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Consider the following migration process based on a closed network of N queues with $K_N$ customers. Each station is a $\cdot$ /M/ $\infty$ queue with service (or migration) rate $\mu$. Upon departure, a customer is routed independently and uniformly at random to another station. In addition to migration, these customers are subject to a susceptible–infected–susceptible (SIS) dynamics. That is, customers are in one of two states: I for infected, or S for susceptible. Customers can swap their state either from I to S or from S to I only in stations. More precisely, at any station, each susceptible customer becomes infected with the instantaneous rate $\alpha Y$ if there are Y infected customers in the station, whereas each infected customer recovers and becomes susceptible with rate $\beta$. We let N tend to infinity and assume that $\lim_{N\to \infty} K_N/N= \eta $ , where $\eta$ is a positive constant representing the customer density. The main problem of interest concerns the set of parameters of such a system for which there exists a stationary regime where the epidemic survives in the limiting system. The latter limit will be referred to as the thermodynamic limit. We use coupling and stochastic monotonicity arguments to establish key properties of the associated Markov processes, which in turn allow us to give the structure of the phase transition diagram of this thermodynamic limit with respect to $\eta$. The analysis of the Kolmogorov equations of this SIS model reduces to that of a wave-type PDE for which we have found no explicit solution. This plain SIS model is one among several companion stochastic processes that exhibit both random migration and contagion. Two of them are discussed in the present paper as they provide variants to the plain SIS model as well as some bounds and approximations. These two variants are the departure-on-change-of-state (DOCS) model and the averaged-infection-rate (AIR) model, which both admit closed-form solutions. The AIR system is a classical mean-field model where the infection mechanism based on the actual population of infected customers is replaced by a mechanism based on some empirical average of the number of infected customers in all stations. The latter admits a product-form solution. DOCS features accelerated migration in that each change of SIS state implies an immediate departure. This model leads to another wave-type PDE that admits a closed-form solution. In this text, the main focus is on the closed stochastic networks and their limits. The open systems consisting of a single station with Poisson input are instrumental in the analysis of the thermodynamic limits and are also of independent interest. This class of SIS dynamics has incarnations in virtually all queueing networks of the literature. [ABSTRACT FROM AUTHOR]

BACKGROUND: Increasing rates of maternal opioid use disorder has led to greater number of opioid exposed newborns (OENs). Maternal enrollment in medication for opioid use disorder (MOUD) program improves short term neonatal outcomes. This study aimed at assessing neurobehavioral outcomes for OENs. METHODS: Retrospective observational cohort study of OENs between Jul 2006 and Dec 2018. Two study groups were identified as initiation of medication for opioid use disorder (MOUD) prior to diagnoses of pregnancy or after. Primary outcome variables were enrollment in and duration of EI services. Secondary outcome variable was diagnoses of a behavioral and/or developmental disorder (BDD) during the study period. RESULTS: Of 242 infants, 113 were enrolled in EI and BDD diagnoses data was available for all infants [age range 6 to 12 years], 82% infants had exposure to maternal MOUD, while 18% were exposed to either maternal prescription non-MOUD opioids or illicit opioids. Maternal MOUD initiation prior to pregnancy was associated with improved short term outcomes for OENs. Almost a third of infants were diagnosed with a BDD with no differences between the two study groups. CONCLUSION: Early initiation of maternal MOUD improved short term outcomes and discharge disposition for OENs. Prolonged in-utero exposure to opioids presents a potential for negative impact on neurodevelopmental and behavioral outcomes. These risks must be considered to increase access and adherence to EI services, as well as to focus on non-opioid based maternal MOUD. Longitudinal studies assessing the safety of MOUD on short and long-term child health outcomes are needed. [ABSTRACT FROM AUTHOR]