# About the Model

This model was developed in collaboration with Carl Bergstrom and Ryan McGee from the University of Washington. This model can be used to simulate infection dynamics of the Omicron variant of SARS-CoV-2 and evaluate the impact that different testing strategies and vaccination have on outbreaks in in primary and secondary schools.

On 2/25/2022, Version 3 of the modeling tool was released.

You can learn more about the model and proactive testing here:

# 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.

# Where can I find information about earlier version

of the model and webapp?

# Model Version 1

Version 1 of our modeling tool is described in more detail in our publication and the modeling tool can be found here.

# 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.

# Omicron Variant Model Updates

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

Vaccine booster status. To model the Omicron wave, we need to account for vaccination status (unvaccinated, one or two doses, or boosted) and prior infection by pre-Omicron strains of the virus. To do so, we have separate compartment layers for boosted and non-boosted individuals. Among individuals who have not received a booster, some have had previous immune exposure to either the vaccine or to natural infection (*R**pre*), while others may be entirely immunologically naive (*S*). Within each booster status class, individuals can be infected and subsequently move through exposed (*E*), presymptomatic (*P*), symptomatic (*I*) or asymptomatic (*A*), and recovered (*R*) states.

In our previous model, we allowed vaccine breakthrough but did not allow reinfection. In the updated model, Omicron can infect those who have recovered from previous strains, albeit at a lower rate than it infects epidemiologically naive individuals. We assume that those who have recovered from Omicron itself will not be reinfected by Omicron within the 90 day simulation period. Omicron also readily infects those who have had one or two doses of the vaccine, but is less likely to infect individuals who have received a booster.

**Figure 1. Compartment diagram**

The compartment diagram then appears in Figure 1. Individuals may be non-boosted (filled compartments) or boosted (open compartments). Booster vaccinations are not administered while the model is running. Non-boosted individuals may have never been vaccinated nor naturally infected previously (*S* compartment), may have recovered from infection with a non-Omicron variant or received one or two vaccine doses (*R**pre* compartment), or may have recovered from Omicron infection (*R* component). For boosted individuals, we draw no distinction between those with previous infection from a non-Omicron variant and those without; both are placed in the *B* compartment.

The new model distinguishes the efficacy of the vaccine against infection from the efficacy against transmission. The former measures how likely a vaccination person is to contract COVID; the latter how likely a vaccinated person infected with COVID is to transmit it. We assume that vaccine and booster doses partially reduce susceptibility to infection by Omicron but do not reduce transmissibility conditional upon infection with Omicron.

We assume that the vaccine boosters reduces susceptibility to infection by Omicron and transmissibility conditional upon infection by the same percentage, as set with the Booster Effectiveness slider.

The range of baseline R0 values has been extended up to 8. We e assume that 90% of teachers and staff have some experience with pre-Omicron SARS-CoV-2, either in the form of vaccination or prior infection. We assume that these individuals are 10% less susceptible to Omicron than non-vaccinated individuals who have not been previously infected. We have added a slider allowing users to specify the fraction of students who have been vaccinated but not booster and/or infected with pre-Omicron strains.

As the Omicron wave sweeps through the population, immunity from prior infection by this strain will be an important driver of the epidemic trajectory. To account for this, we have added a slider specifying the fraction of the population already infected with Omicron.

Vaccine booster status. To model the Omicron wave, we need to account for vaccination status (unvaccinated, one or two doses, or boosted) and prior infection by pre-Omicron strains of the virus. To do so, we have separate compartment layers for boosted and non-boosted individuals. Among individuals who have not received a booster, some have had previous immune exposure to either the vaccine or to natural infection (*R**pre*), while others may be entirely immunologically naive (*S*). Within each booster status class, individuals can be infected and subsequently move through exposed (*E*), presymptomatic (*P*), symptomatic (*I*) or asymptomatic (*A*), and recovered (*R*) states.

In our previous model, we allowed vaccine breakthrough but did not allow reinfection. In the updated model, Omicron can infect those who have recovered from previous strains, albeit at a lower rate than it infects epidemiologically naive individuals. We assume that those who have recovered from Omicron itself will not be reinfected by Omicron within the 90 day simulation period. Omicron also readily infects those who have had one or two doses of the vaccine, but is less likely to infect individuals who have received a booster.

**Figure 1. Compartment diagram**

The compartment diagram then appears in Figure 1. Individuals may be non-boosted (filled compartments) or boosted (open compartments). Booster vaccinations are not administered while the model is running. Non-boosted individuals may have never been vaccinated nor naturally infected previously (*S* compartment), may have recovered from infection with a non-Omicron variant or received one or two vaccine doses (*R**pre* compartment), or may have recovered from Omicron infection (*R* component). For boosted individuals, we draw no distinction between those with previous infection from a non-Omicron variant and those without; both are placed in the *B* compartment.

The new model distinguishes the efficacy of the vaccine against infection from the efficacy against transmission. The former measures how likely a vaccination person is to contract COVID; the latter how likely a vaccinated person infected with COVID is to transmit it. We assume that vaccine and booster doses partially reduce susceptibility to infection by Omicron but do not reduce transmissibility conditional upon infection with Omicron.

We assume that the vaccine boosters reduces susceptibility to infection by Omicron and transmissibility conditional upon infection by the same percentage, as set with the Booster Effectiveness slider.

The range of baseline R0 values has been extended up to 8. We e assume that 90% of teachers and staff have some experience with pre-Omicron SARS-CoV-2, either in the form of vaccination or prior infection. We assume that these individuals are 10% less susceptible to Omicron than non-vaccinated individuals who have not been previously infected. We have added a slider allowing users to specify the fraction of students who have been vaccinated but not booster and/or infected with pre-Omicron strains.

As the Omicron wave sweeps through the population, immunity from prior infection by this strain will be an important driver of the epidemic trajectory. To account for this, we have added a slider specifying the fraction of the population already infected with Omicron.

# 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

**Figure 2. Primary school network model**

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 (Figure 2).

**Figure 3. Secondary school network model**

For our secondary school model, we consider a large school with 2000 students (500 per graduating class), 175 teachers, and 75 staff. 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 (Figure 3).

# Model parameters

The parameter values used in the models are listed in the table below. Some of the parameters are represented by distributions or time-varying functions. These include:

**Figure 4. Distribution of infectious period**

The infectious period varies considerably from individual to individual in our model. We assume a gamma-distributed presymptomatic period with mean duration 1 day and a coefficient of variation of 0.5, and a gamma-distributed additional infectious period of 4 days with a coefficient of variation of 0.4 days. The total infectious period is then a random variable drawn from the distribution shown in Figure 2.

**Figure 5. Distribution of individual transmissiveness**

The infectious period varies considerably from individual to individual in our model. We assume a gamma-distributed presymptomatic period with mean duration 1 day and a coefficient of variation of 0.5, and a gamma-distributed additional infectious period of 4 days with a coefficient of variation of 0.4 days. The total infectious period is then a random variable drawn from the distribution shown in Figure 2.

**Figure 6. Test sensitivities over time**

We assume that the test sensitivities vary over the course of disease progression, as shown in Figure 4.

Parameter

Mean Value

Description

R0

3.5 (range = 1.0-6.0)

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

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

Temporal sensitivity curves for molecular and antigen tests8-12; see Figure 4

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

Testing Compliance

100% for teachers and staff, 75% for students

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

Percent asymptomatic

30% for adults and secondary school students, 40% for primary school students. 9–12

Percentage of individuals infected with SARS-CoV-2 who do not develop symptoms.

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% (range = 60%-90%)

Percentage of vaccinated individuals in which the vaccine takes effect.

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

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.

We describe the limitations of our model in further detail in our paper.

# References

1.3 billion learners are still affected by school or university closures, as educational institutions start reopening around the world, says UNESCO. Published April 29, 2020. Accessed November 9, 2020. https://en.unesco.org/news/13-billion-learners-are-still-affected-school-university-closures-educational-institutions

Tindale L, Coombe M, Stockdale JE, et al. Transmission interval estimates suggest pre-symptomatic spread of COVID-19. Epidemiology. Published online March 6, 2020. doi:10.1101/2020.03.03.20029983

He X, Lau EHY, Wu P, et al. Temporal dynamics in viral shedding and transmissibility of COVID-19. Nat Med. 2020;26(5):672-675.

Wölfel R, Corman VM, Guggemos W, et al. Virological assessment of hospitalized patients with COVID-2019. Nature. 2020;581(7809):465-469.

Ganyani T, Kremer C, Chen D, et al. Estimating the generation interval for coronavirus disease (COVID-19) based on symptom onset data, March 2020. Euro Surveill. 2020;25(17). doi:10.2807/1560-7917.ES.2020.25.17.2000257

Young BE, Ong SWX, Kalimuddin S, et al. Epidemiologic Features and Clinical Course of Patients Infected With SARS-CoV-2 in Singapore. JAMA. Published online March 3, 2020. doi:10.1001/jama.2020.3204

Wikramaratna P, Paton RS, Ghafari M, Lourenco J. Estimating false-negative detection rate of SARS-CoV-2 by RT-PCR. Epidemiology. Published online April 7, 2020. doi:10.1101/2020.04.05.20053355

Kucirka LM, Lauer SA, Laeyendecker O, Boon D, Lessler J. Variation in False-Negative Rate of Reverse Transcriptase Polymerase Chain Reaction-Based SARS-CoV-2 Tests by Time Since Exposure. Ann Intern Med. Published online May 13, 2020. doi:10.7326/M20-1495

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