key: cord-0658397-lr2uac08
authors: Zhang, Yi; Tao, Yudong; Shyu, Mei-Ling; Perry, Lynn K.; Warde, Prem R.; Messinger, Daniel S.; Song, Chaoming
title: Simulating COVID19 Transmission From Observed Movement: An Agent-Based Model of Classroom Dispersion
date: 2021-08-17
journal: nan
DOI: nan
sha: 4e589ce3c668b339659c1d1b08bc891559fa851c
doc_id: 658397
cord_uid: lr2uac08

Current models of COVID-19 transmission predict infection from reported or assumed interactions. Here we leverage high-resolution observations of interaction to simulate infectious processes. Ultra-Wide Radio Frequency Identification (RFID) systems were employed to track the real-time physical movements and directional orientation of children and their teachers in 4 preschool classes over a total of 34 observations. An agent-based transmission model combined observed interaction patterns (individual distance and orientation) with CDC-published risk guidelines to estimate the transmission impact of an infected patient zero attending class on the proportion of overall infections, the average transmission rate, and the time lag to the appearance of symptomatic individuals. These metrics highlighted the prophylactic role of decreased classroom density and teacher vaccinations. Reduction of classroom density to half capacity was associated with an 18.2% drop in overall infection proportion while teacher vaccination receipt was associated with a 25.3%drop. Simulation results of classroom transmission dynamics may inform public policy in the face of COVID-19 and similar infectious threats.

On March 11, 2020, the World Health Organization declared COVID-19 a pandemic and called for coordinated mechanisms to support preparedness and response efforts across health sectors [1] [2] [3] [4] . Predictive models can effectively inform policy to coordinate social policy responses to infectious pandemics [5] . The CDC has embraced the use of mitigation strategies in schools to allow communities to keep preschools and K-12 schools open [6, 7] . Here we focus on simulating infection responses to the novel coronavirus (SARS-CoV-2, which is responsible for COVID-19) based on observed interactions in a classroom setting. To our knowledge, this is the first study to apply Susceptible, Exposure, Infected, Recovered (SEIR) to physically-defined interactions in these contexts. Results may inform policy decisions related to vaccination, school attendance, and school closure and reopening. SEIR models are used to quantify the spread of epidemics at a societal level [8, 9] . SEIR models of COVID-19 quantify growth rate with respect to sample and population characteristics that may change with time [10] . Given their population focus, these models do not characterize individuals as transmission agents, and do not model their activity over time in a physical space inhabited by other agents. As SEIR models do not model granular trans-mission processes, they may be limited in the degree to which they speak to policy initiatives involving the repeated activity of individuals in physical space, such as schools.

While movement data collected by GPS [11, 12] , mobile phone records [13] [14] [15] [16] , and public transportation [17] has been used to investigate the transmission of SARS-CoV-2 and other viruses, the spatial resolution of those sensors is largely limited to ranges of meters to kilometers, which are unsuitable to the study of classroom and other enclosed space transmission. Two recent reports demonstrate the potential of more granular analysis. One analysis modeled reported infection outcomes based on estimates of exposure duration and the physical density of individuals derived from published outbreak data [18] . This report contributed a model of infection saturation in small, relatively defined samples, but did not quantify transmission dynamics between individuals in physical space. Likewise, a recent model of transmission at a conference event quantified the importance of contact duration during which a specific pair of individuals were within an a priori (badge-based) physical range [19] . Similar techniques have been used to study classroom transmissions [20] . In contrast to the current effort, physical distance was not measured continuously nor was the relative physical orientation of individuals modeled.

Data from 2020 indicate that young children may acquire COVID-19 (including confirmed, asymptomatic cases) in childcare settings and subsequently infect their household members [21] . Moreover, contact tracing revealed high levels of transmission between individuals of similar ages, including among children younger than four years of age [22] . Thus there is evidence that young children may serve as vectors for COVID-19, Badge-based technology has been used to explore the temporal dynamics of transmission among highschool students [20] , but it is not clear how physical motion facilitates infection dynamics in classrooms. The current study capitalizes on automated observations of pre-COVID-19

classroom physical interaction to model transmission between children and their teachers.

It quantifies the dynamics of potential classroom outbreaks to help support efforts to mitigate outbreaks. To do so, we simulate the consequences of both half-capacity classrooms and teacher vaccinations on infection outcomes. These outcomes include infection saturation (percent and number of infected individuals) over time, time to the emergence of the first, second, and third symptomatic individual, and infection saturation levels when these symptomatic individuals emerge (a potential trigger of school closure). 

Simulations were based on continuous real-time location tracking of individuals (students and teachers). A total of 34 observation periods occurred for four preschool classes housed in three classrooms in a large urban center in the United States. Observations were conducted in inclusion classrooms for typically developing children and children with developmental disabilities contained in both a standard public county-funded school (Classes C and D) and a county-funded university preschool (Classes A and B). The range of represented class sizes (10-17) and teacher-child ratio are typical of preschool classrooms, as state regulations require preschool classrooms to have fewer than 20 children and teacher:child ratios of at least 10:1 [23] . Ultra-Wide Radio Frequency Identification (RFID) technology such as that embedded in child-worn Ubisense tags Fig. 1 Table 1 .

The study was conducted in accordance with APA ethical standards and was approved by the University of Miami Institutional Review Board (#20160509). All teachers gave their informed consent and received $100 for their participation. Parents gave their informed consent on behalf of their children and received $75 for their participation. Teachers also received a classroom gift of their choice (e.g., a new book shelf). All research was performed in accordance with relevant guidelines and regulations.

We modelled the spread of SARS-CoV-2 in the classrooms using a novel modification of the SEIR model. The typical SEIR model consists of susceptible, exposed, infectious, and recovered individuals where the susceptible individual becomes infectious after close contact with an infectious individual, with infection rate β, while infectious individuals recover at rate γ. For each individual i, the real-time classroom movement data records the position (x i , y i ) and the orientation θ i for every second. To integrate the simplest SEIR model with the movement data, we develop a model to account for the dependence of the infection rate

with σ r ≈ 2m, and σ θ ≈ 45 0 where β max is the maximum pairwise infection rate. That is, the rate of infection falls directly as an exponential function of growth in the square of the radius between the two individuals and their angular distance from face-to-face contact [25] .

A visual illustration of the distance r ij and orientations θ i and θ j is shown in Fig. 2 (a), and the heatmap and the 3D plot of (r ij , θ i , θ j ) with x = r ij cosθ i , y = r ij sinθ j and θ i = 0 are presented in Fig. 2 (b) and (c), respectively. A detailed discussion of the infection function is found in SM Section 4.C and 4.D. The e −λt term accounts for temporal decay, modeling declining transmission over time t which characterizes the decay rate. For airborne transmission, λ = 0.34/hours [26] , whereas droplet transmission decay is functionally instantaneous over a timescale of seconds (see SM Section 4.A for more details).

To calibrate β max of the agent-based SEIR model, we calculate the average infection ratē

Note that the population infection rate ρ 0 is proportional to the density ρ ≡ N A whereas β max is an intrinsic parameter, independent of the social environment. Researchers have estimated the average daily infection rate at the meta-population level dailyβ daily ≈ β 0 , where β 0 = R 0 γ with R 0 and γ being the reproduction number and the recovery rate , respectively. Combining Eq. (2), we calibrate our modeling parameter

The definition of orientation for each pair of children, r ij is the distance between two children, and the orientation for a child i is defined by angle θ i , which is the angle between the direction being faced (red arrow) and the direct line between persons (blue line). A similar concept is applied to child j to obtain θ j . (b) The heatmap and (c) the 3D plot of β(r ij , θ i , θ j ) with x = r ij cos θ j , y = r ij sin θ j and θ i = 0 where child i is positioned at the origin and facing graph right. (d) The transition from infection to becoming infectious was 1 day; the incubation period (from infection to symptomaticity) had a mean of 4 days; the mean recovery time (from infection to infectious to recovered, i.e., non-infectious) had a mean of 10 days.

we posit that the reported R 0 s variations in these works are largely due to virus evolution/mutation, as well as social, political, and environmental between each cohort [27] [28] [29] [30] [31] [32] .

We chose a conservative lower bound of R 0 = 2.0 for our numeric simulations based on reported R 0 values, which typically fall within the 2.0 − 3.0 range [28] [29] [30] [31] [32] [33] . CDC guidelines have suggested defining close contact as being within r = 6 feet of an infected person for a cumulative total of T = 15 minutes or more over a 24-hour period [34] . We thus estimate the daily population density ρ daily = Nc hours after being infected [36] . Once infected, individuals have a 75% probability of becoming symptomatic [37] ; the transition to symptomaticity followed a poisson process with a mean of four days [28, 38, 39] . Infected individuals become non-infectious with a probability γ∆t where γ = 1/10 days −1 at each time step, reflecting a mean 10-day duration [40] . We Likewise, the time to subsequent symptomatic cases would indicate the cost of ignoring a first infection.

To investigate the spread of SARS-CoV-2, we plot the proportion (and number) of infected individuals over one month in Fig. 3 for both full/half classes and non-vaccinated and teacher-vaccinated scenarios. The mean level (red line) and standard deviation of infection (grey area) over simulation runs is presented. The classroom population density, ρ, the number of individuals per square meter of classroom space is calculated for each observation (see SM Section 2 for more details). Figure 3 shows lower infection levels over time in half-sized than full size simulations. Likewise, the teacher vaccinated scenarios yield lower infection levels over time than the not vaccinated scenarios. In addition, Figure 3 suggests an association between the classroom density ρ and infection levels over time. Specifically, scenarios with higher densities produced higher proportions of infected individuals. These findings suggest that classroom density plays an important role in controlling SARS-CoV-2 spread.

To quantitatively investigate the impact of classroom density, we characterize the infection patterns of the numerical simulations using the methodology of Tupper et al [18] .

In the current model, the time-dependent infection number is determined directly by the agent-based simulation (see Fig. 3 Table 1 ). The area of each circle is proportional to the specific classroom density for a specific observation day, and the color represents Classes A-D for the full and half-sized simulations, respectively.

A mixed effects regression model, with observations nested in classes, predicted saturation and transmission likelihood from class size and vaccination (see Fig. 4 ). Half- While the simulation uses the same infection parameters for each individual, we observe transmission heterogeneity due to behavioral differences. For example, different patient zeros lead to different infection patterns based on individual differences in contact with others (i.e., variation in r 1 &θ 1 ) (see SM Section 4.G and Fig. S7 for more details).

To explore the criteria under which in-person schooling might be terminated (school shutdown) after discovery of COVID-19 cases, we investigated the timing of the emergence of infected, symptomatic individuals (children or teachers). We first focused on the probability of a given set of simulations yielding a first, second, or third symptomatic individuals ( Fig. 5 and Table S3 ). The 75% symptomaticity rate used in simulations implies that the first symptomatic case is patient zero in three of four simulations. The emergence of the 1 st , 2 nd , and 3 rd symptomatic case was significantly reduced in the half class scenarios, reflecting sensitivity to classroom density (see Tables S3 and S4 ). The probability of not detecting a second symptomatic individual, for example, was 35.2% and 53.0% for full and halfsized classrooms, respectively. The corresponding probability was 38.7% and 49.5% for not vaccinated and teacher vaccinated scenarios, respectively. The reduction associated with teacher vaccination significantly impacted the emergence of a 2 nd , and 3 rd (but not a 1 st ) symptomatic individual (see SM Section 3 and Table S4 for details).

Next we plotted the time in days until the emergence of the first, second, and third symptomatic individual in Fig. 6 . The 75% symptomaticity rate again implied that the a second symptomatic individual never emerged (see Table S5 ).

To explore the impact of classroom behavior, we categorized classroom time as unstructured (free-play and transitions between activities) or as structured. Structured activities were teacher-led and primarily occurred when children were seated at tables (such as circletime, shared book reading, meal-time, and organized play). Simulations suggest that unstructured time was associated with a higher trajectory of infection than unstructured time,

presumably because individuals were in closer proximity and were more mutually oriented during these periods (see SM Section 4.B for more details). We also find that transmission heterogeneity is naturally encoded in our model since individuals behave differently and their social interactions are inhomogeneous (SM Section 4.G). 

In the face of pandemic threats such as that posed by COVID-19, federal and local governments must adopt policies that weigh the importance of educational opportunities against the risks of infection within classroom settings. School closure reduces transmission risk [45] , but also limits the employment capacity of health-care and other essential employees who are tasked with child care [46] , which reduces economic productivity [47] . Here we explore the impact of less drastic prophylactic measures such as reducing classroom enrollment and teacher vaccination.

Our goal was to describe the development of an agent-based SEIR model to inform decisions and policy on curbing the impact of COVID-19 classroom spread. Existing research examines infection as a product of expected rates of interaction between infected and susceptible individuals, but does not examine the role of individual (agents) in transmission.

With few exceptions [19] , existing models do not describe the actual physical interactions through which individuals infect others. Transmission, however, occurs in physical space over time.

Since their original formulation [48] , SEIR models have been used to model infectious transmission at a population level [8, 9] . The current study's RFID system provided subsecond (4Hz) measures of the physical distance and relative orientation of all individuals in a classroom [19] . Using the resulting data we employed an agent-based model to examine infection in real-time in a real-life system, the preschool classroom. The model uses actual student and teacher interaction behavior as input and estimates infection probabilities based on interpersonal distance and orientation.

The current policy-focused modeling scenarios indicated that classroom density was a key parameter limiting SARS-CoV-2 transmission. Both simulations that reduced classroom enrollment and simulations of teacher vaccination receipt effectively reduced classroom density by decreasing the number of susceptible individuals (see Fig. 4 ). The half class scenario reduced overall infection proportion by 18.2% while teacher vaccination was associated with a 25.3% reduction. In half class scenarios, the proportion of simulations in which a second symptomatic individual never emerged rose from approximately one third to one half of cases (35.2% to 53.0%) while teacher vaccination was associated with a more modest rise (38.7% to 49.5%). The timing of the emergence of these first, second, and third symptomatic cases -canaries in the coal mine -was also delayed in both the half class and the teacher vaccination scenarios.

A limitation of the current modeling approach is the need to make assumptions about the physical and temporal parameters characterizing transmission. Modeling parameters assumed that the time lag to infectiousness was one day and the lag to symptomaticity was four days [34] . Likewise, the precise physical parameters involved in SARS-CoV-2 transmission are not known [49] . The current models, however, allow for flexible modeling of the physical and temporal parameters associated with pathogen spread in enclosed spaces over time. Note, for example, that CDC changes in safe distance recommendation from 6 to 3 feet do not change key model parameters [27, 50] . Differential infectiousness of SARS-CoV-2 variants[51] (e.g., the Delta variant, B.1.617.2), changes in vaccine effectiveness and avail-ability, and mask mandates will all affect infection levels, as will community positivity. We have explored the impact of parameters to the transmission patterns in SM Section 4. That key contribution of the models is their flexibility. Indeed, as our model has been purposely designed to be modular and therefore used as a general framework, it allows inputs to change to reflect a multitude of future scenarios that include different pathogens such as COVID variants and new viral pathogens, and different transmission vectors including airborne and aerosol particles, as well as variation in class room size/density, and vaccination status.

The datasets generated during and/or analysed during the current study are available in the OSF repository, https://osf.io/h7ks8/?view_only=5b03a51e79ff4c57b59c2a814d60dbd3.

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