In this paper, we show that solutions to ordinary differential equations describing the large-population limits of Markovian stochastic epidemic models can be interpreted as survival or cumulative hazard functions when analysing data on individuals sampled from the population. We refer to the individual-level survival and hazard functions derived from population-level equations as a survival dynamical system (SDS). To illustrate how population-level dynamics imply probability laws for individual-level infection and recovery times that can be used for statistical inference, we show numerical examples based on synthetic data. In these examples, we show that an SDS analysis compares favourably with a complete-data maximum-likelihood analysis. Finally, we use the SDS approach to analyse data from a 2009 influenza A(H1N1) outbreak at Washington State University.

In this paper we derive several quasi steady-state approximations (QSSAs) to the stochastic reaction network describing the Michaelis-Menten enzyme kinetics. We show how the different assumptions about chemical species abundance and reaction rates lead to the standard QSSA (sQSSA), the total QSSA (tQSSA), and the reverse QSSA (rQSSA) approximations. These three QSSAs have been widely studied in the literature in deterministic ordinary differential equation (ODE) settings and several sets of conditions for their validity have been proposed. By using multiscaling techniques introduced in [1, 2] we show that these conditions for deterministic QSSAs largely agree with the ones for QSSAs in the large volume limits of the underlying stochastic enzyme kinetic network.In chemistry and biology, we often come across chemical reaction networks where one or more of the species exhibit a different intrinsic time scale and tend to reach an equilibrium state quicker than others. Quasi steady state approximation (QSSA) is a commonly used tool to simplify the description of the dynamics of such systems. In particular, QSSA has been widely applied to the important class of reaction networks known as the Michaelis-Menten models of enzyme kinetics [3,4,5].Traditionally the enzyme kinetics has been studied using systems of ordinary differential equations (ODEs). The ODE approach allows one to analyze various aspects of the enzyme dynamics such as asymptotic stability. However, it ignores the fluctuations of the enzyme reaction network due to intrinsic noise and instead focuses on the averaged dynamics. If accounting for this intrinsic noise is required, the use of an alternative stochastic reaction network approach may be more appropriate, especially when some of the species have low copy numbers or when one is interested in predicting the molecular fluctuations of the system. It is well-known that such molecular fluctuations in the species with small numbers, and stochasticity in general, can lead to interesting dynamics. For instance, in a recent paper [6], Perez et al. gave an account of how intrinsic noise controls and alters the dynamics, and steady state of morphogen-controlled bistable genetic switches. Stochastic models have been strongly advocated by many in recent literature [7,8,9,10,11,12]. In this paper, we consider such stochastic models in the context of QSSA and the Michaelis-Menten enzyme kinetics and relate them to the deterministic ones that are well-known from the chemical physics literature.The QSSAs are very useful from a practical perspective. They not only reduce the model complexity, but also allow us to better relate it to experimental measurements by averaging out the unobservable or difficult-to-measure species. A substantial body of work has been published to justify such QSSA reductions in deterministic models, typically by means of perturbation theory [13,14,15,16,17,18]. In contrast to this approach, we derive here the QSSA reductions using stochastic multiscaling techniques [1,2]. Although our approac...

We present a new method for analysing stochastic epidemic models under minimal assumptions. The method, dubbed dynamic survival analysis (DSA), is based on a simple yet powerful observation, namely that population-level mean-field trajectories described by a system of partial differential equations may also approximate individual-level times of infection and recovery. This idea gives rise to a certain non-Markovian agent-based model and provides an agent-level likelihood function for a random sample of infection and/or recovery times. Extensive numerical analyses on both synthetic and real epidemic data from foot-and-mouth disease in the UK (2001) and COVID-19 in India (2020) show good accuracy and confirm the method’s versatility in likelihood-based parameter estimation. The accompanying software package gives prospective users a practical tool for modelling, analysing and interpreting epidemic data with the help of the DSA approach.

ImportanceIncarcerated individuals are a vulnerable population for severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infection. Understanding SARS-CoV-2 dynamics in prisons is crucial for curbing transmission both within correctional facilities and in the surrounding community.ObjectiveThe purpose of this study was to identify transmission scenarios that could underly rapid, widespread SARS-CoV-2 infection among inmates in Marion Correctional Institution (MCI).DesignPublicly available data reported by the Ohio Department of Rehabilitation and Corrections (ODRC) was analyzed using mathematical and statistical models.SettingWe consider SARS-CoV-2 transmission dynamics among MCI inmates prior to and including April 16, 2020.ParticipantsThis study uses de-identified, publicly available SARS-CoV-2 test result data for MCI inmates.ExposuresInmates at MCI were considered exposed to potential infection with SARS-CoV-2.Main outcome and measuresResults from mass testing conducted on April 16, 2020 were analyzed together with time of first reported SARS-CoV-2 infection among MCI inmates.ResultsRapid, widespread infection of MCI inmates was reported, with nearly 80% of inmates infected within three weeks of first reported inmate case. These data are consistent with i) a basic reproduction number greater than 14, together with a single initially infected inmate, ii) an initial super-spreading event resulting in several hundred initially infected inmates, together with a basic reproduction number of approximately three, and iii) earlier undetected circulation of virus among inmates prior to April.Conclusions and relevanceMass testing data are consistent with extreme transmissibility, super-spreading events, or undetected circulation of virus among inmates. All scenarios consistent with these data attest to the vulnerabilities of prisoners to COVID-19.Key pointsQuestionTo identify transmission characteristics consistent with timing and extent of SARS-CoV-2 infection among inmates in Marion Correctional Institution.FindingsMathematical and statistical modeling finds three scenarios that are consistent with the observed widespread infection in Marion Correctional Institution: i) very high transmissibility corresponding to a basic reproduction number in the double digits, ii) an initial super-spreading event involving exposure of several hundred inmates, iii) undetected circulation of virus prior to the first documented case among inmates.MeaningHigh transmissibility, super-spreading events, and challenges with disease surveillance all attest to the vulnerabilities of prison populations to SARS-CoV-2.

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