Assessing the risk of cascading COVID-19 outbreaks from prison-to-prison transfers.

COVID-19 transmission has been widespread across the California prison system, and at least two of these outbreaks were caused by transfer of infected individuals between prisons. Risks of individual prison outbreaks due to introduction of the virus and of widespread transmission within prisons due to poor conditions have been documented. We examine the additional risk potentially posed by transfer between prisons that can lead to large-scale spread of outbreaks across the prison system if the rate of transfer is sufficiently high. We estimated the threshold number of individuals transferred per prison per month to generate supercritical transmission between prisons, a condition that could lead to large-scale spread across the prison system. We obtained numerical estimates from a range of representative quantitative assumptions, and derived the percentage of transfers that must be performed with effective quarantine measures to prevent supercritical transmission given known rates of transfers occurring between California prisons. Our mean estimate of the critical threshold rate of transfers was 27 individuals transferred per prison per month, with standard deviation 26, in the absence of quarantine measures. Available data documents transfers occurring at a rate of 61 transfers per prison per month. At that rate, estimates of the threshold rate of adherence to quarantine precautions had mean 61%, with standard deviation 32%. While the impact of vaccination and possible decarceration measures is unclear, we include estimates of the above quantities given reductions in the probability and extent of outbreaks. We conclude that the risk of supercritical transmission between California prisons has been substantial, requiring quarantine protocols to be followed rigorously to manage this risk. The rate of outbreaks occurring in California prisons suggests that supercritical transmission may have occurred. We stress that the thresholds we estimate here do not define a safe level of transfers, even if supercritical transmission between prisons is avoided, since even low rates of transfer can cause very large outbreaks. We note that risks may persist after vaccination, due for example to variant strains, and in prison systems where widespread vaccination has not occurred. Decarceration remains urgently needed as a public health measure.

Introduction

As the COVID-19 pandemic continues in the United States, its dynamics in congregate settings of heightened transmission, including prisons, is crucial to understanding its spread, addressing racial disparities in the burden of the disease, and strategizing effective control.

Prisons are often overcrowded, unsanitary, and provide poor health care, and have been the site of many of the most concentrated and brutal outbreaks of the pandemic so far. Prevention of prison outbreaks is essential because standard control measures such as social distancing and self-isolation are not generally available to prison residents. One in five prisoners in the United States has been infected with SARS-CoV-2, compared to one in 20 in the U.S. overall, more than 1700 have died, and prisoners continue to become infected (The Marshall Project, 2020). The New York Times reported on January 29, 2021 that of the ten largest outbreaks in U.S. correctional facilities to date, six of them have been in California state prisons (The New York Times, 2021). Every single one of California’s 35 prisons has reported 200 or more cases (Covid Behind Bars, 2021).

Likely routes of introduction of the disease into prisons are via infected prison staffers, admission of infected prison residents from outside the prison system, and transfers of residents from other prisons. A widely reported outbreak at San Quentin prison in California, which infected over 2200 of the 3563 inmates and killed 28, was caused by a transfer of prisoners from the Correctional Institute for Men in Chino, California (Office of the Inspector General, 2021), and a subsequent outbreak at California Correctional Center in Susanville, California was likely caused by transfer from San Quentin. Multiple outbreaks in winter 2020 appear to have been caused by importation via staff members.

Prison outbreaks can be very large – the outbreak at San Quentin infected over 60% of the prison population, and the outbreak in California’s Avenal State Prison topped 80% – and because of well-known inequities in the criminal justice system, they contribute to racial inequity in the burden of COVID infection (Fortuna et al., 2020, Okonkwo et al., 2020, Lee and Ahmed, 2021, Franco-Paredes et al., 2020). In October 2020, the California Court of Appeals ruled that the California Department of Corrections and Rehabilitation (CDCR) has been guilty of “deliberate indifference” and that California prison populations must be reduced by half to address the ongoing risk of SARS-CoV-2 transmission (Anon, 2020a). This decision reflects the recommendation to decarcerate California prisons to 50% of capacity published by public health experts during the San Quentin outbreak (The Amend Project, 2020). The decision is undergoing appeal, substantial decarceration has not occurred, and multiple outbreaks have occurred in California state prisons in the time since the decision.

While the risks of outbreaks sparked by staff introductions or transfers and spread within prisons due to poor conditions are well known, here we examine the potential danger from another, potentially less apparent risk: the possibility that transmission from prison to prison via transfer of prison residents may be sufficient to lead to uncontrolled spread across the prison system. If such conditions should occur, the disease could be expected to spread to substantially more prisons than otherwise, and infect far more individuals (Fig. 1). The risks to prison residents, staff, and surrounding communities could be considerably increased.

Fig. 1

Scenarios for transmission between individuals and between prisons. A. When the reproduction number between individuals – the mean number of cases caused by a case – is below the critical threshold of , transmission chains are short and outbreaks are small. B. When is above the critical threshold, a large outbreak is possible. C. When the reproduction number between prisons – the mean number of prison outbreaks caused by a prison outbreak – is below the critical threshold of , transmission between prisons may still occur, but spread between prisons will be relatively limited. D. When is above the critical threshold, transmission between prisons can cascade and cause spread throughout the prison system.

The CDCR currently has quarantine and testing policies in place to prevent transfer of infective individuals. Unfortunately, adherence to CDCR policies has not always been universal, and it cannot be assumed that no risky transfers occur.

We have addressed this question by using established theory of disease transmission, specifically a patch model to be defined below, to estimate the threshold rate of transfer associated with supercritical transmission between prisons, and the rate of adherence to transfer policies needed to prevent supercritical transmission at known rates of transfer.

Methods

Data

Numbers of Covid-19 cases in each CDCR facility were obtained from data from California Department of Corrections and Rehabilitation, collected by the UCLA Covid Behind Bars project (Covid Behind Bars, 2021). Overall population sizes as of March 26, 2021 were extracted directly from data released by CDCR (California Department of Corrections and Rehabilitation, 2021c) (California City Correctional Facility population extracted from California Department of Corrections and Rehabilitation (2021b))

Records released in the course of ongoing legal proceedings document the rate of transfers between prisons in the period from September 21 through October 11, 2020 at about 500 individuals transferred per week (Anon, 2020b p. 7, line 8). There are 35 institutions in the California prison system, making that equivalent to approximately transfers per prison per month.

Analysis

A patch model of disease transmission can model a collection of discrete populations in which transmission happens within a population, and at a separate rate between populations. One such approach, the so-called household model (Ball et al., 1997), assumes that the population is divided into many small groups (the households) in which local contacts occur frequently, whereas global contacts may occur between any two individuals in the population, albeit at a much lower rate.

In such a model, there are two types of outbreaks: local ones, in which the infection spreads widely within a single group, but remains confined to that group, and global outbreaks, in which the epidemic spreads among many groups. Global outbreaks are governed by a group-to-group reproduction number whose value is the expected number of groups infected by transmission from a single group: the critical value is , and a large global outbreak is possible if the value is greater than (Fig. 1).

For our purposes, we assumed that all contacts are local, within groups, except when a transfer occurs of an individual from one group to another. We assumed also that the rate of transfer is low enough that an individual will transfer to at most one group while infective. We found (see Appendix A) that the group-to-group reproduction number has the form where is the expected size of a major outbreak at a randomly chosen facility and is the probability that an individual will be transferred to a new site and cause a major outbreak in the new site (see Appendix A for details).

In order to evaluate how prison transfers affect prison-to-prison transmission, we modelled the group-to-group reproduction number in terms of the average number of individuals transferred between prisons. We expressed the probability in terms of the transfer rate, used empirical prison data to obtain upper and lower bound estimates for , and then solved for a threshold rate at which the critical value is equal to one.

We modelled an ensemble of scenarios for these quantities, to cover the range of possibilities. We characterize these as optimistic or pessimistic according to whether they will lead to a lower or higher estimate of within the current modelling framework.

1. Optimistic vs. pessimistic reproduction number. As an estimate of the basic reproduction number within a prison we used the value 8.44 (95% credible interval: 5.00–13.13) estimated from a COVID-19 outbreak in a large urban jail in the U.S. (Puglisi et al., 2020). Because this value is estimated from a setting in which a large outbreak occurred, and conditions in some prisons may be less conducive to transmission than those in which the largest outbreaks have occurred, we took the above number as a pessimistic estimate for . For an optimistic estimate, we calculated the probability that a transfer event leads to transmission between prisons using the more optimistic value of 2.87 (95% CI, 2.39–3.44) that was estimated for a basic reproduction number for COVID-19 in general community transmission (Arif Billah et al., 2020), and cut the probability in half to reflect the possibility that conditions may be better in roughly half of prisons. The factor of one half was chosen arbitrarily, to obtain a conservative bound.

2. Optimistic vs. pessimistic outbreak sizes. We constructed lower and upper bound estimates of the size-weighted mean final outbreak size from reported case counts in CDCR prisons. We calculated outbreak sizes to date by taking sequences of reported resident cases at a prison separated by 14 days or more of no cases as separate outbreaks. Because our model results exclude “outbreaks” that end after only a few cases, we exclude these from our estimation. In Appendix E, we estimate the mean and standard deviation of such small outbreaks by computing the total number infected in a branching process conditioned on extinction.

Using outbreaks of size 3 and larger (Figs. 2, 3), the size-weighted mean outbreak size was cases. Given the realities of asymptomatic infections, incomplete and imperfect testing, and under-reporting, this is likely to underestimate the true sizes of outbreaks. We thus took this value as an estimated lower bound of final outbreak sizes in California prisons. We take the mean overall population of each prison as a conservative upper bound for (see Appendix D), which was as of March 26, 2021.

Fig. 2

Sizes of COVID-19 outbreaks in California prisons as of March 26, 2021. Bars show the number of cases per outbreak (blue) for each outbreak of size 3 or more, and total population (grey) at each prison. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 3

Distribution of sizes of COVID-19 outbreaks in California prisons, up to size 50.

3. Optimistic vs. pessimistic secondary case distribution. Evidence is accumulating that transmission of SARS-CoV-2 has an overdispersed pattern, in which many people cause few or no infections and a relatively large number of infections are caused by a few people (Althouse et al., 2020, Adam et al., 2020, Susswein and Bansal, 2020). This pattern may make the probability lower than it could be, because relatively more people infect nobody, which reduces the likelihood of a major outbreak (Lloyd-Smith et al., 2005). We estimated given this pattern by assuming a negative binomial distribution of secondary cases (mean , shape = 0.5 (Susswein and Bansal, 2020)), as is standard. However, this overdispersed pattern may be caused partially by wide variation in the number of people contacted by individuals socially (Susswein and Bansal, 2020), and it is not clear that this variation in contact structure is possible to the same degree in a prison setting, where individuals’ movements and locations are heavily constrained and regulated. For this reason, we also considered the possibility that secondary cases may be Poisson distributed within the prison setting, though they are more highly dispersed in community transmission.

4. Optimistic vs. pessimistic timing of transmission events. We also considered that the way in which the timing of transmission events is distributed can affect the probability of transmitting to another prison. If transmission tends to occur in bursts, for example driven by exceptional events when multiple people gather, such a burst might happen either before or after an individual is transferred from prison to prison. If transmission events are independent and happen all at different times, on the other hand, it is more likely that at least one of them will occur after a transfer. We model both of these cases.

We used each combination of the above assumptions to estimate a threshold transfer rate for the California prison system, above which transfers may create a risk of global spread of the coronavirus across the prison system. Calculations detailed in Appendix C use branching process approximations to express the probability in terms of the rate of transfer between prisons per person per day for each combination of the above assumptions. These were used, together with our upper and lower bounds on , to numerically solve for the threshold rate at which under each scenario. We transformed to a threshold rate of transfers per prison per month, using conversion factors of 30 days per month and the average number of individuals per prisons in the CDCR system as of March 26, 2021.

We note that this threshold number of transfers concerns potentially infective transfers who are exposed to the prison resident population in the facility where they arrive. Prisons, of course, have policies for quarantine of transferred residents and for testing before transfer to prevent transfer of infected individuals, and these policies are likely to reduce the risk due to transfer.

Transfers between California state prisons are regulated by a policy called the movement matrix (California Department of Corrections and Rehabilitation, 2021a). Residents are tested five days before transfer, rapid tested one day before transfer, and quarantined for 14 days after transfer. Quarantine is in celled housing with a solid door if possible, and otherwise in cohorts of no more than four people. Residents in quarantine are screened for symptoms daily, tested if symptomatic, and isolated if they test positive. All of them are tested after five days post-transfer, again after 12 to 14 days post-transfer, and then released if negative and asymptomatic. Both residents and staff are to wear N95 masks during transfer. Residents who have been diagnosed with COVID-19 and subsequently resolved are considered immune for 90 days and exempt from quarantine and testing. After 90 days they are considered susceptible again and subject to the above measures.

We assume that these procedures are likely to reduce the risk of transmission by transfer substantially. However, compliance with safety policies may not be perfect. For example, the California Inspector General has documented extensive noncompliance with mask guidelines in the California prisons, including during the San Quentin outbreak (Office of the Inspector General, 2020). If protective policies reduce the number of transfers who can potentially transmit the virus by some percentage, it is the number of unprotected transfers that must be compared to the threshold value. If transfers are occurring at a known rate , and the threshold transfer rate for uncontrolled transmission between prisons is , then the percentage of transfers that must be conducted in adherence to the protective policy in order to reduce unprotected transfers to the threshold rate is , or zero if exceeds .

Vaccination and decarceration

Vaccination in the California prison system is underway, with CDCR reporting that 40% of prison residents have received COVID-19 vaccination. A recent legal filing reported that accounting for previously infected prisoners, 76% of incarcerated people may have immunity (Miller, 2021).

Both increasing immunity and decarceration are likely to affect the spread of infections in at least two important ways, firstly by reducing the reproduction number and relatedly the probability that an introduction leads to an outbreak, and second by reducing the sizes of outbreaks if they occur. Both of these changes will affect our estimates of the rate of transfers needed to produce cascading outbreaks.

We look at the relation between increasing immunity and the critical threshold for cascading outbreaks by estimating the threshold transfer rate and associated quarantine adherence rate, as above, while reducing the local reproduction number parameters discussed above () by half (which affects our estimate of how often a transfer causes an outbreak, but not of outbreak size), reducing the characteristic outbreak size by half, and reducing both by half simultaneously.

Results

We first estimated the threshold transfer rate () under all combinations of the above listed model assumptions, without protective measures (Table 1, Fig. 4). The values estimated for ranged from 3.6 to individuals transferred per prison per month, with mean and standard deviation . The generation time distribution used in these estimates was that estimated in a recent meta-analysis (Ferretti et al., 2020): a Weibull distribution with mean 5.5 days and standard deviation 1.8 days (parameters , ).

Table 1

Estimates of critical threshold for supercritical transmission between prisons in individuals transferred per prison per month and needed levels of adherence, given partial adherence with California’s transfer policy.

Optimistic μ	Optimistic R	Optimistic case distribution	Optimistic timing	n^∗	Threshold adherence
Y	Y	Y	Y	92.0	0
N	Y	Y	Y	20.0	67
Y	N	Y	Y	30.0	51
N	N	Y	Y	6.2	90
Y	Y	N	Y	48.0	21
N	Y	N	Y	10.0	83
Y	N	N	Y	22.0	64
N	N	N	Y	4.2	93
Y	Y	Y	N	77.0	0
N	Y	Y	N	18.0	71
Y	N	Y	N	24.0	62
N	N	Y	N	5.6	91
Y	Y	N	N	39.0	36
N	Y	N	N	9.6	84
Y	N	N	N	16.0	75
N	N	N	N	3.6	94

Fig. 4

Estimated threshold transfer rates and levels of adherence to policy. (Left) threshold number of individuals transferred per prison per month, under multiple scenarios (Table 1), (Right) rate of adherence to transfer policy needed to reach threshold number of transfers, under the assumption of 61 total transfers per prison per month. Box plots display median and inter-quartile range.

We converted our threshold estimates to the percentage of transfers that must be conducted in compliance with the safety policy in order to achieve the threshold number of unprotected transfers or below, given a total of transfers per prison per month (Table 1, Fig. 4). Our estimates of this threshold rate of adherence to quarantine precautions ranged widely but clustered in the upper third of the percentage scale, with mean % and standard deviation %.

As a look at the sensitivity of our estimates to reductions in risk due to vaccination and/or decarceration, we estimated the same quantities while reducing the probability of an outbreak, the size of outbreaks, or both, by half (Fig. 5). We estimated that while under our most optimistic assumptions the risk of cascading outbreaks is reduced substantially at transfers per facility per month, to the point where quarantine measures could be ignored entirely without exceeding the estimated threshold (which could of course cause substantial risks other than cascading outbreaks), the change in the median and mean estimates is much more modest. We estimated the mean threshold transfers per prison per month () at , , and respectively when reducing the assumed parameter by half for estimation of the probability of an outbreak, reducing the characteristic outbreak size by half, and both (standard deviation , , ). The mean estimate of threshold rate of adherence to policy was %, %, and % respectively, with standard deviation , , .

Fig. 5

Estimated threshold transfer rates and levels of adherence to policy given vaccination and/or decarceration, as in previous figure with reductions in reproduction number and/or outbreak size assumed. (Left) threshold number of individuals transferred per prison per month, under multiple scenarios as above, (Right) rate of adherence to transfer policy needed to reach threshold number of transfers, under the assumption of 61 total transfers per prison per month. Box plots display median and inter-quartile range.

Discussion

We have constructed a range of values of a threshold rate of mixing between prisons above which transmission between prisons is likely to be supercritical. Supercriticality between prisons means that a prison outbreak is expected to produce more than one other prison outbreak on average, potentially leading to uncontrolled spread throughout the prison system.

We estimate that the reported rate of transfers that has been occurring in the California prison system has likely exceeded this threshold. We estimate that at these rates of transfers, the quarantine precautions must be highly effective and rates of compliance must be high to avoid risk of supercritical transmission between prisons. The rate of outbreaks occurring in California prisons suggests that supercritical transmission may already have occurred or may be occurring.

We offer these estimates as a way of assessing one of the multiple risks posed by infectious disease transmission in the prison system. It is important to note that this threshold cannot be understood as providing a safe or acceptable rate of transfers, since transfer rates below the critical threshold can still cause huge outbreaks in multiple prisons. We are discussing an additional risk beyond the clear dangers of prisons’ unsafe conditions and spread due to transfers: the risk that in addition to having multiple large and deadly prison outbreaks of COVID-19, the rate of transfers could be sufficient to cause uncontrolled spread across a large portion of the prison system. This situation would likely lead to a great deal more harm to prison residents and staff than even the known risks of multiple large prison outbreaks, and could place communities throughout the state at risk as well.

The programme of vaccination that is underway in the California prison system is crucial in reducing transmission and saving lives, and will likely help to end the pandemic more broadly as prison transmission poses risks to communities beyond the prison walls. We caution that substantial risks may continue to exist in the CDCR system, as spread of the SARS-CoV-2 virus can still occur, prison conditions continue to be overcrowded and unsanitary, and the effects of variant strains are yet unknown, not to mention the other diseases currently circulating and the potential of future emerging pandemics. Our results and methods may also be relevant to other prison systems where vaccination is not yet widespread. Decarceration remains a crucial public health measure to bring disease spread under control.

While these estimates are necessarily imprecise due to limited availability of data, such that risks could in fact be lower than we have estimated, we note as well that in addition to the mechanism of transfer of prison residents considered here, transmission between prison facilities may also be occurring resulting from travel of infected staff who work at multiple facilities. For this reason, the risk of uncontrolled transmission between prisons may in fact have been higher than we have estimated here.

These results have a number of limitations. We have assumed that individuals are removed from the epidemic process at the end of their infective period, as we consider the final size of each local epidemic, and thus do not account for the possibility of reinfection. In using a branching process, we have implicitly assumed a very large number of local communities, so that at least initially, each global transmission is to a new site, and ignores the possibility of a second epidemic in the same location. This assumption is reasonable in the context of prisons, where there are indeed many sites. Branching processes are thus most relevant to modelling emerging pathogens, novel strains, and regions with lower rates of vaccination/acquired immunity. We have thus limited ourselves to retrospective conclusions about supercritical transmission in the California prison system; such methods could, however, inform decisions in the face of e.g. vaccine resistant variants. The assumption of a homogeneous rate of transfer per individual across all prisons may be limiting as heterogeneity may be important; with sufficient data on site-to-site transfer rates, this could potentially be addressed by dividing facilities into classes with class-specific transfer rates and using a multi-type branching processes, but that is beyond the scope of this study.

This approach is applicable to analysis of risk due to transmission between sites in a variety of hotspot settings of transmission including but not limited to prisons. Transfer, migration, and mixing between sites may be important sources of risk in other locations of high transmission as well, such as jails, ICE facilities, skilled nursing care facilities, meat packing plants, and other agricultural operations.

CRediT authorship contribution statement

Todd L. Parsons: Designed the research, Performed the research, Wrote the paper. Lee Worden: Designed the research, Performed the research, Wrote the paper.

Declaration of Competing Interest

Lee Worden received partial support from the office of the federal receiver, J. Clark Kelso, for research not including this project. Otherwise the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.