Mass RT-PCR testing of all inmates and partial testing of staff at MCI was conducted on April 16, 2020. The total number of inmates and the number of inmates and staff testing positive for SARS-CoV-2 over time were obtained from public ODRC reports (7). Results from early SARS-CoV-2 tests were available with a slight time-lag, so we accumulate the cases reported at MCI over April 16–23, 2020 as a single mass testing data point assigned to April 16, which was the date of mass testing. The mass testing event received significant media coverage and was reported widely in numerous news articles (4–6).

We use a compartmental susceptible-exposed-infectious-recovered (SEIR) model of SARS-CoV-2 dynamics in MCI. Such compartmental models have been used extensively in the literature because they tend to provide a good approximation to the process of disease spread (31). Assuming a well-mixed population, under the standard SEIR model, the proportions of individuals in the susceptible (S_t), exposed (E_t), infectious (I_t), and recovered (R_t) compartments as a function of time t satisfy the following system of differential equations:

where the positive parameters β, α, and γ denote the infection rate, incubation rate, and recovery rate, respectively.

We derive a likelihood function for observing n positives out of N incarcerated individuals on day u as follows: Using the DSA approach of (29, 30, 32, 33), we interpret S_t as an improper survival function. The mathematical justification for such an interpretation is provided by the Sellke construction by which the function S_t can be identified as the limiting probability of an initially susceptible individual not getting infected by time t. Note that the function S_t satisfying (1) is indeed a decreasing function and, when properly scaled, we set S₀ = 1. However, unlike proper survival functions that vanish at infinity (i.e., decrease to zero in the limit), the function S_t → S_∞ > 0 as t → ∞ so it is an improper survival function. However, we make it a proper survival function by conditioning on ever being infected. Given observation up to time T > 0, the time T_E that an initially susceptible individual becomes infected and enters the E compartment follows the conditional probability density function

where τ_T = 1 − S_T. The time T_I to becoming infectious has the conditional density

and the recovery time T_R has the conditional density

Note that the random variables T_E, T_I − T_E, and T_R − T_I are mutually independent and that T_I − T_E and T_R − T_I have exponential distributions with rates α and γ, respectively (29). The parameter ρ is the initial proportion of infectious individuals.

Mass testing yields a number of individuals who test positive and a total number of tests administered on the day of mass testing. To use these data, let T_N denote the time when virus first becomes undetectable in an individual. We then describe the epidemic process by the pair of random variables (T_E, T_N). Let ε = T_N − T_E. Then, the probability of an individual testing positive on the day of the mass testing (at time u) is given by

We fix ε = 21 days, corresponding to detectable virus for 3 weeks following an individual becoming infectious (34, 35). We set 1/α = 5.1/log(2) days [corresponding to a median incubation period of 5.1 days (36)] and assume a mean infectious period 1/γ of 5.6 days (37).

If n out of N individuals test positive on the day of the mass testing u, the log-likelihood function is given by

with the probability of testing positive p_u as described in Equation (5). Note that the above likelihood function is a consequence of the functional law of the large numbers for Poisson processes and the Sellke construction.

The crux of the DSA method is that it allows one to interpret functions that describe the large-population limiting proportions of individuals in different compartments as probabilistic quantities, such as survival functions or probability density functions of transfer times from one compartment to another. This change in perspective has a number of statistical advantages. For instance, it makes available the entire toolkit of survival analysis, by virtue of which it can account for censoring, truncation and aggregation of data in a natural way. Variations of the DSA method have been recently applied to analyze not only COVID-19, but also the 2001 foot-and-mouth disease (FMD) outbreak in the United Kingdom (30) and multiple waves of the 2018–2020 Ebola epidemic in the Democratic Republic of Congo (32). It is important to note that the date considered in this article are from the first phase of the pandemic when vaccines were not yet available. Nevertheless, the DSA method is capable to incorporating vaccination regimes. For instance, the method was applied to assess the potential impact of vaccination in Israel in (38). See also Klaus et al. (39) where the method was applied to COVID-19 data in the state of Ohio, USA.

According to ODRC reports (7), the first identified COVID-19 case at MCI was an infected staff member on March 29. Following this initial case, precautions such as cohorting and modified movement were enacted in order to restrict mixing and reduce transmission. As stated in the publicly available ODRC report from March 30, 2020:

The first COVID-19 case among inmates was identified on April 3. Mass RT-PCR testing of all inmates and partial testing of staff was conducted on April 16. By April 20, SARS-CoV-2 infection had been identified in 79% (1,950/2,453) of inmates and 35% (154/446) of staff. These numbers come directly from data on the ODRC website. We take the May 5, 2020 listing of 2,453 inmates at MCI as the denominator. There is a lag of a few days between when mass testing occurred (April 16) and when jumps in case counts are reported in the ODRC data (April 18–19 for inmates and April 20 for staff), which may reflect delay in data entry. Figure 1 shows a time series of reported COVID-19 cases at MCI.

The basic reproduction number R₀ is one of the key parameters in models of infectious diseases (31). It is defined as the expected number of secondary cases generated by an infected individual in a population where all individuals are susceptible to infection. When R₀ > 1, disease can spread rapidly and cause a large epidemic with positive probability. When R₀ < 1, the spread of disease dies out stochastically and a large epidemic cannot occur. R₀ can also be used to calculate the so-called “herd immunity threshold” for interventions like vaccination that effectively reduce the susceptible population.

To examine which values of the basic reproduction number R₀ are consistent with the rapid spread of COVID-19 observed at MCI, we use the SEIR model (1). In order for mass screening to identify 80% of the population as positive for COVID-19 on April 16, at least 80% of the population must have been infected by that date. Figure 2 shows the time needed to infect 80% of the population in the SEIR model as a function of R₀ and the initial number of exposed individuals (E₀). While R₀ values of two or larger are able to eventually infect 80% or more of the population, this can take on the order of months for modest values of R₀. Reproduction numbers >14 are needed before outbreaks originating from a single exposed individual are able to generate a 3-week cumulative incidence consistent with that reported for MCI.

An alternative explanation is that the outbreak involved more than one initially infected prisoner. Figure 2 shows that, for a fixed R₀ value, increasing E₀ decreases the time needed to infect 80% of the population. However, an initial condition of E₀ > 563 is needed for an outbreak with R₀ = 3 to infect 80% of the population within 3 weeks.

A third possibility is that SARS-CoV-2 was circulating among prisoners prior to April 3. Figure 3 shows the log-likelihood (6) for observing the mass testing results in MCI according to the SEIR model (1) as a function of R₀ and the outbreak onset date, with E₀ fixed at one. The outbreak onset date and R₀ are unidentifiable from the mass testing data alone, with the “wishbone” shape running diagonally across Figure 3 corresponding to pairs of outbreak onset and R₀ that are almost equally likely given the observed data. Outbreak onsets in late March or later correspond to R₀ > 10, while earlier outbreak onsets correspond to smaller R₀ values. Note that onset dates prior to March are required to give R₀ values of less than five.

In general, the larger the value of the parameter R₀, the more difficult it is to control the epidemic. Our analysis is consistent with this. Both the first and the third possibilities explained above suggest that the R₀ values consistent with the reported data must be extremely high, calling attention to the explosive potential for COVID-19 transmission in prisons. Both the second and the third possibilities underscore the implausibility of the reported disease introduction date and/or the initial amount of infection, calling attention to the need for more reliable monitoring and reporting of infection in correctional facilities.