Estimating Covid-19 Dynamics and Clinical Detection Rate using an Approximation to the Discrete Stochastic SEIR Model
Ben J. Brintz
Division of Epidemiology
This presentation will not include equations
Statisticians may feel unsettled by the lack of notation to remember
The Covid-19 pandemic highlighted the challenge of imperfect detection in disease surveillance
- Case reports undercount true infections due to asymptomatic cases and lack of access to testing
- Only the sickest got diagnosed with Covid early in the pandemic
- “We must rely on data from people who are sick enough to get themselves tested, which is a bit like trying to understand exercise trends among average Americans by surveying the participants of a marathon”
- Utah Hero Project
Various approaches have been developed to estimate clinical detection rates
- Back-calculation methods estimate the number of infections from hospitalizations and deaths
- Seroprevalence studies estimate the proportion of the population that has been infected by testing for antibodies
- Compartment models estimate the number of infections from observed cases and the distribution of the incubation period
- E.g. SIR (Susceptible-Infectious-Recovered) models
Covid-19’s incubation period requires an extension to the SIR compartment model
- SIR model assumes that individuals transition directly from susceptible to infectious
- SEIR model adds an exposed compartment to account for the incubation period (e.g. 5 days)
- SEIR models don’t directly account for imperfect detection
Our stochastic SEIR model acounts for imperfect detection
- Individuals transition from susceptible to exposed to infectious
- We fit a model to observed county-level counts of new cases
- We assume incidence cases (E -> I, we call EI) are imperfectly detected
- Currently assumed to be constant across time.
- But it does allow all dynamics parameters to vary over time and space
Our study has a number of novelties in the stochastic SEIR model space
Novelty #1
A novel computational solution to estimating dynamics in a discrete transmission model
- We used a continuous approximation in STAN with non-informative priors (Bayesian Hamiltonian Monte Carlo)
- The binomial distribution is a reasonable way to describe dynamics between compartments but STAN doesn’t allow discrete priors
- We approximate the binomial distribution with the inverse logit of a normal distribution to sample more efficiently and leverage the strength of gradient-based algorithms
A novel computational solution to estimating dynamics in a discrete transmission model
Novelty #2
Model estimation allows for different start times of infections across health-districts
Note
This observed data aren’t smooth SEIR curves
Novelty #3
Estimating an evolving transmission rate using a hierarchical AR(1) process
Estimating an evolving transmission rate using a hierarchical AR(1) process
Novelty #4
Estimating an evolving recovery rate and incubation using beta-binomial transitions
Some Results
The Team
Questions?
Model Formulation
\[ \begin{aligned} I_{if_1} &= c,\\[6pt] EI_{if_i} &= I_{if_i},\\[6pt] S_{if_i} &= N_i - EI_{if_i}. \end{aligned} \]
Model Formulation
For (t > f_{i}):
\[ \begin{aligned} SE_{it} &\sim \text{Binomial}\!\Bigl(S_{it-1},\,1 - \exp\!\Bigl(-\tfrac{\beta_{it}I_{it-1}}{N}\Bigr)\Bigr),\\[6pt] EI_{it} &\sim \text{BetaBinomial}\!\Bigl(E_{it-1},\,1 - \exp(-\eta_i),\,\rho_{EI}\Bigr),\\[6pt] IR_{it} &\sim \text{BetaBinomial}\!\Bigl(I_{it-1},\,1 - \exp(-\gamma_i),\,\rho_{IR}\Bigr),\\[6pt] \end{aligned} \]
\[ \begin{aligned} I_{if_1} &= c,\\[6pt] EI_{if_i} &= I_{if_i},\\[6pt] S_{if_i} &= N_i - EI_{if_i}. \end{aligned} \]
Model Formulation
\[ \begin{aligned} S_{it} &= S_{it-1} - SE_{it},\\[6pt] E_{it} &= \begin{cases} SE_{it}, & \text{if } t = f_i + 1,\\[6pt] E_{it-1} + SE_{it} - EI_{it}, & \text{if } t > f_i + 1, \end{cases}\\[6pt] I_{it} &= I_{it-1} + EI_{it} - IR_{it},\\[6pt] ii_{it} &\sim \text{Binomial}\!\Bigl(EI_{it},\,p\Bigr) \end{aligned} \]
\[ \begin{aligned} SE_{it} &\sim \text{Binomial}\!\Bigl(S_{it-1},\,1 - \exp\!\Bigl(-\tfrac{\beta_{it}I_{it-1}}{N}\Bigr)\Bigr),\\[6pt] EI_{it} &\sim \text{BetaBinomial}\!\Bigl(E_{it-1},\,1 - \exp(-\eta_i),\,\rho_{EI}\Bigr),\\[6pt] IR_{it} &\sim \text{BetaBinomial}\!\Bigl(I_{it-1},\,1 - \exp(-\gamma_i),\,\rho_{IR}\Bigr),\\[6pt] \end{aligned} \]
\[ \begin{aligned} I_{if_1} &= c,\\[6pt] EI_{if_i} &= I_{if_i},\\[6pt] S_{if_i} &= N_i - EI_{if_i}. \end{aligned} \]
Model Formulation
\[ \begin{aligned} \log{\beta_{it}} &\sim \text{Normal}(B_{it}, \tau^2) \\ B_{it} &= \phi B_{i,t-1} + \sigma \epsilon_{it}, t>1 \end{aligned} \]
\[ \begin{aligned} S_{it} &= S_{it-1} - SE_{it},\\[6pt] E_{it} &= \begin{cases} SE_{it}, & \text{if } t = f_i + 1,\\[6pt] E_{it-1} + SE_{it} - EI_{it}, & \text{if } t > f_i + 1, \end{cases}\\[6pt] I_{it} &= I_{it-1} + EI_{it} - IR_{it},\\[6pt] ii_{it} &\sim \text{Binomial}\!\Bigl(EI_{it},\,p\Bigr) \end{aligned} \]
\[ \begin{aligned} SE_{it} &\sim \text{Binomial}\!\Bigl(S_{it-1},\,1 - \exp\!\Bigl(-\tfrac{\beta_{it}I_{it-1}}{N}\Bigr)\Bigr),\\[6pt] EI_{it} &\sim \text{BetaBinomial}\!\Bigl(E_{it-1},\,1 - \exp(-\eta_i),\,\rho_{EI}\Bigr),\\[6pt] IR_{it} &\sim \text{BetaBinomial}\!\Bigl(I_{it-1},\,1 - \exp(-\gamma_i),\,\rho_{IR}\Bigr),\\[6pt] \end{aligned} \]
\[ \begin{aligned} I_{if_1} &= c,\\[6pt] EI_{if_i} &= I_{if_i},\\[6pt] S_{if_i} &= N_i - EI_{if_i}. \end{aligned} \]
Model Formulation
\[\begin{aligned} ii_{it} &\sim \text{Binomial}\!\Bigl(EI_{it}, p \Bigr) \\ \end{aligned} \]
\[ \begin{aligned} \log{\beta_{it}} &\sim \text{Normal}(B_{it}, \tau^2) \\ B_{it} &= \phi B_{i,t-1} + \sigma \epsilon_{it}, t>1 \end{aligned} \]
\[ \begin{aligned} S_{it} &= S_{it-1} - SE_{it},\\[6pt] E_{it} &= \begin{cases} SE_{it}, & \text{if } t = f_i + 1,\\[6pt] E_{it-1} + SE_{it} - EI_{it}, & \text{if } t > f_i + 1, \end{cases}\\[6pt] I_{it} &= I_{it-1} + EI_{it} - IR_{it},\\[6pt] ii_{it} &\sim \text{Binomial}\!\Bigl(EI_{it},\,p\Bigr) \end{aligned} \]
\[ \begin{aligned} SE_{it} &\sim \text{Binomial}\!\Bigl(S_{it-1},\,1 - \exp\!\Bigl(-\tfrac{\beta_{it}I_{it-1}}{N}\Bigr)\Bigr),\\[6pt] EI_{it} &\sim \text{BetaBinomial}\!\Bigl(E_{it-1},\,1 - \exp(-\eta_i),\,\rho_{EI}\Bigr),\\[6pt] IR_{it} &\sim \text{BetaBinomial}\!\Bigl(I_{it-1},\,1 - \exp(-\gamma_i),\,\rho_{IR}\Bigr),\\[6pt] \end{aligned} \]
\[ \begin{aligned} I_{if_1} &= c,\\[6pt] EI_{if_i} &= I_{if_i},\\[6pt] S_{if_i} &= N_i - EI_{if_i}. \end{aligned} \]