Orange County, CA COVID-19 Situation Report Start Date - End Date
Last updated: 2020-08-26
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Knit directory: uci_covid_modeling/
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Introduction
The goal of this report is to inform interested parties about dynamics of SARS-CoV-2 spread in Orange County, CA and to predict (if feasible) the outbreak epidemic trajectory. Our approach is based on fitting a mechanistic model of SARS-CoV-2 spread to multiple sources of surveillance data. More specifically, we use daily numbers of new cases and deaths, while taking into account changes in the total number of tests reported on each day.
Executive summary
- 95% Bayesian credible interval for the basic reproductive number \(R_0\) is (0.37, 0.8).
- 95% Bayesian credible interval for the total number of infections that had occurred between June 20, 2020 and July 25, 2020 is (13,000, 110,000). The number of reported cases in this period was 23,000.
- 95% Bayesian credible interval for the infection-to-fatality ratio (IFR), defined as a fraction of deaths among the total number of infections, is (0.002, 0.013).
Since we rely on a mechanistic model, it is important to acknowledge limitations of this model. We will try to be transparent about our assumptions in the hope to receive feedback about their realism. So if something does not look right to you, please get in touch with us to help us improve our model.
Model inputs
Our method takes three time series as input: daily new tests, case counts, and deaths. However, we find daily resolution to be too noisy due to delay in testing reports, weekend effect, etc. So we aggregated/binned the three types of counts in 3 day intervals. These aggregated time series are shown below.
Model structure
We assume that all individuals in Orange County, CA can be split into 6 compartments: S = susceptible individuals, E = infected, but not yet infectious individuals, \(\text{I}_\text{e}\) = individuals at early stages of infection, \(\text{I}_\text{p}\) = individuals at progressed stages of infection (assumed 20% less infectious than individuals at the early infection stage), R = recovered individuals, D = individuals who died due to COVID-19. Possible progressions of an individual through the above compartments are depicted in the diagram below.
Mathematically, we assume that dynamics of the proportions of individuals in each compartment follow a set of ordinary differential equations corresponding to the above diagram. These equations are controlled by the following parameters:
- Basic reproductive number (\(R_0\))
- mean duration of the latent period
- mean duration of the early infection period
- mean duration of the progressed infection period
- probability of transitioning from progressed infection to death, rather than to recovery (i.e., IFR)
We fit this model to data by assuming that case counts are noisy realizations of the actual number of individuals progressing from \(\text{I}_\text{e}\) compartment to \(\text{I}_\text{p}\) compartment. Similarly we assume that observed deaths are noisy realizations of the actual number of individuals progressing from \(\text{I}_\text{p}\) compartment to \(\text{D}\) compartment. A priori, we assume that death counts are significantly less noisy than case counts. We use a Bayesian estimation framework, which means that all estimated quantities receive credible intervals (e.g., 80% or 95% credible intervals). Width of these credible intervals encode the amount of uncertainty that we have in the estimated quantities.
Latent cumulative deaths, cumulative incidence, and prevalence
We report estimated trajectories of latent cumulative deaths, incidence (i.e., total infections and deaths that have occurred by some prespecified time point), and prevalence . These trajectories represent our estimation of accumulation of unobserved/hidden number of infections and deaths in the Orange County population. We also project these three trajectories 4 weeks into the future.
The main takeaways are:
- Cases underestimate the total number of infections by a factor that ranges with 95% probability between 0.57 and 4.9. This means that we estimate that the total number of infections between June 20, 2020 and July 25, 2020 is between 13,000 and 110,000. The number of reported cases in this period was 23,000.
- 95% Bayesian credible interval of death underreporting factor is (0.52, 0.97).
Prior and posterior distributions of model parameters
Next we report prior and posterior distributions of key model parameters. Prior distributions represent our beliefs about these parameters before seeing/analyzing data. Posterior distributions encode our updated beliefs after seeing/analyzing data.
The main takeaways are:
- 95% Bayesian credible interval for the basic reproductive number \(R_0\) is (0.37, 0.8). The posterior probability that \(R_0\) is above 1.0 is 0%.
Reported deaths and positive test rate forecast
We report predictive distribution of observed deaths and positive test rate during the observation interval (blue shaded area with black dots in the plot below). In addition, we show a four week ahead forecast for both quantities. Our forecast assumes that interventions (physical distancing orders, mask recommendations, etc.) stay at the status quo, and that there are 2100 tests in each three day interval in the future. We plan to archive our forecasts and retrospectively measure their predictive ability/skill.
Modeling limitations
- Some model parameters are not well identified, so we rely on scientifically sound prior information (please check our assumptions and help us improve our priors!). We will include sensitivity to prior analyses as an appendix in future reports.
- The model assumes a well mixed population with no geographic or age structure.
- Basic reproductive number (\(R_0\)) is constant. However, implementation of physical distancing and school closures should gradually reduce it. Similarly, relaxing current physical distancing measures will start increasing \(R_0\).
Planned model extensions
- We plan to add H = hospitalized and ICU compartments to the model. This extension will allow us to inform the model with hospitalization and occupied ICU beds time series, in addition to already used test, case, and death counts. Equally importantly, we will be able to forecast hospital and ICU occupancy.
- We will allow \(R_0\) to vary with time and will attempt to estimate its changes non-parameterically.
- We would like to add age structure into the model. This is not a high priority at the moment, because we suspect we don’t have enough data to estimate additional parameters that will need to be added to the model.
Appendix
Sensitivity to Prior for \(R_0\)
We examine how sensitive our conclusions about \(R_0\) to our prior assumptions by repeating estimation of all model parameters under different priors for this parameter. The priors are listed in the titles of the figures. Although the prior distribution of \(R_0\) does have some effect on its posterior (as it should), the our results and conclusions are not too sensitive to a particular specification of this prior.
Sensitivity to prior for fraction not susceptible
We examine how sensitive our conclusions about \(R_0\) to our prior assumptions by repeating estimation of all model parameters under different priors for the parameter controlling how many people are not susceptible initially. This prior changes depending on the time period, so we adjust by changing the prior mean to be twice as large or one half as large as the default prior. As we would expect, changing this prior changes the number of people we estimate will become infected or are currently infectious. However, it seems to have little impact on the posterior of \(R_0\).
R version 4.0.2 (2020-06-22)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 10 x64 (build 18362)
Matrix products: default
locale:
[1] LC_COLLATE=English_United States.1252
[2] LC_CTYPE=English_United States.1252
[3] LC_MONETARY=English_United States.1252
[4] LC_NUMERIC=C
[5] LC_TIME=English_United States.1252
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] patchwork_1.0.1 scales_1.1.1 tidybayes_2.1.1 forcats_0.5.0
[5] stringr_1.4.0 dplyr_1.0.0 purrr_0.3.4 readr_1.3.1
[9] tidyr_1.1.0 tibble_3.0.3 ggplot2_3.3.2 tidyverse_1.3.0
[13] here_0.1 lubridate_1.7.9 workflowr_1.6.2
loaded via a namespace (and not attached):
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[4] httr_1.4.1 rprojroot_1.3-2 rstan_2.21.2
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[10] R6_2.4.1 DBI_1.1.0 colorspace_1.4-1
[13] ggdist_2.2.0 withr_2.2.0 gridExtra_2.3
[16] tidyselect_1.1.0 prettyunits_1.1.1 processx_3.4.3
[19] curl_4.3 compiler_4.0.2 git2r_0.27.1
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[49] stringi_1.4.6 whisker_0.4 yaml_2.2.1
[52] pkgbuild_1.1.0 plyr_1.8.6 grid_4.0.2
[55] blob_1.2.1 parallel_4.0.2 promises_1.1.1
[58] crayon_1.3.4 lattice_0.20-41 haven_2.3.1
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[67] reprex_0.3.0 glue_1.4.1 evaluate_0.14
[70] V8_3.2.0 RcppParallel_5.0.2 modelr_0.1.8
[73] vctrs_0.3.2 httpuv_1.5.4 cellranger_1.1.0
[76] gtable_0.3.0 assertthat_0.2.1 xfun_0.15
[79] broom_0.7.0 coda_0.19-3 later_1.1.0.1
[82] ellipsis_0.3.1