ( t ) No. of exposed pedestrian (in %) V = V = e x p (- | u - v | ) Figure 6: Number of pedestrian with an increased probablity of being ex-posed (in %) vs time. First row bi-directional flow without obstacle (left)and with obstacle (right). Second row: uni-directional with obstacle.19igure 7: Macroscopic density ρ of pedestrian dynamics at times t =10 , , ,
40 with influence of contact time for uni-direction flow with fixedobstacle. 20
Figure 8: Positions of pedestrian and obstacle at t = 8 s, s (first row) and t = 18 s, s (second row). Red indicate infected, green indicate susceptibles,blue indicate probably exposed pedestrians.21 C en t e r o f m a ss - - . - - . - - . X-velo. comp. of obstacle
Figure 9: The velocity of the obstacle along the center of mass.22
Concluding Remarks
We have presented a multi-group macroscopic pedestrian flow model combin-ing a dynamic model for pedestrians flows and a SEIS based kinetic diseasespread model. A meshfree particle method to solve the governing equations ispresented and used for the computation of several numerical examples analyz-ing different situations and parameters. The dependence of the solutions and,in particular, the dependence of the number of exposed pedestrians on ge-ometry and parameters is investigated and discussed and shows qualitativelyconsistent results. Findings indicate, that in particular, in bi-directional flowit is important to take into account the contact time for a realistic descriptionof the flow. This is a realistic qualitative behavior, which sheds a new lightfor the design of emergency exits in the presence of a pandemic.
Acknowledgment
This work is supported by the German research foundation, DFG grant KL1105/30-1 and by the DAAD PhD program MIC.
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