Archetypal analysis of COVID-19 in Montana, USA, March 13, 2020 to April 26, 2022

Abstract

Infectious disease data can often involve complex spatial patterns intermixed with temporal trends. Archetypal Analysis is a method to mine complex spatio-temporal data, and can be used to discover the dynamics of spatial patterns. The application of Archetypal Analysis to epidemiological data is relatively new, and here we present one of the first applications on COVID-19 data from March 13, 2020 to April 26, 2022, for the counties of Montana, USA. We present three views of the data set decomposed with Archetypal Analysis. First, we evaluate the entire 56 county data set. Second, we use a mutual information calculation to remove counties whose dynamics are mainly independent from the other counties, reducing the set to 17 counties. Finally, we analyze the top ten counties in terms of population size to focus on the dynamics in the large cities in the state. For each data set, we analyze four significant disease outbreaks across Montana. Archetypal Analysis uncovers distinct spatial patterns for each outbreak and demonstrates that each has a unique trajectory across the state.

Fig 1. COVID-19 cases in Montana, USA.

Running weekly average of COVID-19 cases plotted for all 56 Montana counties. The four outbreaks are seen here: Initial (Sept. 2020-March 2021), Early Delta (April-Aug. 2021), Delta (July—Sept. 2021), and Omicron (Jan. 2021-April 2022).

Fig 2. Scree plot.

Residual sum of squares vs. number of archetypes in the 56 county data set. Used to select the number of archetypes in the decomposition.

Fig 3. Ten archetype set for all 56 counties.

Presented as β color-coded counties in the map of Montana. The first archetype [a] is nearly zero, as it captures the “no disease” state, and acts to “turn-off” the infection/spread in each county. The rest are in order: [b] archetype 2, [c] archetype 3, [d] archetype 4, [e] archetype 5, [f] archetype 6, [g] archetype 7, [h] archetype 8, [i] archetype 9, [j] archetype 10. High β values are considered those greater than 1.5, mid-size between 0.75 and 1.5, and below 0.75 as low. Figure created with ArcGiS: Esri, HERE, Garmin, FAO, NOAA, USGS, EPA.

Fig 4. Total mutual information for all 56 counties in Montana, USA.

Total mutual information (y-axis) across all counties in increasing order.

Fig 5. Total mutual information for the largest MI counties.

Total mutual information in the top 17 total MI counties in increasing order.

Fig 6. Time series of COVID-19 case counts.

COVID-19 cases for the 17 largest total MI counties in Montana, USA.

Fig 7. Scree plot.

Residual sum of squares vs. number of archetypes being computed for the High MI County Set. Used to select the number of archetypes in the decomposition. Note: the RSS for 1 archetype is 3178, not seen with this y-range, which chosen to show the decay at larger numbers.

Fig 8. Nine archetype set for the high MI county set.

Presented as color-coded counties in map of Montana. The first archetype is not included, as it captures the “no disease” state, and acts to “turn-off” the disease in each county. The rest are presented in order: [a] archetype 2, [b] archetype 3, [c] archetype 4, [d] archetype 5, [e] archetype 6, [f] archetype 7, [g] archetype 8, [h] archetype 9. Note that high β values are considered those greater than 1.5, mid-size are between 0.75 and 1.5, low are below 0.75. Figure created with ArcGiS: Esri, HERE, Garmin, FAO, NOAA, USGS, EPA.

Fig 9. Reconstruction of high MI county data set with 9 archetypes.

Original data are plotted in blue, and the reconstruction of the data in red.

Fig 10. Reconstruction coefficient (α) time series for the 9 archetype decomposition of the 17 county data set.

Bars across top are color-coded to show the dominant archetype during that time period.

Fig 11. Time series of major county components in the archetypes.

As labeled: Initial, Early Delta, Delta and Omicron outbreaks, 17 county data set.

Fig 12. Data time series: COVID case numbers for the large population county data set.

Fig 13. Scree plot.

Residual sum of squares vs. number of archetypes for the Large Population County Set. Used to select the number of archetypes in the decomposition.

Fig 14. The six archetypes for the large population county data set.

Note that the first archetype is not included, as it captures the “no disease” state, and acts to “turn-off” the outbreak in each county. The rest are in order: [a] archetype 2, [b] archetype 3, [c] archetype 4, [d] archetype 5, [e] archetype 6. Figure created with ArcGiS: Esri, HERE, Garmin, FAO, NOAA, USGS, EPA.

Fig 15. Reconstruction of large population county data set with 6 archetypes.

Data are plotted in blue, reconstruction in red.

Fig 16. Reconstruction coefficient (α) time series.

The 6 archetype decomposition of the 10 county set. Bars across top are color-coded to show the dominant archetype during that time period.

Fig 17. Time series of major county components in the archetypes.

As labeled: Initial, early Delta, Delta and Omicron Phases, 10 county data set.