The Dynamics of Respiratory Infectious Diseases in Incarcerated and Free-Living Populations: A Simulation Modeling Study

Abstract

Background:

Historically, correctional facilities have had large outbreaks of respiratory infectious diseases like COVID-19. Hence, importation and exportation of such diseases from correctional facilities raises substantial concern.

Methods:

We developed a stochastic simulation model of transmission of respiratory infectious diseases within and between correctional facilities and the community. We investigated the infection dynamics, key governing factors, and relative importance of different infection routes (e.g., incarcerations and releases versus correctional staff). We also developed machine learning metamodels of the simulation model, which allowed us to examine how our findings depended on different disease, correctional facility, and community characteristics.

Results:

We find a magnification-reflection dynamic: a small outbreak in the community can cause a larger outbreak in the correction facility, which can then cause a second, larger outbreak in the community. This dynamic is strongest when community size is relatively small as compared to the size of the correctional population, the initial community R-effective is near one, and initial prevalence of immunity in the correctional population is low. The timing of the correctional magnification and community reflection peaks in infection prevalence are primarily governed by the initial R-effective for each setting. Because the release rates from prisons are low, our model suggests correctional staff may be a more important infection entry route into prisons than incarcerations and releases; in jails, where incarceration and release rates are much higher, our model suggests the opposite.

Conclusions:

We find that across many combinations of respiratory pathogens, correctional settings, and communities, there can be substantial magnification-reflection dynamics, which are governed by several key factors. Our goal was to derive theoretical insights relevant to many contexts; our findings should be interpreted accordingly.

Background

Correctional facilities have historically had high rates of respiratory infectious diseases.¹ For example, the tuberculosis (TB) incidence rate ratio between prisoners and the general population has been estimated to be 4.1 and 26.9 for the North American and South American regions, respectively;² the COVID-19 incidence rate ratio between prisoners and the general population has been estimated to be over 5 in the United States.^3,4 As such, there is substantial concern regarding importation and exportation of such diseases from correctional facilities.^1,5,6

Despite the concern, there has been relatively little empirical research on this topic. Several studies have examined the relationship between infectious diseases in correctional facilities and the general population in specific contexts using ecological approaches. For example, Reinhart and Chen analyzed data on COVID-19 cases in Cook County Jail and the general population in Chicago; they found jail-community cycling was a significant predictor of community cases at the ZIP code level.^7,8 The authors also analyzed county-level data for the US on COVID-19 cases and jail incarceration rates; they found a negative association between jail decarceration and cases in the general population.⁹ Kajeepeta et al. analyzed county-level data for the US on cause-specific mortality and jail incarceration rates; they found a positive association between jail incarceration rate and mortality from infectious diseases in the general population.¹⁰

Several simulation modeling studies have examined this relationship in specific contexts. Basu et al. modeled TB transmission within and between an institution, such as a correctional facility, and the general population and evaluated the effectiveness of control strategies.¹¹ Mabud et al. combined observational and simulation modeling approaches to model TB transmission within and between prisons and communities in Brazil and evaluate the effectiveness of control strategies.¹² Other studies have examined this relationship for non-respiratory infectious diseases such as hepatitis C virus (HCV) and human immunodeficiency virus (HIV).¹³ For example, Stone et al. used simulation modeling to assess how incarceration contributed to HCV incidence in people who inject drugs (PWID) in Perry County and evaluated the effectiveness of control strategies.¹⁴ In all these specific contexts, the authors identified the potential for substantial spillover of infections from populations in correctional facilities to those in communities.

Recently, a study examined this relationship using a genomic approach in a specific context. Walters et al. amassed genomic epidemiologic data to quantify spillover of TB from prisons in Brazil.¹⁵

While these studies make important contributions in relationship to the specific diseases and contexts they analyze, it is important to understand the infection dynamics of systems in which communities are linked to congregate settings more generally. Specifically, it is unclear in which contexts infection transmission between correctional facilities and communities might be most problematic, what factors govern this relationship, and the relative importance of different routes of infection introduction (e.g., incarcerations and releases versus correctional staff). To address these knowledge gaps, we developed a stochastic simulation model and used it to investigate these questions across diseases, correctional facilities, and communities. The goal of the present analysis was to derive theoretical insights relevant to many contexts. As such, our results should not be interpreted as forecasting infections in a specific context but rather as illustrating contexts in which transmission between correctional facilities and communities may be particularly problematic and the factors that influence the strength of this relationship.

Methods

Overview

We developed a stochastic simulation model of infections within and between a correctional facility and community. We used this model to examine the size, timing, and moderators of a magnification-reflection dynamic across different diseases, correctional facilities, and communities. We also developed metamodels of the simulation model. We used these metamodels to conduct probabilistic sensitivity analyses and interpret the simulation model. All analyses were programmed in R version 4.0.3,^16,17 with input data and statistical code for replication and extension of our analysis available via GitHub concurrent with publication.

Simulation model

We developed a stochastic susceptible-infectious-detected-recovered (SIDR) compartmental model¹⁸ of infections within and between a correctional facility and community (Figure 1). We tracked the susceptible, infected, detected, and recovered counts for three populations including (1) incarcerated people, (2) correctional staff, and (3) the rest of the free-living population.

Figure 1/. Model schematic.

We developed a stochastic model of infections within and between a correctional facility and community using the tau-leap method. We considered three populations: (1) inmates, (2) correctional staff, and (3) community members. For these populations, we tracked health states including susceptible (S), infected (I), detected (D), and recovered (R). Epidemiological parameters governed disease progression shown by the solid arrows. Movement parameters governed movements between populations, shown by the dashed arrows, including the incarcerations and release rate (e) and the fraction of time staff spend in the correctional facility (f) versus in the community.

For population movement, correctional staff spend some fraction of their time (f) at the correctional facility versus in the community. Incarcerations and releases occur at some rate (e) by which people are transferred between the correctional facility and free-living population.

For transitions between health states, infections occur based on homogeneous mixing of the populations at the correctional facility and community, respectively, based on betas (β). The betas account for both the contact rate and probability of infection per contact. Detections occur based on detection rates (d); detected people have substantially reduced risk of transmitting to others and are ineligible for incarceration/release until their infections resolve. The detection rate is based on the incubation period of the disease accounting for asymptomatic people. Recoveries from the infected and detected states occur based on recovery rates (ν).

Model outcomes included the disease prevalence over time and total number of infections with and without connection between the community and the correctional facility. They also included the time to the correctional facility magnification peak and time to the community reflection peak. The time to correctional facility magnification peak is the time until correctional infection prevalence is at a maximum over the model period. The time to community reflection peak is the time until community infection prevalence is at a maximum from the correctional facility magnification peak time until the end of the model period.

We simulated outcomes for a correctional facility and community for 1 year using a time step of a tenth of a day. The number of people simulated depended on the size of the correctional facility and community, which varied across analyses. Since the model is stochastic, we simulated outcomes 200 times per analysis and results are based on the outcome distributions. We used the tau-leap method, which is an approximate method for stochastic simulation based on Gillespie’s algorithm.¹⁹ For these sets of analyses, there was no parameter uncertainty (i.e., parameters were fixed at particular levels). Table S1 demonstrates this number of simulations (200) is sufficient to achieve stable estimates of expected outcomes.

Model inputs

We obtained estimates of model parameters and associated measures of uncertainty from the published literature (Tables S2–S4). We considered six diseases including COVID-19 caused by the SARS-CoV-2 wild-type strain, COVID-19 caused by the SARS-CoV-2 Delta variant, pertussis, SARS, and smallpox. For each disease, we obtained estimates of β, the incubation period, percent asymptomatic infections, and ν. In the base case analysis, β for the correctional facility was twice that of the community based on prior literature; however, we considered a wide range of alternate scenarios in one-way and probabilistic sensitivity analyses. For correctional facilities, we considered a prison and a jail. Typically, a jail is a smaller-sized facility with fewer incarcerated people and correctional staff, and has a higher rate of incarcerations and releases than a prison. For each of these correctional settings, we obtained estimates of the typical number of incarcerated people, correctional staff, and incarcerations and releases per day. For communities, we obtained an estimate of the number of people in a typical zip code. In all cases, we seeded the community but not the correctional facility with initial infections.

Simulation model analyses

We performed several analyses for each of the 10 cases (combinations of one of five diseases with a community linked to a prison or jail). First, we examined the existence of a magnification-reflection dynamic. We projected the expected per capita infection prevalence over the model period when the community and correctional facility are connected versus not connected.

We next performed one-way sensitivity analyses to identify key factors governing the magnitude of the magnification-reflection dynamic, time to the correctional facility magnification peak, and time to the community reflection peak. The magnitude of the magnification-reflection dynamic was equal to the increase in per capita infections over the model period when the community and correctional facility are connected versus not connected. We projected the expected outcomes when each parameter value was varied one at a time.

Then, we examined the relative importance of different infection pathways by which the magnification-reflection dynamic occurs (e.g., staff introducing infections vs. infection introductions via incarcerations and releases). We projected the reduction in expected total infections over the model period with no resident movement (i.e., no incarcerations and releases), no staff movement, and neither movement.

Metamodel analyses

The metamodels served two primary purposes. First, they enabled probabilistic sensitivity analyses to be conducted across diseases, correctional facilities, and communities. Since the stochastic simulation model must be run many times for each parameter set to get stable estimates, such analyses would otherwise be very computationally costly. In all metamodel analyses, we account for both first and second order uncertainty. Second, they allowed for interpretation of the simulation model.

To develop the metamodels, we generated datasets using the simulation model that subsequently enabled metamodel training and evaluation. Uniform distributions were constructed for disease, correctional facility (for prisons and jails separately), and community parameters based on the literature (Table S3). These distributions were sampled 50,000 times using orthogonal sampling²⁰ for prisons and jails, respectively, to create training datasets. Likewise, they were sampled 10,000 times to create validation and test datasets. For all datasets, we used the simulation model to simulate outcomes 50 times per parameter set. Outcomes included the mean per capita difference in the number of infections with and without connection between the community and the prison or jail, mean time to the correctional facility magnification peak, and mean time to the community reflection peak. Outcomes also included the mean number of infections with normal population movement, no resident movement (i.e., no incarcerations and releases), no staff movement, and neither movement.

We created a total of 14 metamodels including two for each of the seven outcomes for the prison and jail cases. For each case, we trained candidate metamodels that have previously been suggested as potentially useful for health services research²¹ including generalized linear models, deep learning models, random forests, and gradient boosting models using the training dataset. Hyperparameter tuning was conducted via grid search using the validation dataset. The best model for each case was selected based on performance on the test dataset as determined by the mean absolute error (MAE). All models were trained and evaluated using the H20 package in R. Table S5 provides additional details on the models and hyperparameters considered.

We interpreted the metamodels by constructing partial dependence plots (PDPs) and variable importance plots. These plots provide information on the marginal effects of the variables on the outcomes and their relative importance in predicting outcomes. To generate these, we used disease-specific datasets, which we created separately. For disease-specific parameters, we sampled from disease-specific distributions (Table S4). For correctional facility and community parameters, we used the same sampling scheme as we did to generate the datasets to train the metamodels. We also used the metamodels and these disease-specific datasets to examine the degree to which different infection pathways influenced the outcomes across diseases, correctional facilities, and communities via probabilistic sensitivity analysis.

Results

Simulation model

Results for the existence of a magnification-reflection dynamic and its key drivers were generally consistent across diseases and correctional setting (i.e., prisons versus jails). As such, we focus here on the results for the SARS-CoV-2 wild-type strain and prison case. Results from the other cases are included in the supplement. Despite their similarity, there were some differences, which we discuss.

We projected expected per-capita COVID-19 prevalence over time with and without connection between a prison and community (Figure 2). With connection, infections can be transmitted between the populations. We find that, without connection, the community outbreak dies out over time. However, with connection, the community outbreak causes a large prison outbreak, which then causes a second, larger community outbreak. We term this the magnification-reflection dynamic as infections imported from the community are magnified in the prison and then reflected back into the community. We repeated this analysis for the other cases (Figure S1), all of which demonstrated magnification-reflection dynamics to some extent. These dynamics were stronger for those cases with an initial community R_e value approximately equal to one, which is explored further in the subsequent analyses.

Figure 2/. Existence of a magnification-reflection dynamic in a specific setting.

For our base case analysis of COVID-19 caused by the SARS-CoV-2 wild-type strain in a prison and community the size of a typical zip code, we projected the expected per capita infection prevalence over time with and without connection between the prison and community. With connection, infections can be transmitted between the populations. Time refers to the days following the first infection in the community. This dynamic also exists for other diseases and settings (Supplement S1).

To identify key factors governing the magnification-reflection dynamic, we conducted one-way sensitivity analyses in which we varied each parameter over ranges to reflect the heterogeneity in correctional facilities and communities. We projected the difference in expected per capita COVID-19 prevalence over time with and without connection between the prison and community. See Figure S2A for a tornado diagram summarizing the effects of all factors. Three factors are especially important in determining the magnitude of the magnification-reflection dynamic: community size, initial community R_e, and initial prevalence of immunity in the correctional population (Figure 3). The magnification-reflection dynamic is greatest when: (1) community size is small since the share of total incident infections that comes from the correctional facility is largest; (2) initial community R_e value is approximately equal to one as the community epidemic is then shifted from an extinction regime to a growth regime by the increase in incidence from the prison; and (3) initial prison immunity is low such that the prison epidemic grows faster and larger. For the other cases we considered, the overall magnitude of the magnification-reflection dynamic differed but the key governing factors were generally consistent (Figures S2A, Figure S3).

Figure 3/. Drivers of the magnification-reflection dynamic.

For our base case analysis of COVID-19 caused by the SARS-CoV-2 wild-type strain in a prison and community the size of a typical zip code, we varied parameters one at a time to reflect diversity in prisons and communities. We projected the difference in expected infection prevalence (per 100k people) over time with and without connection between the prison and community. Shown here are three key drivers. These drivers were consistent for other diseases and settings (Supplement S2).

We also conducted one-way sensitivity analyses for the expected time of the correctional magnification and community reflection peaks. See Figure S2B for tornado diagrams summarizing the effects of all factors on the expected time of the correctional magnification peak across cases. The timing of this peak was largely governed by factors dictating the initial R_e values in the correctional facility and community. Other factors, such as population sizes and the incarceration and release rate, had relatively little effect. See Figure S2C for tornado diagrams summarizing the effects of all factors on the expected time of the community reflection peak across cases. As with the timing of the correctional magnification peak, the timing of this peak was largely governed by factors dictating the initial R_e values in the correctional facility and community. In no case did the timing of the correctional magnification and community reflection peaks coincide.

To investigate the relative importance of different infection routes by which the magnification-reflection dynamic occurs, we projected the percent reduction in expected infections with no incarcerations and releases, with no staff movement, and with neither (Figure 4). Across diseases, for prisons, eliminating staff movement results in a much greater reduction in infections than eliminating incarcerations and releases because prisons have large staff populations and release rates from prison are low due to long prison sentences. However, the opposite occurs for jails because jails have smaller staff populations and stays in jails are much shorter than in prisons. For both prisons and jails, the two routes increase each other’s effect. Hence, when neither operates, a greater reduction in infection occurs than the sum of the reductions from the routes individually.

Figure 4/. Relative importance of infection routes.

For various diseases and settings, we projected the percent reduction in expected infections with no resident movement (i.e., no incarcerations and releases), no staff movement, and neither movement. The error bars show 90% credible intervals. Abbreviations: COVID-19 WT, COVID-19 caused by the SARS-CoV-2 wild-type strain; COVID-19 Delta, COVID-19 caused by the SARS-CoV-2 Delta variant.

We found some differences across diseases in terms of the effects of eliminating movements. The effects of eliminating movements were greatest for those diseases with an initial community R_e value approximately equal to one. The effects of eliminating both movements as opposed to one alone was greatest for those diseases with high initial community R_e values. This likely occurs as, for these diseases, each infected person tends to infect many others so even the less influential movement route can still facilitate importation of the disease into the correctional facility.

Metamodels

All metamodels performed well based on the MAE and root mean squared error (RMSE) (Table S6). Table S7 shows the selected metamodels and hyperparameters. Using the metamodels to examine governing factors of the magnification-reflection dynamic across diseases, correctional facilities, and communities—accounting for uncertainty in each—we find that those identified above for the exemplar diseases and settings remain robust.

The direction and effect sizes from the PDPs for the magnitude of the magnification-reflection dynamic were consistent with those identified from the one-way sensitivity analyses (Figure 5 for select variables, Figure S4A for all variables). The magnification-reflection dynamic was strongest when community size was small, the initial community R_e was near one, and initial prevalence of immunity in the correctional population was low. For community size, the effect was highly non-linear and the magnitude of the magnification-reflection dynamic increased rapidly when community size decreased below roughly 15 or 70 times the size of the prison or jail population, respectively. The influence of number of staff was more substantial than the incarceration and release rate for prisons; however, the opposite was true for jails. The partial dependence plots for peak timings were also consistent with the one-way sensitivity analyses (Figures S4B–C). In both cases, peaks occurred later when the community R_e was near one and the correctional R_e was around one-and-a-half.

Figure 5/. Partial dependence plots.

We trained machine learning models to predict the difference in expected infections (per 100k people) with and without connection between the correctional facility and community for prisons and jails, respectively. We generated partial dependence plots, which show the marginal effect of each variable on the mean response (i.e., mean model outcome, which is in units of infections per 100k people). Shown here are results for the three key variables from Figure 3 in addition to staff population and incarceration and release rate. See Supplement S4A for partial dependence plots for all variables. Abbreviations: COVID-19 WT, COVID-19 caused by the SARS-CoV-2 wild-type strain; COVID-19 Delta, COVID-19 caused by the SARS-CoV-2 Delta variant.

The influence of different variables from the variable importance plots was consistent with both the one-way sensitivity analyses and partial dependence plots for the magnitude of the magnification-reflection dynamic and peak timings (Figure S5).

Finally, the relative importance of staff transmission for prisons and incarcerations and releases for jails in driving the magnification-reflection dynamic remained robust during probabilistic sensitivity analysis using the metamodels (Figure S6).

Discussion

There can be a substantial magnification-reflection dynamic across many combinations of respiratory pathogens, correctional settings, and communities. This dynamic is strongest when community size is relatively small as compared to the size of the correctional population, the initial community R_e is near one, and initial prevalence of immunity in the correctional population is low. Both the correctional magnification and community reflection peaks occur later when the initial correctional R_e is near one-and-a-half and community R_e is near one. Under a wide range of plausible assumptions, staff are a more important infection entry route into prisons than incarcerations and releases; however, for jails, the relative importance of the entry routes is reversed.

We extend prior literature by providing conceptual understanding of the dynamics of respiratory epidemics in incarcerated and free-living populations. Previous empirical studies of specific pathogens and settings document relationships between case rates in correctional facilities and epidemiological outcomes in communities to which they are connected; likewise prior models have focused on particular combinations of pathogens, correctional settings, and communities.^7–15 Specifically, by using a model and exploring its dynamics systematically across a wide range of parameters representing pathogens, correctional settings, and communities, we identify a magnification-reflection dynamic, key governing factors, and the relative importance of different infection routes. These dynamics are particularly concerning with novel respiratory infectious diseases as populations are at substantially elevated risk due to low levels of initial immunity from the absence of past outbreaks and vaccination. However, these dynamics can also occur with existing respiratory infectious diseases, particularly in the high-risk settings we identified. Our findings have relevance for those using models for prediction and for control. Existing models typically consider correctional facilities or the general population but not both. However, as we have shown, these systems are not isolated and what happens in one can have important implications for the other.

Our analysis has several limitations. First, we assumed homogenous mixing of the populations in the correctional facility and community, respectively. In reality, people likely have specific contact networks. Second, we did not account for previously-incarcerated people being at elevated risk of reincarceration. However, since our model time horizon was relatively short, this is unlikely to qualitatively influence our results. Third, while we analyzed an isolated system of a single correctional facility connected to a single community, this system actually exists within a broader context of other correctional facilities and communities. Such simplifications allowed us to observe general trends and generate insights across diseases, correctional facilities, and communities. However, due to them, our models should not be used to provide infection projections in specific contexts for planning purposes.

For future work, it would be helpful to include possible magnification-reflection dynamics in quantitative prediction and control of infectious diseases in specific contexts. To our knowledge, this rarely occurs in practice. However, our findings suggest correctional facilities and communities may want to account for the status of the other in their projections and planning. In this way, they may be able to prevent outbreaks before they occur or make them less severe. It would also be helpful for future studies to generate additional data on model parameters. For example, there is relatively little data on betas in different correctional settings. Finally, it would be helpful for future studies to test our theoretical modeling results using empirical data.

To conclude, we find that across many combinations of respiratory pathogens, correctional settings, and communities, outbreaks in correctional facilities and communities can echo back and forth via the magnification-reflection dynamic, which is governed by several key factors. This finding has important implications for policy makers and public health departments that seek to predict and control the spread of infectious diseases. Moreover, it highlights how it is important to not think of prisons and jails as isolated systems—with infectious diseases, the health of community members depends on that of all other community members including those incarcerated in prisons and jails.

Highlights.

• We find a magnification-reflection dynamic: a small outbreak in a community can cause a larger outbreak in a correction facility, which can then cause a second, larger outbreak in the community.

• For public health decision makers considering contexts most susceptible to this dynamic, we find that the dynamic is strongest when the community size is relatively small, initial community R-effective is near one, and initial prevalence of immunity in the correctional population is low; the timing of the correctional magnification and community reflection peaks in infection prevalence are primarily governed by the initial R-effective for each setting.

• We find correctional staff may be a more important infection entry route into prisons than incarcerations and releases; however, for jails, the relative importance of the entry routes may be reversed.

• For modelers, we combine simulation modeling, machine learning metamodeling, and interpretable machine learning to examine how our findings depend on different disease, correctional facility, and community characteristics; we find they are generally robust.