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About the Model

This model was developed is model was developed in collaboration with Carl Bergstrom and Ryan McGee from the University of Washington. It can be used to simulate infection dynamics of SARS-CoV-2 and evaluate the impact different mitigation strategies have on outbreaks in primary and secondary schools. This model is used in our pre-print “Model-driven mitigation measures for reopening schools during the COVID-19 pandemic”.

On August 16th, 2021, Version 2 of the modeling tool was released.

For a detailed description of the model and methods, please refer to our pre-print and supplementary material:

SEIRS+ Model

SEIR models are epidemiological models that are used to model the spread of disease in a population. Standard SEIR models are compartmental models, meaning they track the proportion of the population in different disease states over time. SEIR models include compartments for susceptible (S), exposed (E), infectious (I), and recovered (R) disease states.

The SEIRS+ model is an extended SEIR model, which incorporates the effects of stochastic dynamics, network structure, SARS-CoV-2 testing, and additional interventions in a population.

Further information and code for the SEIRS+ model framework can be found at: https://github.com/ryansmcgee/seirsplus

Where can I find information about earlier version
of the model and webapp?

Released August 16th, 2021

Model Version 2

Version 2 of our modeling tool is described in more detail in our publication and the modeling tool can be found here.

Released February 25th, 2022

Model Version 3

Version 3 of our modeling tool is described in more detail in our publication and the modeling tool can be found here.

Primary versus secondary schools

The dynamics of SARS-CoV-2 transmission differ substantially between primary schools and secondary schools for three principal reasons: (1) children (under age 10) appear to have different epidemiological characteristics from adolescents (ages 10-19), (2) primary and secondary schools have different organizational structures, and (3) vaccines were authorized for these age groups at different times, so that as of March 2022, most adolescents are eligible for boosters whereas children are not.

Primary schools are often structured into more stable cohorts, with groups of students assigned to a single teacher for their entire day. In contrast, secondary school students typically move from classroom to classroom and thus encounter multiple teachers and groups of students over the course of a single day.

For these reasons, we have developed two distinct models for primary versus secondary schools, each with parameters chosen to reflect these critical differences

Model network structure

The contact structures of schools differ from other settings. For our primary school model, we simulate a medium-sized school of 480 students with 24 teachers and 24 additional staff (Table 2). Each class comprises one teacher and 20 students that interact with one another. Additionally, each teacher interacts with a handful of other teachers and staff, and students that share the same household (as calibrated by United States (US) census data) are connected. Most of the contacts that an individual makes in the school population are with the students and teacher in their own class, and disease transmission within a class is more likely than between classes.


For our secondary school model, we consider a medium-sized school with 800 students (200 per graduating class), 125 teachers, and 75 staff (Table 3). We define network layers for students, teachers, and staff using the FARZ network generation algorithm, which allows us to calibrate epidemiologically-important network properties (e.g., community structure, assortativity, clustering coefficient) to values consistent with studies of secondary school contact networks. A FARZ community network layer is generated for each grade, with students belonging to one or more communities of about 10 individuals. 80% of each student’s contacts are with students in the same grade, and 80% of those within-grade contacts are with students in their own communities. Students that share a household (as calibrated by US census data) are connected as well. Interactions between teachers and staff are represented by another FARZ network layer with a total of six communities. Finally, students are connected with six random teachers with whom they have classes, with students in the same grade being more likely to share teachers.

Model parameters

The parameter values and descriptions used in the model are listed in the table below.

Table 1. Table of parameters included in model

Parameter

Mean Value

Description

R0

1.5

The R0, or reproductive number, is the expected average number of secondary infectious cases produced by a single infectious case. This level of baseline transmissibility (R0=3.5) assumes that basic mitigation strategies, such as mask-wearing and social distancing, are in place.

Student Susceptibility

60% for primary school students, 100% for secondary school students

Children 10 and younger are less susceptible to infection than older children and adults.

Latent period

3.0 days

The time from exposure to when the individual becomes infectious to others.

Presymptomatic infectious period

2.2 days 2,3

The time period during which an infected individual is infectious to others. For symptomatic cases, this includes the presymptomatic period.

Infectious period

6.2 days 3–6

The time period during which an infected individual is infectious to others. For symptomatic cases, this includes the presymptomatic period.

Test sensitivity

75% while presymptomatic, 90% during first 3 days of infectious period, and decreasing thereafter.7,8

Probability that a single test will correctly identify an infectious individual as having SARS-CoV-2.

Testing Compliance

100% for teachers and staff, 75% for students

Probability that an individual will comply with testing, if any.

Percent symptomatic who self-quarantine

20%

Percentage of symptomatic individuals who develop sufficient symptoms (i.e., fever) that they call in sick and stay home from work.

Test turnaround time

1 day

Length of time between testing and isolation for individuals who receive positive results.

Isolation Time

10 days 13,14

Isolation time for individuals who receive a positive test result, self-isolate due to symptoms, or quarantine in response to a known positive contact.

Vaccine Effectiveness

90%

Percentage of vaccinated individuals in which the vaccine takes effect.

Table 2. Model assumptions for primary schools

Age of students (range)

5 to 10 years

Number of grades

6 (Kindergarten through 5th)

Classes per grade

4

Students per class (teacher)

20

Number of students

480 (#grades x #classes/grade x #students/class)

Number of teachers

24 (#grades x #classes/grade x 1)

Number of staff

24

Student-student connections

Well-connected within classroom; Household (siblings) connected

Table 3. Model assumptions for secondary schools

Age of students (range)

13 to 18 years

Number of grades

4 (9th through 12th)

Students per grade

200

Number of students

800 (#grades x #students/grade)

Number of teachers

125 (medium schools), 175 (large schools)

Number of staff

75

Student-student connections

Connected to 10 other students on average; Household (siblings) connected

Limitations

Modeling can be extremely important to help us understand epidemic progression, however, all models have assumptions, limitations, and biases that make them imperfect estimates. While we do our best to pick the most accurate and evidence-based parameters about SARS-CoV-2 disease spread, estimates for these parameters vary and may change as we learn more about the SARS-CoV-2 virus. Because these parameter choices can have significant impacts on model outcomes, we cannot guarantee our choices are always correct, and any results produced by this model should not be interpreted to predict exact numbers of cases or outcomes. Furthermore, epidemic spread is itself a stochastic process. Even holding all parameters constant, a particular transmission chain may fizzle out quickly or explode into a large outbreak according to the whims of chance. We represent this element of uncertainty by running 1000 simulations for each parameter set and showing the range of results there in using “jitter plots”.

References

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Treibel TA, Manisty C, Burton M, et al. COVID-19: PCR screening of asymptomatic health-care workers at London hospital. Lancet. 2020;395(10237):1608-1610.

Nishiura H, Kobayashi T, Miyama T, et al. Estimation of the asymptomatic ratio of novel coronavirus infections (COVID-19). Int J Infect Dis. 2020;94:154-155.

Byambasuren O, Cardona M, Bell K, Clark J, McLaws M-L, Glasziou P. Estimating the extent of true asymptomatic COVID-19 and its potential for community transmission: systematic review and meta-analysis. medRxiv. Published online 2020. https://www.medrxiv.org/content/10.1101/2020.05.10.20097543v1.abstract

CDC. Duration of isolation and precautions for adults with COVID-19. Published December 1, 2020. Accessed December 16, 2020. https://www.cdc.gov/coronavirus/2019-ncov/hcp/duration-isolation.html

CDC. Duration of isolation and precautions for adults with COVID-19. Published December 1, 2020. Accessed December 16, 2020. https://www.cdc.gov/coronavirus/2019-ncov/hcp/duration-isolation.html

CDC. Options to reduce quarantine for contacts of persons with SARS-CoV-2 infection using symptom monitoring and diagnostic testing. Published December 2, 2020. Accessed December 16, 2020. https://www.cdc.gov/coronavirus/2019-ncov/more/scientific-brief-options-to-reduce-quarantine.html