Population structure
There are N households in the population, and a single educational institution (either a school or a school, dependent on scenarios to be introduced later) with M rooms and a maximum capacity dependent on the scenario being tested. Effective contacts between individuals occur within each household, as well as rooms and common areas (entrances, bathrooms, hallways, etc.) of the institution. All groups of individuals (households and rooms) in the model are assumed to be well-mixed.
Each individual (agent) in the model is assigned an age, household, room in the childcare facility and an epidemiological status. Age is categorical, so that every individual is either considered a child (C) or an adult (A). Epidemiological status is divided into stages in the progression of the disease; agents can either be susceptible (S), exposed to the disease (E), presymptomatic (an initial asymptomatic infections period P), symptomatically infected (I), asymptomatically infected (A) or removed/recovered (R), as shown in Fig. 1b.
In the model, some children in the population are enrolled as students in the institution and assigned a classroom based on assumed scenarios of classroom occupancy while some adults are assigned educator/caretaker roles in these classroom (again dependent on the occupancy scenario being tested). Assignments are made such that there is only one educator per household and that children do not attend the same institution as a educator in the household (if there is one), and vice versa.
Interaction and disease progression
The basic unit of time of the model is a single day, over which each attendee (of the institution) spends time at both home and at the institution. The first interactions of each day are established within each household, where all members of the household interact with each other. An asymptomatically infectious individual of age i will transmit the disease to a susceptible housemate with the age j with probability (beta ^H_{i,j}), while symptomatically infectious members will self-isolate (not interact with housemates) for a period of 14 days.
The second set of interpersonal interactions occur within the institution. Individuals (both students and educators) in each room interact with each other, where an infectious individual of age i transmits the disease to some susceptible individual of age j with probability (beta ^C_{i,j}). To signify common areas within the building (such as hallways, bathrooms and entrances), each individual will then interact with every other individual in the institution. There, an infectious individual of age j will infect a susceptible individual of age i with probability (beta ^O_{i,j}).
To simulate community transmission (for example, public transport, coffee shops and other sources of infection not explicitly modelled here), each susceptible attendee is infected with probability (lambda _S). Susceptible individuals not attending the institution in some capacity are infected at rate (lambda _N), where (lambda _N>lambda _S) to compensate for those consistent effective interactions outside of the institution that are neglected by the model (such as workplace interactions among essential workers and members of the public).
Figure 1b shows the progression of the illness experienced by each individual in the model. In each day, susceptible (S) individuals exposed to the disease via community spread or interaction with infectious individuals (those with disease statuses P, A and I) become exposed (E), while previously exposed agents become presymptomatic (P) with probability (delta). Presymptomatic agents develop an infection in each day with probability (delta), where they can either become symptomatically infected (I) with probability (eta) or asymptomatically infected (A) with probability (1-eta).
The capacity of the sole educational institution in the model is divided evenly between 5 rooms, with class size and student-educator ratio governed by one of three basic scenarios: seven students and three educators per room (7 : 3), eight students and two educators per room (8 : 2), and fifteen students and two educators per room (15 : 2). Classroom assignments for children can be either randomized or grouped by household (siblings are put in the same class).
Symptomatically infected agents (I) are removed from the simulation after 1 day (status R) with probability (gamma _I), upon which they self-isolate for 14 days, and therefore no longer pose a risk to susceptible individuals. Asymptomatically infected agents (A) remain infectious but are presumed able to maintain regular effective contact with other individuals in the population due to their lack of noticeable symptoms; they recover during this period (status R) with probability (gamma _A). Disease statuses are updated at the end of each day, after which the cycles of interaction and infection reoccur the next day.
The actions of symptomatic (status I) agents depend on age and role. Individuals that become symptomatic maintain a regular schedule for 1 day following initial infection (including effective interaction within the institution, if attending), after which they serve a mandatory 14-day isolation period at home during which they interaction with no one (including other members of their household). On the second day after the individual’s development of symptoms, their infection is considered a disease outbreak centred in their assigned room, triggering the closure of that room for 14 days. All individuals assigned to that room are sent home, where they self-isolate for 14 days due to presumed exposure to the disease. Symptomatically infected children are not replaced, and simply return to their assigned classroom upon recovery. At the time of classroom reopening, any symptomatic educator is replaced by a substitute for the duration of their recovery, upon which they reprise their previous role in the institution; the selection of a substitute is made under previous constraints on educator selection (one educator per household. with no one chosen from households hosting any children currently enrolled in the institution).
Parameterisation
The parameter values are given in Supplementary Table S4. The sizes of households in the simulation was determined from 2016 Statistics Canada census data on the distribution of family sizes42. We note that Statistics Canada data only report family sizes of 1, 2 or 3 children: the relative proportions for 3+ children were obtained by assuming that (65 %) of families of 3+ children had 3 children, (25%) had 4 children, (10%) had 5 children, and none had more than 5 children. Each educator was assumed to be a member of a household that did not have children attending the school. Again using census data, we assumed that (36%) of educators live in homes with no children, where an individual lives alone with probability 0.282, while households hosting 3, 4, 5, 6, and seven adults occur with probability 0.345, 0.152, 0.138, 0.055, 0.021 and 0.009 respectively. Others live with (ge 1) children in households following the size and composition distribution depending on the number of adults in the household. For single-parent households, a household with a single child occurs with probability 0.169, and households with 2, 3, 4 and 5 children occur with probabilities 0.079, 0.019, 0.007 and 0.003 respectively. With two-parent households, those probabilities become 0.284, 0.307, 0.086, 0.033 and 0.012.
The age-specific transmission rates in households are given by the matrix:
$$begin{aligned} begin{bmatrix} beta ^H_{1,1} &{} beta ^H_{1,2} beta ^H_{2,1} &{} beta ^H_{2,2} end{bmatrix} equiv beta ^H begin{bmatrix} c^H_{1,1} &{} c^H_{1,2} c^H_{2,1} &{} c^H_{2,2} end{bmatrix}, end{aligned}$$
(1)
where (c^H_{i,j}) gives the number of contacts per day reported between individuals of ages i and j estimated from data28 and the baseline transmission rate (beta ^H) is calibrated. To estimate (c^H_{i,j}) from the data in Ref.28, we used the non-physical contacts of age class 0–9 years and 25–44 years of age with themselves and one another in Canadian households. Based on a meta-analysis, the secondary attack rate of SARS-CoV-2 appears to be approximately (15 %) on average in both Asian and Western households43. Hence, we calibrated (beta ^H) such that a given susceptible person had a (15 %) chance of being infected by a single infected person in their own household over the duration of their infection averaged across all scenarios tested. As such, age specific transmission is given by the matrix
$$begin{aligned} beta ^Hcdot begin{bmatrix} 0.5378 &{} 0.3916 0.3632 &{} 0.3335 end{bmatrix}. end{aligned}$$
(2)
To determine (lambda _S) we used case notification data from Ontario during lockdown, when schools, workplaces, and schools were closed44. During this period, Ontario reported approximately 200 cases per day. The Ontario population size is 14.6 million, so this corresponds to a daily infection probability of (1.37 times 10^{-5}) per person. However, cases are under-ascertained by a significant factor in many countries. We assumed an under-ascertainment factor of 8.45 based on an empirical estimate of under-reporting45, meaning there are actually 8.45 times more cases than reported in Ontario, giving rise to (lambda _S = 1.16 times 10^{-4}) per day; (lambda _N) was set to (2cdot lambda _S). We emphasize that this number may fall later in the pandemic as testing capacity increases, although some individuals may still never get tested–especially schoolchildren, who are often asymptomatic.
The age-specific transmission rates in the school rooms is given by the matrix
$$begin{aligned} begin{bmatrix} beta ^C_{1,1} &{} beta ^C_{1,2} beta ^C_{2,1} &{} beta ^C_{2,2} end{bmatrix} equiv beta ^C begin{bmatrix} c^C_{1,1} &{} c^C_{1,2} c^C_{2,1} &{} c^C_{2,2} end{bmatrix} equiv beta ^C begin{bmatrix} 1.2356 &{} 0.0588 0.1176 &{} 0.0451 end{bmatrix}, end{aligned}$$
(3)
where (c^C_{i,j}) is the number of contacts per day reported between age i and j estimated from data28. To estimate (c^C_{i,j}) from the data in Ref.28, we used the non-physical contacts of age class 0–9 years and 20–54 years of age, with themselves and one another, in Canadian schools. Epidemiological data on secondary attack rates in educational institutions are rare, since childcare centres and schools were closed early in the outbreak in most areas. We note that contacts in families are qualitatively similar in nature and duration to contacts in schools with small group sizes, although these contacts are generally more dispersed among the larger groups in rooms than among the smaller groups in households. On the other hand, rooms may represent equally favourable conditions for aerosol transmission, as opposed to close contact. Hence, we assumed that (beta ^C = alpha _C beta ^H), with a baseline value of (alpha _C = 0.75) based on more dispersed contacts expected in the larger room group, although we varied this assumption in sensitivity analysis.
To determine (beta ^O) we assumed that (beta ^O = alpha _O beta ^C) where (alpha _O ll 1) to account for the fact that students spend less time in common areas than in their rooms. To estimate (alpha _O), we note that (beta ^O) is the probability that a given infected person transmits the infection to a given susceptible person. If students and staff have a probability p per hour of visiting a common area, then their chance of meeting a given other student/staff in the same area in that area is (p^2). We assumed that (p=0.05) and thus (alpha _O = 0.0025). The age-specific contact matrix for (beta ^O) was the same as that used for (beta ^C) (Eq. 3).
Model initialization
Upon population generation, each agent is initially susceptible (S). Individuals are assigned to households as described in the “Parameterisation” section, and children are assigned to rooms either randomly or by household. We assume that parents in households with more than one child will decide to enroll their children in the same institution for convenience with probability (xi =80%), so that each additional child in multi-child households will have probability (1-xi) of not being assigned to the institution being modelled.
Households hosting educators are generated separately. As in the “Parameterisation” section, we assume that (36%) of educators live in adult-only houses, while the other educators live in houses with children, both household sizes following the distributions outlined in the “Parameterisation” section. The number of educator households is twice that required to fully supply the school due to the replacement process for symptomatic educators outlined in the “Disease Progression” section.
Initially, a proportion of all susceptible agents (R_{init}) is marked as removed/recovered (R) to account for immunity caused by previous infection moving through the population. A single randomly chosen school attendee is chosen as a primary case and is made presymptomatic (P) to introduce a source of infection to the model. All simulations are run until there are no more potentially infectious (E, P, I, A) individuals left in the population and the institution is at full capacity. All results were averaged over 2000 trials.
Estimating β
H
Agents in the simulation were divided into two classes: “children” (ages 0–9) and “adults” (ages 25–44). Available data on contact rates28 was stratified into age categories of width 5 years starting at age 0 (0–5, 5–9, 10–14, etc.). The mean number of contacts per day (c_{i,j}^H) for each class we considered (shown in Eq. 2) was estimated by taking the mean of the contact rates of all age classes fitting within our presumed age ranges for children and adults.
For (beta ^H) calibration, we created populations by generating a sufficient number of households to fill the institution in each of the three tested scenarios; 15 : 2, 8 : 2 and 7 : 3. In each household, a single randomly chosen individual was infected (each member with equal probability) by assigning them a presymptomatic disease status P; all other members were marked as susceptible (disease status S). In each day of the simulation, each member of each household was allowed to interact with the infected member, becoming exposed to the disease with probability given in Eq. 2. Upon exposure, they were assigned disease status E. At the beginning of each subsequent day, presymptomatic individuals proceeded to infected statuses I and A, and infected agents were allowed to recover as dictated by Fig. 1b and Supplementary Table S4. This cycle of interaction and recovery within each household was allowed to continue until all infected individuals were recovered from illness.
We did not allow exposed agents (status E) to progress to an infectious stage (I or A) since we were interested in finding out how many infections within the household would result from a single infected household member, as opposed to added secondary infections in later days. At the end of each trial, the specific probability of infection ((pi _n)) in each household (H_n) was calculated by dividing the number of exposed agents in the household ((E_n)) by the size of the household (|H_n|) less 1 (accounting for the member initially infected). Single occupant households ((|H_n|=1)) were excluded from the calculation. The total probability of infection (pi) was then taken as the mean of all (pi _n), so that
$$begin{aligned} pi =frac{1}{D}sum _{n}pi _n=frac{1}{D}sum _{|H_n|ge 2}frac{E_n}{|H_n|-1}, end{aligned}$$
(4)
where D represents the total number of multiple occupancy households in the simulation. This modified disease simulation was run for 2000 trials each of different prospective values of (beta ^H) ranging from 0 to 0.21. The means of all corresponding final estimates of the infection rate were taken per value of (beta ^H), and the value corresponding to a infection rate of (15%) was interpolated.
Simplifying assumptions
Our model makes simplifying assumptions that may influence its predictions. For instance, we assume that classrooms are homogeneously mixing and did not take social structure into account. Social structure might slow the spread of COVID-19 in classrooms. We also assumed that public health authorities will respond to a confirmed case by closing the classroom, although in practice, they may keep the class running if they think the case does not represent an infection risk to children or adults. This would reduce the number of student-days lost to closure. Similarly, we did not account for potential contacts between school children outside of classes, although students of a classroom that has been closed may still interact with their classmates outside of school. Other simplifying assumptions are mentioned in the “Discussion” section.
Source: Ecology - nature.com
