"What are the odds?" I Am Exposed to a COVID-19 Positive Individual

Note: This post is on the likelihood of being exposed to an infectious COVID-19 positive individual over a certain number of "close contact" community interactions, not the likelihood of being infected by a COVID-19 positive individual over a certain number of interactions

It's actually very straight forward to calculate the "probability of being exposed to someone who is infectious with COVID-19" using publicly available data, based on the number of in-person, "close contact" interactions, as defined by the CDC:

Close contact is defined as being within 6 feet for at least a period of 10 minutes to 30 minutes or more depending upon the exposure. In healthcare settings, this may be defined as exposures of greater than a few minutes or more. Data are insufficient to precisely define the duration of exposure that constitutes prolonged exposure and thus a close contact.

I will illustrate this using Florida data, which I've analyzed before, using Miami-Dade as the region in my example. Of course, this comes with some assumptions around independence and homogeneity of the population you're interacting with. Data you need:

• # of confirmed positive cases in the past 10 days: 10,396
• 10 days being the cutoff since that is the current estimate as to how long individuals "with mild to moderate COVID-19 remain infectious"
• % of tests positive: 8.37%
• This data is usually available on a daily or weekly basis; weekly average suffices for this exercise
• Population in region: 2,716,940
• Census data, last yearly estimate
• # of "close contact" interactions: 10 (in this example)

First the number of "true infections" needs to be estimated. Youyang Gu provides this formula to estimate that:

• # of confirmed positives * (16 * sqrt(% positive) + 2.5) =
• 10,396 * (16 * sqrt(8.37%) + 2.5) = 74,110
• This implies roughly 86% of actual cases are not confirmed, or ~6 in 7 (in this example)

From here, presumably the the confirmed positive cases are quarantining as directed, leaving the unknown cases as potential interactions:

• # of unknown positive cases = # of true infections - # of confirmed positive cases
• 74,110 - 10,396 = 63,714

Next, calculate the percent of the population you're interacting with (in this case Miami-Dade County) that are unknowingly actively positive and likely infectious:

• % of population unknowingly actively positive and likely infectious = # of unknown positives / population
• 63,714 / 2,715,940 = 2.345%

An additional rough estimate for this would be to use the # of confirmed daily cases per 100,000 people like so:

• % of population (rougher estimate) unknowingly actively positive and likely infectious = (# of confirmed daily cases per 100,000 people * 5) / 10,000
• (46 * 5) / 10,000 = 2.3%

Finally, the probability that none of the people you had "close contact" with were unknowingly, actively infectious is:

• (1 - % population unknowingly positive) ^ # of interactions
• (1 - 2.345%) ^ 10 = 78.9%
• Conversely, 1 or more positive interactions = 1 - above
• 1 - 78.9% = 21.1%

Again, this does not estimate how likely it is to be INFECTED. The runaway nature of exponential growth is why eliminating superspreader events is so important; as an illustration, here's how the probability of exposure changes with higher incidence rates in the community vs number of interactions:

And here are the formulas described above:

where:

t = true infections in region (see below)

k = known infections in region (lab confirmed)

p = population of region

n = number of close contact in-person interactions (CDC definition)

t is estimated by:

where:

k = known infections in region (lab confirmed)

r = test positivity rate in region