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:

 

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

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

 

SEIRS+ model

This interactive modeling tool uses a version of the SEIRS+ model, which was developed by Ryan McGee, Carl Bergstrom, and colleagues at the University of Washington.

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.

 

Omicron variant model updates

  • Vaccine booster status. 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 (Rpre), 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.
  • Reinfection dynamics. 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 (Rpre 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. 

  • Vaccine efficacy. 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.
  • Booster effectiveness. 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.
  • Growth Kinetics. The latent and presymptomatic periods for the virus have been shortened to account for the faster growth kinetics of the Omicron variant.
  • Parameter ranges. 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. 
  • Protection due to prior Omicron infection. 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

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 2. Primary 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).

Figure 3. Secondary school network model

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:

  •  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 4. Distribution of infectious period
  • We assume that individuals vary in infectiousness, and represent this as an overdispersed distribution of transmissiveness (R0 values, effectively) across individuals. We assume a coefficient of variation of 2.0, which gives us approximately an “80-20” rule whereby the most transmissive 20% of the population are responsible for 80% of the transmissions. The distribution of individual transmissiveness values is as shown in Figure 3.

Figure 5. Distribution of individual transmissiveness
  • We assume that the test sensitivities vary over the course of disease progression, as shown in Figure 4.

Figure 6. Test sensitivities over time

Table 1. Table of parameters included in model

Parameter

Mean Value

Description

R0

6.0 (default, can be adjusted)1-4

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=6.0) assumes that basic mitigation strategies, such as mask-wearing and social distancing, have largely been lifted.

Mean latent period

2.0 days 5-7

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

Mean presymptomatic infectious period

1.0 days 5-7

The period when an individual infected with SARS-CoV-2 is contagious but has not yet developed symptoms.

Mean infectious period

5.0 days 5-7

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 fas shown in Figure 68-12

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

Testing compliance for teachers and staff

100%

Probability that a teacher or staff member will comply with a proactive testing program, if any.

Testing compliance for students

50% (default, can be adjusted)

Probability that a student will participate in a proactive testing program, if any.

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.

Percent asymptomatic

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

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 for molecular tests

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

Isolation Time 

5 days

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

Booster Effectiveness

50% (range 10-90%)

Percentage reduction in per-encounter infection probability for boosted individuals.13-15

 

Table 2. Model assumptions for primary schools

Primary School Structure (n=528)

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

Secondary School Structure (n=1,000)

Age of students (range) 

13 to 18 years

Number of grades

4 (9th through 12th)

Students per grade

200

Number of students

2000 (#grades x #students/grade)

Number of teachers

175

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 therein using “jitter plots”.

 

References

  1. Nishiura H, Ito K, Anzai A, Kobayashi T, Piantham C, Rodríguez-Morales AJ. Relative Reproduction Number of SARS-CoV-2 Omicron (B.1.1.529) Compared with Delta Variant in South Africa. J Clin Med Res. 2021;11(1). doi:10.3390/jcm11010030
  2. Ito K, Piantham C, Nishiura H. Relative instantaneous reproduction number of Omicron SARS-CoV-2 variant with respect to the Delta variant in Denmark. J Med Virol. Published online December 30, 2021. doi:10.1002/jmv.27560
  3. UK Health Security Agency. SARS-CoV-2 Variants of Concern and Variants under Investigation in England – Technical Briefing 31. UKHSA; 2021. Accessed January 18, 2022. https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/1042367/technical_briefing-31-10-december-2021.pdf
  4. World Health Organization. Enhancing Response to Omicron SARS-CoV-2 Variant: Technical Brief and Priority Actions for Member States, Update #5. WHO; 2022. Accessed January 18, 2022. https://www.who.int/docs/default-source/coronaviruse/2022-01-07-global-technical-brief-and-priority-action-on-omicron—corr2.pdf?sfvrsn=918b09d_23
  5. Jansen L, Tegomoh B, Lange K, et al. Investigation of a SARS-CoV-2 B.1.1.529 (Omicron) Variant Cluster – Nebraska, November-December 2021. MMWR Morb Mortal Wkly Rep. 2021;70(5152):1782-1784.
  6. Brandal LT, MacDonald E, Veneti L, et al. Outbreak caused by the SARS-CoV-2 Omicron variant in Norway, November to December 2021. Euro Surveill. 2021;26(50). doi:10.2807/1560-7917.ES.2021.26.50.2101147
  7. Active epidemiological investigation on SARS-CoV-2 infection caused by Omicron variant (Pango lineage B.1.1.529) in Japan: preliminary report on infectious period. Accessed January 18, 2022. https://www.niid.go.jp/niid/en/2019-ncov-e/10884-covid19-66-en.html
  8. Siddiqui ZK, Chaudhary M, Robinson ML, et al. Implementation and Accuracy of BinaxNOW Rapid Antigen COVID-19 Test in Asymptomatic and Symptomatic Populations in a High-Volume Self-Referred Testing Site. Microbiol Spectr. 2021;9(3):e0100821.
  9. Schuit E, Veldhuijzen IK, Venekamp RP, et al. Diagnostic accuracy of rapid antigen tests in asymptomatic and presymptomatic close contacts of individuals with confirmed SARS-CoV-2 infection: cross sectional study. BMJ. 2021;374:n1676.
  10. Smith RL, Gibson LL, Martinez PP, et al. Longitudinal Assessment of Diagnostic Test Performance Over the Course of Acute SARS-CoV-2 Infection. J Infect Dis. 2021;224(6):976-982.
  11. Levine-Tiefenbrun M, Yelin I, Uriel H, et al. SARS-CoV-2 RT-qPCR Test Detection Rates Are Associated with Patient Age, Sex, and Time since Diagnosis. J Mol Diagn. Published online November 23, 2021. doi:10.1016/j.jmoldx.2021.10.010
  12. Miller TE, Garcia Beltran WF, Bard AZ, et al. Clinical sensitivity and interpretation of PCR and serological COVID-19 diagnostics for patients presenting to the hospital. FASEB J. 2020;34(10):13877-13884.
  13. Collie S, Champion J, Moultrie H, Bekker LG, Gray G. Effectiveness of BNT162b2 Vaccine against Omicron Variant in South Africa. N Engl J Med. Published online December 29, 2021. doi:10.1056/NEJMc2119270
  14. Tseng HF, Ackerson BK, Luo Y, et al. Effectiveness of mRNA-1273 against SARS-CoV-2 omicron and delta variants. bioRxiv. Published online January 8, 2022. doi:10.1101/2022.01.07.22268919
  15. UK Health Security Agency. SARS-CoV-2 Variants of Concern and Variants under Investigation in England Technical Briefing: Update on Hospitalisation and Vaccine Effectiveness for Omicron VOC-21NOV-01 (B.1.1.529). UKHSA; 2021.