Age-Varying Susceptibility to the Delta Variant (B.1.617.2) of SARS-CoV-2

This decision analytic model investigates susceptibility to the Delta variant of SARS-CoV-2 vs previous variants by age group.


A. Overview
In a Susceptible-Exposed-Infectious-Recovered (SEIR) model, the force of infection , the rate at which susceptible individuals are infected (i.e exposed), is a crucial factor. The age-specific force of infection in age group at discrete time could be written as: where is the probability that a contact between a susceptible in age group and infectious person leads to infection, means contact matrix at discrete time ( is the number of contacts an individual of age group makes with those of age group per unit time at discrete time ), is the number of individuals in age group .
is the number of individuals who become infectious before the symptom onset; is the number of individuals who are both symptomatic and infectious; is the number of individuals who are infectious but never developed any symptoms. We suppose that the relative infectiousness of the is half of or . Here is in age group 1,2, ⋯ , .
We try to estimate during the third and fourth waves in South Korea and denote it as hereafter. If we could observe the number of exposed individuals at time for age group and the number of total infectious individuals, the likelihood of the parameters could be easily derived. However what we could observe is only the number of diagnosed (i.e quarantined) individuals for age group ∈ at time . In addition, asymptomatic infection which is the notable feature of COVID-19 should also be reflected in the model. To resolve these difficulties, we use a Bayesian approach. In particular, we develop an efficient MCMC (Markov Chain Monte Carlo) algorithm in which the exposed date, symptom onset date and transmission onset date for all quarantined individuals are imputed with an assumption that there are 16% asymptomatic individuals. We explain the details of our Bayesian method in the following three subsections.

B. Data, Model and Posterior
The data we used in the analysis is daily numbers of quarantined individuals for each age group from 15 October to 22 December 2020 (3rd wave) and from 27 June to 21 August 2021 (4th wave).
To estimate , we are going to impute the exposed dates, symptom onset dates and transmission onset dates of all quarantined individuals conditional on given quarantined dates. For this purpose, we need a probability model which relates the exposed dates, symptom onset dates and transmission onset dates to the quarantined dates. For each symptomatic individual, the quarantined date is sum of the exposed date( ), the incubation period( ) and the period for symptom onset to diagnostic delay( ). In addition, these individuals start infecting other susceptibles from the transmission onset date ( ) For each asymptomatic individual, the quarantined date is sum of exposed date ( ), latent period( ) and period for transmission onset to quarantined date( ). We assume that the latent period distribution of asymptomatic individuals is same as that of symptomatic individuals. Also the quarantined time distribution of of asymptomatic individuals is set to be the exponential distribution with mean 1/1.7, which satisfies 0.01 where is defined as the recovered date.

Symptomatic cases
Finally, the quarantined date is defined as where is equal to 0 when the individual is symptomatic and 1 when asymptomatic.We set 1 0.16.
For individual , let , , , , , and , , , , . Let be the observed data which consist of the daily numbers of quarantined individuals. Our strategy to estimate is to generate and iteratively from their conditional posterior distributions | , and | , , respectively, where and denotes except .
We could describe as,