Modeling Social Distancing and Quantifying Epidemic Disease Exposure in a Built Environment

Abstract

As we transition away from pandemic-induced isolation and social distancing, there is a need to estimate the risk of exposure in built environments. We propose a novel metric to quantify social distancing and the potential risk of exposure to airborne diseases in an indoor setting, which scales with distance and the number of people present. The risk of exposure metric is designed to incorporate the dynamics of particle movement in an enclosed set of rooms for people at different immunity levels, susceptibility due to age, background infection rates, intrinsic individual risk factors (e.g., comorbidities), mask-wearing levels, the half-life of the virus and ventilation rate in the environment. The model parameters have been selected for COVID-19, although the modeling framework applies to other airborne diseases.

The performance of the metric is tested using simulations of a real physical environment, combining models for walking, path length dynamics, and air-conditioning replacement action. We have also created a visualization tool to help identify high-risk areas in the built environment. The resulting software framework is being used to help with planning movement and scheduling in a clinical environment ahead of reopening of the facility, for deciding the maximum time within an environment that is safe for a given number of people, for air replacement settings on air-conditioning and heating systems, and for mask-wearing policies. The framework can also be used for identifying locations where foot traffic might create high-risk zones and for planning timetabled transitions of groups of people between activities in different spaces. Moreover, when coupled with individual-level location tracking (via radio-frequency tagging, for example), the exposure risk metric can be used in real-time to estimate the risk of exposure to the coronavirus or other airborne illnesses, and intervene through air-conditioning action modification, changes in timetabling of group activities, mask-wearing policies, or restricting the number of individuals entering a given room/space. All software are provided online under an open-source license.

I. Introduction

The COVID-19 outbreak started in late 2019 and was declared a pandemic by the World Health Organization in March 2020. It has affected all countries across the globe. The COVID-19 coronavirus is known to spread mainly through close person-to-person interaction, mainly through an airborne vector [1]. As vaccines roll out, there is hope for society to return to a certain degree of normalcy. While it is important for businesses and schools to reopen, to reduce the risk of further waves, it is important to ensure that they reopen safely, particularly as variants evolve, and vaccine uptake faces obstacles such as supply restrictions or distrust of science and authorities.

Extensive research has been done on the mechanisms of spreading infectious diseases and many models have been proposed. One class of models is called the compartmental model [2]. In such models, the population of interest is divided into compartments, and assumptions are made about the rate of transfer from one compartment to another. The susceptible-infected-recovered (SIR) model [3], susceptible-infected-susceptible (SIS) model, susceptible-exposed-infected-recovered (SEIR) model [4], [5], and susceptible-exposed-infected-susceptible (SEIS) models are some examples of compartmental models. Based on the nature of the infection being studied, such as whether immunity can be developed against the disease or whether there is a period of time when individuals can be exposed before becoming infected (or symptomatic), different models can be used [6]. For example, the SIR model was used to model an outbreak of influenza [7] and SARS [8], [9]. Alrabaiah et al. [10] modeled the COVID-19 coronavirus as a modified SEIR model. Deterministic compartmental models assume that the compartment population is homogeneous. While this might be a reasonable assumption when an epidemic has progressed, the assumption does not hold at the beginning of an epidemic when the number of infected individuals is small. Thus the pattern of contact between people is the dominating factor determining the transmission of infection. Therefore the model should include such stochastic effects [6], [2]. Some statistical approaches have included these effects in disease spreading models using transport networks as a placeholder for human mobility [11], [12]. Compartmental models are known for their simplicity and interpretability, but they rely on parameters such as the average inter-individual transmission rate or the reproduction rate of the epidemic (${\mathcal{R}}_0$), which are difficult to estimate in the real world. Models which take into account heterogeneous interaction patterns between people have high computational complexity. Goscé et al. proposed a middle ground that uses the knowledge of crowd behavior to improve compartmental models [13]. There are also multiple research, which have focused on using network theory to study spread of diseases on networks [14], [15], [16]. These research have helped in improving accuracy of disease modeling at large scales. There are also interesting studies, which use wearable sensors to estimate disease transmission routes rather than relying on models [17].

Another class of epidemic models are agent-based models, which model the interactions of individuals at a simplified level of abstraction using simulated stochastic environments of populations of such agents. In this study, we propose an agent-based model to study the risk of being exposed to the coronavirus in a contained built environment. Several other researchers have proposed relevant models. Bazant et al. [18] introduced a framework to help understand the number of people that can safely remain in a built environment for a specific duration, based on activities being performed by the individuals, whether they are wearing masks or not, size of the room, level of ventilation in the room and other variables. Baxter et al. [19] proposed an agent-based simulation model to predict the number of infections and deaths for various scenarios and dates of reopening of schools.

Various other works which focus on safe reopening have also been proposed. D’Amico el al. [20], investigate the steps taken by a hospital in New York to restart neurosurgical efforts, and in an effort to construct a framework to help with similar reopenings, the required milestones were outlined. Harris et al. [21] argues that there is a stark contrast in reopening guidelines; some focus on microscopic details while others are too abstract. Therein, the author investigates the most salient status indicators of COVID-19, which will help decide whether to loosen or tighten certain restrictions. Ingrassia et al. [22] study the steps required to safely reopen simulation facilities (which are training spaces for health professionals and students to learn procedures in a safe environment). They have also addressed the effects of reopening. Zhang et al. [23] used an interrupted time-series analysis to study the changes in the number of confirmed cases, hospitalization, and deaths after reopening in eleven countries and forty US states. Kaufman et al. [24] estimate and compare the number of COVID-19 infection and death cases after reopening using data from states that used an evidence-based reporting strategy and stated that reopened indoor dining without implementing a mask mandate.

Herein, a model-based approach is proposed to study the risk of being exposed to the coronavirus in a closed space or built environment. We propose a novel social distancing metric, which helps quantify how well social distancing is being practiced in a space. We also propose a risk metric that quantifies the risk of an individual getting exposed to COVID-19. Since it is not advisable to have multiple subjects in a built environment to collect data to verify our metrics, we introduce a simulation, which captures this data instead, and uses it to verify our metrics. Various factors such as age, underlying diseases, whether masks are being worn, diffusion of particles in the space, ventilation rate, and half-life of the coronavirus in the particular space are taken into account. The contamination risk in the space is visualized by using a heat-map representation.

The proposed framework can be used to answer questions such as: whether the flow of people within the space has to be changed to ensure better social distancing; if the number of people in the space needs to be reduced or can be increased safely; if the ventilation rate needs to be increased, all to minimize the risk of exposure of individuals to the coronavirus. Using the developed simulation software, this framework can also be used to anticipate scenarios that might cause higher risks and to find proactive solutions. The visualization tool also provides an easy way to identify the high-risk hot spot areas of the building, which can be used to install appropriate ventilations or to advise reduced capacity areas.

The proposed framework has been applied to a particular built environment, as a case study, within the Cognitive Empowerment Program (CEP) run by Emory University and Georgia Institute of Technology. The program aims to understand the behaviors and improve people’s lives diagnosed with mild cognitive impairment (MCI), which is the early stage of dementia. The members of this program routinely visit the Executive Park 6 (EP6) building on the Emory campus to engage in various activities to help improve their mental functioning. Since the outbreak of the pandemic, these sessions at EP6 were paused. The current study seeks a fact-based approach to help with a safer reopening of EP6. The developed models and simulations are applicable to other buildings, including shopping centers, schools, libraries, nursing homes, fitness centers, etc.

As part of the CEP study, the hereby proposed social distancing and risk metrics have been implemented to be combined with live positions of individuals in the built environment of EP6 to estimate compliance to social distancing and risk of individuals towards getting infected by an airborne infection in real-time. In the extended study, Bluetooth position sensors and cameras are used to acquire the real-time positions of individuals within this therapeutic space.

II. Methods

The proposed methods consist of 1) a social distancing and risk metric, and 2) a simulation framework for epidemic disease spread through person-to-person contact and infectious suspending airborne particles. A heat-map visualization is provided to visualize how the infectious particles spread in the built environment and identify the regions in the environment which have a higher concentration of infectious particles at a given time. This visualization can help plan the flow of people through the environment and plan a timetable for people in the building.

A. Average social distancing metric

The social distancing metric for the entire building is used to quantify the level of social distancing being practiced in the environment. This metric is not subject-dependent and evaluates the overall social distancing quality in a built environment, as it depends on the number of people in the environment and the distances between them. The risk of infection is highest when people are positioned close to each other. As the distances increase, the risk of getting infected directly by another individual decreases rapidly. The risk depends on the concentration of virions present, which recent research has suggested that it decays in a Gaussian manner with distance from the infected individual, thus reducing exponentially with distance [25]. This is captured in the social distancing metric which is high when people are close to each other and decreases exponentially as the distances between them increase. It also scales linearly with the number of people in the environment.

The proposed social distancing metric ϕ is defined as follows:

$$\varphi =\sum _{i,j;i\ne j}^N{e}^{-(d_{ij}^2-{\alpha }^2)∕{\alpha }^2}$$ (1)

where N is the total number of people in the space or built environment, d_ij is the distance between person i and person j, and α is the recommended social distance to be maintained (e.g. 6ft). Accordingly, the closer people are together, the higher the social distancing metric is. The α² term in the exponential numerator ensures that when two people are α meters apart, the social distancing metric reduces to 1. Therefore, when N people are α meters apart from one another, the social distancing metric equals N. This gives a reference for the social distancing metric values. This metric is calculated for each time step separately.

B. Individual risk of exposure modeling

The social distancing metric alone does not capture the risk of exposure due to the diffusion of infectious particles. Thus, we also introduce an individual risk metric, which quantifies the risk of being exposed to the virus in the environment as an individual moves in the environment. It considers the dynamics of infectious particle movement due to diffusion, age of people, presence of underlying medical conditions, immunity levels, mask-wearing levels, background infection rates, the half-life of the virus, and ventilation rate of the air conditioning in the environment. The higher the value of the risk metric, the higher the risk is for a particular person. This metric is based on the COVID-19 studies, which have shown that the major source of virus transmission is through inhalation of aerosol droplets exhaled by an infected person [18].

To model residual infectious particles, which remain in the environment as infectious individuals move around, we use a model with environmental memory. The model is presented for a 2D environment to simplify the notations, but they are mutatis mutandis applicable to 3D.

The terms used in the model are defined as follows:

• Γ{γ_k(p_k); k = 1,…, K}: the environment (built, natural, or a combination), which is gridded into a mesh of K units of arbitrary shape (but “small” in size and equal in surface/volume) denoted by γ_k and centered at Cartesian coordinates p_k = (u_k, v_k)^T;

• $\mathcal{N}({\gamma }_k)$: the set of γ_k and its direct spatial neighbor units, e.g., each unit has eight direct neighbors if Γ has a rectangular shape mesh, or six neighbors when the mesh is hexagonal;

• s_i: subject i (i = 1,…, N) with spatial Cartesian coordinates x_i(t) = [a_i(t), b_i(t)]^T at time t;

Algorithm 1 Indirect risk of exposure through environment

$$\begin{gathered} \begin{array}{cc}\hfill & \mathrm{Input}:\text{A population of subjects}{s}_i(i=1,\dots ,n),\text{with}\hfill \\ \hfill & \\ \text{initial locations}{\mathit{x}}_i(t_0)\text{and a personalized list of}\hfill \\ \hfill & \\ \text{infection-related factors}\hfill \\ \hfill & \mathrm{Input}:\text{Initial particle concentration}\tilde{C}({\mathit{p}}_k)=C({\mathit{p}}_k;t_0)\hfill \\ \hfill & \\ 1:\mathbf{for}t=t_0\dots T\mathbf{do}\hfill \\ \hfill & \\ 2:\mathbf{for}i=1\dots N\mathbf{do}\hfill \\ \hfill & \\ 3:\tilde{C}({\mathit{x}}_i(t))=\tilde{C}({\mathit{x}}_i(t))+E_i(t)\{\text{Exhale particles}\}\hfill \\ \hfill & \\ 4:\mathbf{end}\mathbf{for}\hfill \\ \hfill & \\ 5:\mathbf{for}\mathbf{all}{\mathit{p}}_k(k=1,\dots ,K)\mathbf{do}\hfill \\ \hfill & \\ 6:\tilde{C}({\mathit{p}}_k)=\lambda \mathcal{D}(\tilde{C}({\mathit{p}}_k))\{\text{Diffuse particles}\}\hfill \\ \hfill & \\ 7:\mathbf{end}\mathbf{for}\hfill \\ \hfill & \\ 8:\mathbf{for}\mathbf{all}{\mathit{p}}_k(k=1,\dots ,K)\mathbf{do}\hfill \\ \hfill & \\ 9:\tilde{C}({\mathit{p}}_k)=\omega \tanh [\mu \tilde{C}({\mathit{p}}_k)]\{\text{Saturate concentration}\}\hfill \\ \hfill & 10:C({\mathit{p}}_k;t+1)=\tilde{C}({\mathit{p}}_k)\{\text{Copy for next iteration}\}\hfill \\ \hfill & 11:\mathbf{end}\mathbf{for}\hfill \\ \hfill & 12:\mathbf{for}i=1\dots N\mathbf{do}\hfill \\ \hfill & 13:I_i(t+1)=\lambda I_i(t)+{\kappa }_i\tilde{C}({\mathit{x}}_i(t))\{\text{Inhale particle}\}\hfill \\ \hfill & 14:R_i(t+1)=\tanh ({\sigma }_i{I}_i(t+1))\{\text{Infection risk by}\hfill \\ \hfill & \\ \text{time}t+1\}\hfill \\ \hfill & 15:{\mathit{x}}_i(t+1)={\mathit{f}}_i({\mathit{x}}_i(t);t)\{\text{Move around}\}\hfill \\ \hfill & 16:\mathbf{end}\mathbf{for}\hfill \\ \hfill & 17:\mathbf{end}\mathbf{for}\hfill \end{array} \end{gathered}$$

• x_i(t+1) = f_i(x_i(t); t): the discrete-time position dynamics of s_i (e.g., a random walk or a Lévy walk), rounded to the nearest environment unit center γ_k(p_k);

• C(p_k; t): concentration of infectious particles at position p_k at time t (0: no particles, 1: saturated with particles);

• $\tilde{C}(\mathbf{x})$: particle concentration function at spatial position x, used as a variable for spatial concentration at an arbitrary time.

With the above definitions, we propose Algorithm 1 to simulate the risk of exposure, as healthy individuals move and breathe between infectious individuals in the environment Γ. Accordingly, the environment has (short-term) memory and acts as an intermediate medium between the infectious and healthy subjects. Throughout this algorithm, nonlinear transforms of the form y = a tanh(x/δ) have been used to model saturation effects between variables x and y, such as the infectious particle concentration. While various functions can be used to model saturation effects, the advantage of the tanh(·) is its simplicity and differentiability, which would allow us to extend this algorithm to be used in machine learning models in the future. In all cases, the saturation level a has the same dimension as the target variable y, δ has the same dimension as the input variable x and controls the saturation point of the functional. In a stochastic framework, taking δ several times greater than the standard deviation (STD) of x guarantees that the nonlinearity only affects the outliers of x. For very large δ (δ ≫ STD(x)), the transformation simplifies to a linear map y ≈ ax/δ.

Additional elements and parameters used in Algorithm 1 are:

• $\mathcal{D}(C({\mathit{p}}_k;\cdot ))$: denotes a spatially smoothed version of C(p_k; ·), due to spatial diffusion of the particles:

  $$\mathcal{D}(C({\mathit{p}}_k;\cdot ))=\sum _{j,{\gamma }_j({\mathit{q}}_j)\in \mathcal{N}({\gamma }_k)}\frac{w_{kj}C({\mathit{q}}_j;\cdot )}{W_k}$$ (2)
  where w_kj is a weight vector as a decaying function of ∥p_k – q_j∥, the distance between the unit γ_k(p_k) and its neighbors, and W_k = ∑_j w_kj. For example, we can take:

  $$w_{kj}=\{\begin{array}{cc}\frac{d{T}_s}{\Vert {\mathit{p}}_k-{\mathit{q}}_j{\Vert }^2}\hfill & j\ne k\hfill \\ 1-\sum _{m\ne k}{w}_{km}\hfill & j=k\hfill \end{array}\phantom{\}}$$ (3)

  where d is the spatial diffusion coefficient and T_s is the simulation sampling time. Note that dT_s should be small enough to guarantee w_kk > 0, which is the stability condition for the finite time-space discretization of the corresponding partial differential equation.

  Equation (3) has been derived by discretizing the two-dimensional heat/diffusion equation:

  $$\frac{\partial U}{\partial t}=d\left(\frac{{\partial }^{2}U}{\partial x^2}+\frac{{\partial }^{2}U}{\partial y^2}\right)$$ (4)

  where the spatial diffusion coefficient d depends on the coronavirus particle radius, mass, ambient temperature and pressure and air viscosity. This value has been noted to be very small for COVID-19 and therefore the virion motion in the horizontal direction occurs due to air convection [26]. According to the literature, the average diffusion coefficients of molecules in the air range from 10 to 130 cm²/s when eddies and air turbulence are considered [27]. We assume our environment does not have significant air currents and assume a diffusion coefficient of 30 cm²/s in our simulations.

• 0 < λ < 1 is the vanishing factor of the particles, due to air change and natural decay¹, which can be formulated as follows:

  $$\lambda =(0.5{)}^{\frac{T_s}{T_h}}{\rho }^{\frac{T_s}{T_e}}$$ (5)

  where T_s is the simulation sampling time, T_h is the half-life period of the particles due to natural decay or becoming ineffective, and ρ is the rate of air change over T_e seconds. The half-life of the virus T_h depends on the UV index, ambient temperature and relative humidity in the environment [28], [29]. It can be calculated using an online available tool provided by the United State’s Department of Homeland Security [30].

• μ: the concentration saturation factor which determines how quickly saturation is reached, at each grid point of the simulated environment.

• ω: Particle concentration correction factor to ensure the dimensions of the concentration term are retained. It has the same dimensions as the concentration term. This value is set to 1 (with appropriate dimension) in all experiments in this work.

• I_i(t): the total amount of infectious particles inhaled by s_i by time t. At the beginning of the algorithm, we initiate I_i(0) to 1 if the individual is infected and 0 if the individual is not infected.

• κ_i: the fraction of floating infectious particles inhaled by s_i at its current position in space;

• R_i(t): the risk of being infected due to inhalation of infectious particles by time t, which we consider as the individualized risk metric defined as follows:

  $$R_i(t)=\tanh ({\sigma }_i{I}_i(t))$$ (6)

  where σ_i is the individualized risk factor, which is a probabilistic term that generally depends on factors such as age, immunity co-morbidities, etc. For the sake of simplicity we consider three major factors: age, immunity and co-morbidities, as independent random variables, which impact the individualized risk [31], [32]. Thus we can take σ_i = ν_iξ_iζ_i as a product of the immunity factor ν_i ∈ [0, 1] (0: immune due to vaccination or anti-body, 1: non-immune), the age risk factor ξ_i ∈ [0, 1] (0: no risk, 1: highest risk), and the background condition risk factor ζ_i ∈ [0, 1] (0: no background illnesses, 1: highly susceptible due to background illness). According to (6), the risk metric depends on the number of infected particles inhaled by the individual and their health status. The risk of infection is a monotonically increasing function of the amount of particles inhaled by a person and whether the virus particles can sustain in the host. The tanh(·) indicates the fact that the risk of infection saturates beyond a certain amount of inhaled particles (controlled by the saturation factor σ_i). Therefore, for a healthy individual, the risk of infection is initially zero. It monotonically increases and saturates to 1, as they inhale in an infected environment, at their individualized risk factors. To calculate the risk factor associated with age, we followed the Centers for Disease Control and Prevention guideline for risk for COVID-19 hospitalization by age group [33]. We normalized the risk of hospitalization values to lie between 0 and 1 and used it as the age risk factor;

• E_i(t): the amount of infectious particles exhaled by s_i at their current position at time t. We propose the following form for this function:

  $$E_i(t)={\beta }_i{\eta }_i{R}_i(t)$$ (7)

  where β_i ∈ [0, 1] is the mask-wearing factor (0: “perfect” mask, 1: unmasked). Asadi et al. studied the efficacy of various mask types for preventing the spread of the coronavirus [34]. They reported that while breathing, 0.31 particles per second are exhaled when no mask is worn. When a surgical mask or KN95 mask is worn, this value drops to 0.06 and 0.07 particles per second, respectively. Based on these studies, we normalized these values with respect to the no mask condition to correlate the mask-wearing factor to types of masks, giving us mask-wearing factors of 1, 0.19 and 0.22 for no mask, surgical mask, and KN95 mask, respectively (assuming that the masks are worn perfectly). According to (7), the amount of exhaled particles is proportional to the risk metric (the chance of being infected), which in turn depends on the amount of inhaled particles. Thus the amount of exhaled particles depends on the number of inhaled particles and the state of the host, but it saturates at a certain point, indicating the fact that an infected individual spreads particles according to their status of infectedness.

• η_i: the infectiousness factor of s_i (0: uninfectious, 1: highly infectious).

Algorithm 1 is used in the sequel to simulate risk of exposure.

C. Simulation

We have developed software to simulate human motion in an arbitrary built environment (here, the EP6 building). The configuration of this software for performing a simulation consists of two phases: 1) User-defined inputs on the built environment and characteristics of people in the simulation, and 2) planning the paths of people in the simulation.

1). User-defined configurations:

The inputs provided by the software user are:

• Floor plan: The floor plan of the building is provided by the user in RGB image format. It is assumed that black pixels represent walls or other obstacles and that the background is white. For non-black and white floor plan images, a threshold is applied to classify each pixel as either obstacle or background. All the 2D positions (x and y coordinates) of obstacle pixels are stored in a list.

• The total number of people required in the simulation is provided by the user as an integer.

• Any number of “start regions” can be set by the user in an interactive way. A graphical window of the floor plan permits the user to select rectangles as start regions. Start regions are defined as areas within the built environment from which a certain number of people start walking towards their destination. The exact start pixel is randomly chosen by the software from the start area.

• Similarly, any number of destination regions can be selected by the user. Destination regions are areas in the environment where a certain number of simulated people walk into, from their start point. The exact destination pixel is chosen randomly by the software from this area. An example of start and destination regions selected by a user is shown in Fig. 1(a)

• Certain areas of the environment can be “closed off” or blocked by the user. This is also done in an interactive way as above. Some obstacles can be removed, such as a removable wall partition. An example of this is shown in Fig. 1(b). This is a practical feature for the EP6 building, in which the building design is such that some internal walls/partitions can be moved.

• Pauses in the motion of individuals along a given path can be added to simulate instances where people stop to talk to others, rest, pause for thought, or hesitate because they unsure of where to go.

• Information about the individuals in the simulation includes the stage of MCI progression; whether or not the individual is immune due to vaccine (either 0 or 1, where 0 indicates immunized and 1 indicates not immunized); age risk factor (a number between 0 and 1, where 1 indicates high risk due to age); background illness risk factor (a number between 0 and 1, where 1 indicates high risk due to underlying illness); the effectiveness of mask (a number between 0 and 1, where 1 is no mask and 0 is perfect mask); and how infectious the individual is (a number between 0 and 1, where 1 indicates highly infectious).

• Information about the environment includes the percentage of air changed in a specific amount of time and the half-life of the virus in the environment.

Fig. 1.

Floor plan of the EP6 building demonstrating (a) the start and destination regions selected by the user. Blue rectangles show the start regions and the yellow rectangles show destination regions; (b) the prohibited areas set by the user (highlighted in red)

D. Path planning

After choosing the start/destination points for each person, a random path between these two points is generated using a modified version of the Rapidly-Exploring Random Trees (RRT) algorithm [35]. RRT is a randomized data structure that helps find a random path between the start and end points, while avoiding obstacles. In our implementation, we allow for any point in the vicinity of the chosen destination point to be considered a destination point. Specifically, any point that is less than or equal to P pixels away from the chosen destination point can be considered a destination point. P is a hyper-parameter. As P is increased, the probability of reaching a destination close to the assigned destination point reduces, but the time required to find a path decreases.

RRT works by building a tree by adding randomly sampled pixel positions from the image. A sampled pixel is added to the closest node of the tree. If an obstacle exists between the sampled point and its closest node, the sample is discarded. Once a point is added to the tree, it is not sampled again. The points on the image are sampled from a uniform distribution.

We have modified the original RRT algorithm to adapt it for the application of interest, as detailed in the sequel.

Research has shown that pedestrians follow a Levy walk in crowded places [36]. Levy walk is when stride lengths are sampled from a Levy distribution.

A Levy walk is positive and left-skewed (cf. Fig. 2). Therefore, most stride lengths are small and some will be longer. The property that makes a Levy walk suitable for modeling pedestrian movements is that in crowded places people tend to take multiple small strides to avoid obstacles and a few long strides as they walk towards their destination. We hypothesize that a similar behavior will be shown by people in a built environment, except that the center of the distribution will shift to the right. Thus the average strides will be longer in a building which is not too crowded when compared with stride lengths in a crowd. On the other hand, the MCI population has the tendency to feel confused and get lost or forget their destination while walking. We hypothesize that this would result in a more random walking pattern with no specific goal. This results in shorter average stride lengths but with less variance in stride lengths. Thus, for the MCI population, we hypothesize that the movement will be a combination of Levy and Brownian motion. In a Brownian motion, the strides are independent identically distributed (as in the Levy walk), but the stride lengths are sampled from a Gaussian distribution.

Fig. 2.

Mixture of Levy and Brownian distributions with μ=0.3.

Most individuals in our simulation scenario are not completely healthy or completely cognitively impaired. We, therefore, introduce a tuning factor μ_mci (between 0 and 1, 0 being a healthy subject and 1 corresponding to severe cognitive impairment) which is used to decide how impaired a person’s cognition level, and thus its walking behavior is. The Montreal Cognitive Assessment (MoCA) [37] is a test used to determine the degree of cognitive impairment of a subject. This test can classify people as having mild cognitive impairment, moderate cognitive impairment, and severe cognitive impairment. To model these impairment levels, herein we define μ_mci from 0 to 0.3 as healthy, 0.4 to 0.7 as mild cognitive impairment, and 0.8 to 1 as moderate cognitive impairment. We do not consider the case of severe cognitive impairment since this population is not present in our studied population. The parameter μ_mci is next used to combine the Levy and Gaussian distributions in a weighted manner to create a mixture distribution, as shown in Fig. 2.

To make our simulation more realistic, we incorporate the above assumption during path planning. As stated above, the classical RRT samples points from a uniform distribution. In our modified RRT algorithm, instead of sampling points from the image directly, we sample path lengths (L) from the detailed mixture distribution. Once a path length L is obtained, we sample all pixels at a distance L from the previous pixel position in a uniform manner. This newly sampled point is added to the RRT tree.

III. Simulation Results

Experiments to stress-test the social distancing and risk metrics are described below. Initially, we used a simple square space with no obstacles to ensure the metrics increase monotonically with the number of people in the space, time spent in the space, and the inverse distance between people. We then simulated multiple scenarios in a real built environment, EP6, in which clinicians and patients will be present in the coming months.

A. Experiment 1

This experiment aimed to show how the social distancing and risk metrics scale with the number of people in a space and the distance between individuals. We simulated a varying number of individuals (N) distributed with equal spacing around a circle of circumference 8 m, walking towards the center of the circle. The step size for each ‘frame’ of the simulation was set to 0.52 m. All individuals reversed trajectory when they were 1 m away from the center of the circle and returned to their original positions. Multiple such simulations were run with different values of N = 4, 10, 30.

Fig. 3 shows the average social distancing metric values for N = 4, 10 and 30 individuals, with respect to the average distance between individuals. Fig. 4 shows the average risk metric values versus the average distance between individuals for N = 4, 10, 30. Accordingly, the social distancing metric increases as people get closer to one another and decreases as they move away from each other. The effective width of the plot is larger for more people, indicating that the risk due to social distancing is higher when there are more people in the environment. The risk metric increases over time since it accounts for residual infectious particles in the environment. It is also seen that both metric values increase as the number of people in the environment increases.

Fig. 3.

Average social distancing metric for N=2, 4, 10 as it varies with average distance between individuals, normalized by N (N – 1) to make the metric at different population densities comparable. The normalization is performed only for visualization purposes, to compare the social distancing metric for different population densities. The normalization term N(N – 1) stems from the fact that the algorithm loops over all combinations of people, i.e., non-identical i and j indexes in (1).

Fig. 4.

Average risk metric for N=2, 4, 10 as it varies with average distance between individuals.

B. Experiment 2

This experiment aimed to evaluate how the social distancing and risk metrics vary when N people remain in the same position for time τ, for example, when they are sitting in a classroom, waiting room, having lunch, or relaxing in a communal area. We set N = 5 for all experiments. These people are placed along the circumference of a circle of radius d, where we vary d. We use d = 1m, 2m, and 6m. The 5 people are at the same position for τ = 60 seconds.

Fig. 5 shows the variation of average risk metric when 5 individuals are positioned on the circumference of circles of radii 1, 2, and 6 meters, for 60 seconds. This experiment shows that the risk metric increases with time and decreases when the distance between people increases.

Fig. 5.

Average risk metric when 5 individuals are positioned on the circumference of a circle of radius = 1m, 2m and 6m.

C. Experiment 3

In this set of experiments, we show how the social distancing and risk metrics change in the EP6 environment with ventilation rate, type of masks worn, number of people in the space, and people’s vaccination status. Just for illustration, in all cases, we consider 20% of the population to be infected at time t = 0.

1). Effect of ventilation rate:

To show how different ventilation rates within a built environment can affect the risk metric, we ran three simulations with air change rates of 20%, 50%, and 80% every 10 minutes (the percentages and time duration for air exchange are real values, which match the HVAC systems in EP6). For all three simulations, 30 people with an average age of 68 years of age were considered. The mask wearing factor, background condition risk factor and infectiousness factor were set to 0.4, 0.7 and 0.7 respectively. All people were considered to have mild cognitive impairment and were not vaccinated. A virus half-life of 9 minutes was assumed.

Fig. 6 shows the variation of average risk metric over time as the 30 people move in the environment, at the three air change rates. Accordingly, the risk metric increases over time as expected. It also increases as the rate of air change decreases, but this increase is not significant. Note that the risk metric values also depend on the initial positions of the infected individuals. If infected individuals are distributed around the environment, the risk metric increases quickly.

Fig. 6.

Average risk metric variation as 30 people move around the EP6 environment when the air change rate is 20%, 50% and 80% every 10 minutes.

2). Effect of mask effectiveness:

This experiment shows how different mask effectiveness changed the values of the risk metric. Once again, 30 people with an average age of 68 years were considered in the simulations. The background condition risk factor and infectiousness factor were set to 0.7. All people were considered to have mild cognitive impairment and were not vaccinated. Three scenarios were simulated with all people wearing no masks (mask-wearing factor of 1), all wearing surgical mask (mask-wearing factor of 0.3) and all wearing near perfect KN95 masks (mask-wearing factor of 0.1). Air change percentage of 50% every 10 minutes and virus half-life of 9 minutes were considered.

Fig. 7 shows the resulting risk metrics. It is seen that the mask-wearing factor has a significant influence on the risk metric. The risk metric reduces to approximately one-fourth when the mask-wearing factor goes from 1 (no masks) to 0.3, and it drops to close to 0 when the mask-wearing factor is 0.1 (i.e., close to perfect mask).

Fig. 7.

Risk metric variation for mask wearing factors of 1 (no mask), 0.3 and 0.1 (near perfect mask) as 30 people move around the EP6 environment.

3). Effect of number of people:

This experiment shows how the social distancing and risk metrics change with a change in the number of people in EP6. Three sub-experiments were run with 5, 15, and 30 people in the environment. All the people in all three simulations had average ages of 68 years and we used mask wearing factor, background condition risk factor and infectiousness factor of 0.4, 0.7 and 0.7 respectively. Air change percentage of 50% every 10 minutes and virus half-life of 9 minutes were considered.

Figs. 8 and 9 show the variation of the average social distancing metric and the average risk metric when the number of people in the simulation is changed, with the above scenario. Accordingly, the risk metric increases by more than twice when the number of people in the environment increases from 5 to 15. The increase is less drastic when the number of people increases from 15 to 30. The social distancing metric, in general, is also higher when there are more people in the environment since there is less space available to move around.

Fig. 8.

Social distancing metric as it varies over time when number of people in the EP6 environment is 5, 15 and 30 people.

Fig. 9.

Risk metric as it varies over time when number of people in the EP6 environment is 5, 15 and 30 people.

4). Effect of vaccination status:

The effect of the percentage of the population vaccinated in the EP6 environment is shown in this experiment. Three sub-experiments were run with all people not vaccinated, 70% of the population vaccinated, and all people vaccinated. The rationale for choosing this percentage is based on research, which has shown that if (approximately) 70% of the population are vaccinated, herd immunity will be reached considering a vaccine efficacy of 95% [38]. Thirty people were considered for each simulation with an average age of 68 years. The mask wearing factor, background condition risk factor and infectiousness factor are set to 0.4, 0.7 and 0.7 respectively. Air change percentage of 50% every 10 minutes and virus half-life of 9 minutes were considered.

Fig. 10 shows the variation in risk metric over time for cases when 0%, 70% and 100% of the population are vaccinated. Vaccination or immunity to the virus significantly decreases the risk metric. When 70% of the population is immune the risk metric drops to close to zero. This is more than 3 times smaller than the case when 0% of the population is vaccinated. As an extreme case, when 100% of the population is immune, the risk metric becomes zero, since the virus cannot infect any individual.

Fig. 10.

Risk metric variation when 0%, 70% and 100% of the population in the EP6 environment are vaccinated.

The heat-maps for all the above cases discussed are shown in Fig. 11. The heat-maps support the results obtained for the risk metrics. It is seen that practicing good masking and immunization of the population significantly decreases the risk. The heat-map for when 100% of the population is vaccinated is white, showing that there is no risk for the individuals.

Fig. 11.

Heat-maps of the risk metrics obtained in experiment 3 when air change percentage, mask wearing factor, number of people, vaccination status are varied.

D. Experiment 4 - Super-spreader events

A super-spreader event is an event in which a large number of people get infected by a small number of infected people. In COVID-19 related studies, super-spreader events in which around 10% of infectious cases caused up to 80% of the transmissions have been considered as one of the most significant factors of the pandemic transmission [39], [40]. The number of coronavirus virions in an infected individual at peak infection is between 10⁹ and 10¹¹ [41]. We consider a person to be infected if the number of virions in their body is greater than or equal to 10⁷ virions.

In this experiment, we found the time taken for an event to become a super-spreader event for all the scenarios of Experiment 3. We sampled random positions from a uniform distribution for 30 people in the EP6 environment and initialized 10% of the population to be infected according to the above definition. We ran the simulation until 80% of the population was infected. These simulations were run multiple times for each scenario and the average time taken for the events to become a super-spreader event was noted. Table I shows the time it takes for a gathering to become a super-spreader event for all the scenarios detailed in Experiment 3.

TABLE I.

Time required for a gathering to become a super-spreader event (SSE)

Experiment heading	Details	Average time to SSE
Ventilation rate	ventilation rate = 20%	17 minutes
	ventilation rate = 50%	18 minutes
	ventilation rate = 80%	18 minutes
Mask wearing	Mask wearing factor=0.1	2 hours
	Mask wearing factor=0.3	17 minutes
	Mask wearing factor=1	7 minutes
Number of people	5 people	25 minutes
	15 people	20 minutes
	30 people	17 minutes
Vaccination	0% people immune	17 minutes
	70% people immune	10 days
	100% people immune	∞

IV. Discussion

A. Impact of number of people

As expected, the results demonstrate that the social distancing metric and the risk metric increase with an increase in the number of people in an environment. They also decrease when the distance between people increases. In all three considered cases (with 5, 15, and 30 people in the EP6 environment), it took 17 to 18 minutes for the event to become a super-spreader event.

B. Impact of dwell time

The social distancing metric does not change when a fixed number of people remain in the same spot for some duration, as the number of people N and the distance between them remains constant. The risk metric, however, increases with time. Even when people start to move away from each other, the risk metric increases due to the presence of residual infectious particles in the environment, which accumulate over time.

C. Impact of ventilation rate

The social distancing metric is not affected by the ventilation rate. It is seen from the results that changing the air change percentage does not significantly impact the risk metric. This can be explained by how the air change rate is used in (3) of Algorithm 1. Keeping the half-life value and simulation sampling time as 9 minutes and 1 second respectively, the value of the vanishing factor λ is approximately 0.996 for an air change percentage of 20%, 0.997 for an air change percentage of 50% and 0.998 for an air change percentage of 80%. In this work, we have assumed relatively non-turbulent air with no significant air currents (which matches the air conditioning system of the studied EP6 building). The air currents generally help to diffuse and clear infectious particle concentration faster. Due to this assumption, in our case, the concentration of infectious particles is affected more by the concentration of people and their distribution in the built environment. Thus the impact of changing the air change percentage is low. Varying the ventilation rate did not alter the time it took to make an event a super-spreader. However, it is anticipated that at high throughput environments, ventilation would have an increased effect.

D. Impact of mask-wearing factor

From the results, it is seen that wearing masks has a very significant impact on reducing the spread of infectious particles. A mask-wearing factor of 1 (no mask) results in a significantly higher risk metric. It has the second biggest impact on reducing the risk metric after vaccination. Wearing a close to perfect mask significantly increased the time it took for an event to turn into a super-spreader.

E. Impact of vaccination

Vaccinating 70% of the population drops the risk metric close to zero. When the entire population is vaccinated, the risk metric drops completely to zero. This factor has the highest impact on reducing the risk metric. When 70% of the population was vaccinated, the simulation showed that it would take 10 days for an event to become a super-spreader, in contrast to 17 minutes, when no one was immune.

In all the risk metric plots, it is seen that there is a higher rate of increase towards the last few time steps. This corresponds to the fact that towards the end of the simulation period, people reach their destination and group together in rooms. This results in smaller distances between people and thus higher risk metric values. This is visually represented by the red areas of the heat-map in Fig. 11.

V. Conclusion and Future Work

In this work, an open source model was implemented in Python, which quantifies social distancing and exposure risks in built environments. The model is intended to assist in the planning for reopening indoor facilities, and track risks as the facilities serve an increasing number of individuals. A novel social distancing metric is introduced, which quantifies how well social distancing is being practiced, together with a novel risk metric and mathematical model which quantifies the risk of being exposed to the virus in a given indoor space. An algorithm to estimate the risk of infection from an airborne infection was also introduced. The algorithm provides a simple mechanism for exchanging infections between individuals in a built environment while considering factors such as diffusion of infectious particles, ventilation rate in the built environment, overall risk to an individual based on age and comorbidities. It provides a risk metric that can be used to quantify the risk to an individual.

The results highlight the importance of mask-wearing and vaccination. These two factors have the highest impact in reducing the risk of infection and slowing down or ending the pandemic

This work considered uniform ventilation and airflow in the built environment. However, in practice, this is not the case. The rates of airflow and ventilation depend on the positions of vents, the presence of open windows, and human movement through the space. This fact should be considered in more complex models. The model described in this work can thus be made more realistic by considering factors such as airflow patterns, air convection, location of air inflow and outflow vents, air movement through windows and doors, and air movement patterns caused due to movement of humans. In addition, vertical gradients in airflow can also be considered. Adding these elements would contribute to a more accurate model. By developing an accurate method to model airflow patterns, not only in a built environment but also in outdoor or hybrid settings, this model can be generalized to other environments such as elevators, public transport, and outdoor spaces.

Further, the simulation of people’s movement in indoor spaces can be modeled more accurately by empirically estimating the distribution of path lengths for a healthy population and a population with MCI. We aim to undertake this task in the future by using cameras and Bluetooth sensors to track the trajectories and path length distributions of a healthy versus MCI population. We also aim to design a more complete model for movement of individuals using this empirical data.

The current method of validation of the model relies on simulated data. In the future we plan to conduct controlled experiments where a number of people will remain in the built environment for a certain duration of time and will move in the environment under predefined motion patterns to further validate our model and to calibrate its parameters.

The quantitative results reported in this research can generally vary based on the path and positions of individuals and the position of infected individuals, which are chosen randomly. This motivates the use of Monte Carlo simulations with the developed software in a per building/scenario basis, in order to arrive at a quantitatively reliable conclusion for planning purposes.

Software to visualize these metrics in the spaces within a built environment is also provided online, so that a user may visualize zones in a building that may pose a higher risk of infection to its occupants. In order to provide tools for others to build upon this work, the source code is provided under an open-source license [42].

Herein, we developed a model for contagious disease spread within built environments, which can be used for calculated-risk reopening of public buildings after the pandemic. Nevertheless, models— no matter how accurate— are only ‘slices of reality’ and oversimplifications of the system under study. Therefore, the hereby developed model is expected to be extended and improved in the future, as it is applied to real data.