We built a new stochastic model, in particular a continuous-time Markov chain (CTMC) , of COVID-19 transmission to estimate the impact of correctional facilities on disease transmission and re-emergence (Web Figure 1). We calibrated our model to describe the spread of COVID-19 in communities with correctional facilities that house 800 inmates and employ 420 correctional officers, based on the configuration of the largest correctional facilities in New County County. York. . For such a facility, we measured how testing and quarantining infected inmates at 0.0, 0.5, 1.0 times the rate that occurs for the general population impacts the spread of COVID -19 within the correctional facility and within the local community. Additionally, to reflect the different population densities of rural and urban communities near correctional facilities, we also considered small, medium, and large communities of 5,000, 10,000, and 20,000 people, respectively, in addition to the period average incarceration of 25 days in prisons  2.6 years in state prisons , as well as the effects of social distancing and vaccination (Web Annex 1). For each scenario, we calculated 10,000 stochastic realizations over 1 year using Gillespie’s algorithm.
In the stochastic model, we considered a separate population as the community (C), incarcerated persons (P), and correctional workers (W), where correctional workers are defined as civilian employees or volunteers who reside at the outside the correctional facility. We further subdivided each section of the population by COVID-19 infection status using indices to denote susceptible (S), latently infected (E), asymptomatic infected (A), symptomatic infected (I), recovered from infection (R), acquired immunity from vaccination (V), hospitalized due to infection (H), premature death due to infection (D) and quarantined (Q ).
To account for the difference in transmission risks between community members, correctional officers, and incarcerated individuals, our model included different rates of COVID-19 infection. The rate of individuals susceptible to contracting COVID-19 in the community is given by the strength of infection:
or βC.W. is the rate of transmission of COVID-19 in the community, VSearly is the total size of the community, Oearly is the total size of correctional workers, and α(you) is the impact of public health control measures, such as face masks and social distancing, on mitigating the spread of COVID-19 in the community. Similarly, the rate of acquisition of COVID-19 by susceptible correctional workers is given by the strength of infection:
or βWP is the transmission rate of COVID-19 among correctional workers, and Pearly is the total size of the prison population. Finally, the rate of incarcerated people likely to acquire COVID-19 is given by the strength of infection:
To reflect the influence of social distancing efforts on community transmission of COVID-19, we considered distinct phases of 1) presocial distancing and 2) social distancing, as well as the introduction of vaccination. These phases are reflected in the rate of new infections occurring across the function