An empirically adjusted approach to reproductive number estimation for stochastic compartmental models: A case study of two Ebola outbreaks.
TL;DR: A new quantity dubbed the “empirically adjusted reproductive number” with several advantages is introduced with more complete use of the underlying compartmental dynamics than common alternatives, use as a potential diagnostic tool to detect the presence and causes of intensity process underfitting, and the ability to provide timely feedback on disease spread.
Abstract: The various thresholding quantities grouped under the "Basic Reproductive Number" umbrella are often confused, but represent distinct approaches to estimating epidemic spread potential, and address different modeling needs. Here, we contrast several common reproduction measures applied to stochastic compartmental models, and introduce a new quantity dubbed the "empirically adjusted reproductive number" with several advantages. These include: more complete use of the underlying compartmental dynamics than common alternatives, use as a potential diagnostic tool to detect the presence and causes of intensity process underfitting, and the ability to provide timely feedback on disease spread. Conceptual connections between traditional reproduction measures and our approach are explored, and the behavior of our method is examined under simulation. Two illustrative examples are developed: First, the single location applications of our method are established using data from the 1995 Ebola outbreak in the Democratic Republic of the Congo and a traditional stochastic SEIR model. Second, a spatial formulation of this technique is explored in the context of the ongoing Ebola outbreak in West Africa with particular emphasis on potential use in model selection, diagnosis, and the resulting applications to estimation and prediction. Both analyses are placed in the context of a newly developed spatial analogue of the traditional SEIR modeling approach.
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TL;DR: This work demonstrates the application of approximate Bayesian inference to spatially heterogeneous Susceptible-Exposed-Infectious-Removed (SEIR) stochastic epidemic models and discusses the practical implementation of these techniques via the open source ABSEIR package for R.
28 citations
TL;DR: It is found that it is possible for even small amounts of intermixing between populations to positively impact the control of an epidemic on a more global scale.
Abstract: Several West African countries - Liberia, Sierra Leone and Guinea - experienced significant morbidity and mortality during the largest Ebola epidemic to date, from late 2013 through 2015. The extent of the epidemic was fueled by outbreaks in large urban population centers as well as movement of the pathogen between populations. During the epidemic there was no known vaccine or drug, so effective disease control required coordinated efforts that include both standard medical and community practices such as hospitalization, quarantine and safe burials. Due to the high connectivity of the region, control of the epidemic not only depended on internal strategies but also was impacted by neighboring countries. In this paper, we use a deterministic framework to examine the role of movement between two populations in the overall success of practices designed to minimize the extent of Ebola epidemics. We find that it is possible for even small amounts of intermixing between populations to positively impact the control of an epidemic on a more global scale.
10 citations
TL;DR: This work forms discrete time stochastic susceptible‐infected‐removed models with linear (standard) and nonlinear mass action structures to mimic varying mixing levels to demonstrate the sensitivity of the basic reproduction number to these mathematical structures of the force of infection.
Abstract: The standard mass action, which assumes that infectious disease transmission occurs in well-mixed populations, is popular for formulating compartmental epidemic models. Compartmental epidemic models often follow standard mass action for simplicity and to gain insight into transmission dynamics as it often performs well at reproducing disease dynamics in large populations. In this work, we formulate discrete time stochastic susceptible-infected-removed models with linear (standard) and nonlinear mass action structures to mimic varying mixing levels. Using simulations and real epidemic data, we demonstrate the sensitivity of the basic reproduction number to these mathematical structures of the force of infection. Our results suggest the need to consider nonlinear mass action in order to generate more accurate estimates of the basic reproduction number although its uncertainty increases due to the addition of one growth scaling parameter.
9 citations
TL;DR: A Bayesian hierarchical modeling framework is developed to analyze spatio-temporal datasets both for individual level spread and as aggregate population level trends and one beneficial product of this model is smoothed estimates for individuals.
Abstract: Glaucoma is the second leading cause of blindness in the USA. A visual field test (perimetry) is used to sample and quantitate visual field function in preselected regions in the eye. These regions can be considered a spatial field with replications across independently measured individuals. At return visits, a new set of visual field measurements is obtained producing a subject specific spatio-temporal dataset. We develop a Bayesian hierarchical modeling framework to analyze these spatio-temporal datasets both for individual level spread and as aggregate population level trends. Our model extends previous research utilizing a dimension reduction matrix and individual specific latent variables. Human characteristics are incorporated into the model to help explain glaucoma progression. One beneficial product of our model is smoothed estimates for individuals. We also specify how progression rates are computed for monitoring purposes so that clinicians can track changes and predict forward in time. Copyright © 2016 John Wiley & Sons, Ltd.
7 citations
Cites background from "An empirically adjusted approach to..."
...It is also important to note that different spatial regions may have different disease transmission rates, but the regions are not truly independent of each other as living objects are not stationary and infectious hosts often interact with susceptible individuals across locations [5]....
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22 Oct 2021
TL;DR: In this article, the authors compared different mobility data sources by constructing spatial weight matrices at a variety of spatial resolutions and further compared the results through hierarchical clustering, taking into account distance between spatial units as well as spatial covariates.
Abstract: The COVID-19 pandemic starting in the first half of 2020 has changed the lives of everyone across the world. Reduced mobility was essential due to it being the largest impact possible against the spread of the little understood SARS-CoV-2 virus. To understand the spread, a comprehension of human mobility patterns is needed. The use of mobility data in modelling is thus essential to capture the intrinsic spread through the population. It is necessary to determine to what extent mobility data sources convey the same message of mobility within a region. This paper compares different mobility data sources by constructing spatial weight matrices at a variety of spatial resolutions and further compares the results through hierarchical clustering. We consider four methods for constructing spatial weight matrices representing mobility between spatial units, taking into account distance between spatial units as well as spatial covariates. This provides insight for the user into which data provides what type of information and in what situations a particular data source is most useful.
7 citations
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TL;DR: Copyright (©) 1999–2012 R Foundation for Statistical Computing; permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and permission notice are preserved on all copies.
Abstract: Copyright (©) 1999–2012 R Foundation for Statistical Computing. Permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and this permission notice are preserved on all copies. Permission is granted to copy and distribute modified versions of this manual under the conditions for verbatim copying, provided that the entire resulting derived work is distributed under the terms of a permission notice identical to this one. Permission is granted to copy and distribute translations of this manual into another language, under the above conditions for modified versions, except that this permission notice may be stated in a translation approved by the R Core Team.
272,030 citations
"An empirically adjusted approach to..." refers methods in this paper
...This simple structure is compared to several expanded versions which include temporal basis splines of varying degrees of freedom, produced with the splines R package (R Core Team, 2013)....
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TL;DR: The focus is on applied inference for Bayesian posterior distributions in real problems, which often tend toward normal- ity after transformations and marginalization, and the results are derived as normal-theory approximations to exact Bayesian inference, conditional on the observed simulations.
Abstract: The Gibbs sampler, the algorithm of Metropolis and similar iterative simulation methods are potentially very helpful for summarizing multivariate distributions. Used naively, however, iterative simulation can give misleading answers. Our methods are simple and generally applicable to the output of any iterative simulation; they are designed for researchers primarily interested in the science underlying the data and models they are analyzing, rather than for researchers interested in the probability theory underlying the iterative simulations themselves. Our recommended strategy is to use several independent sequences, with starting points sampled from an overdispersed distribution. At each step of the iterative simulation, we obtain, for each univariate estimand of interest, a distributional estimate and an estimate of how much sharper the distributional estimate might become if the simulations were continued indefinitely. Because our focus is on applied inference for Bayesian posterior distributions in real problems, which often tend toward normality after transformations and marginalization, we derive our results as normal-theory approximations to exact Bayesian inference, conditional on the observed simulations. The methods are illustrated on a random-effects mixture model applied to experimental measurements of reaction times of normal and schizophrenic patients.
13,884 citations
"An empirically adjusted approach to..." refers methods in this paper
...This was established by requiring that the Gelman and Rubin diagnostic (Gelman and Rubin, 1992) was less than or equal to 1....
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...This was established by requiring that the Gelman and Rubin diagnostic (Gelman and Rubin, 1992) was less than or equal to 1.02 for all model parameters....
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TL;DR: In this article, the authors considered the problem of finding a causal factor which appears to be adequate to account for the magnitude of the frequent epidemics of disease which visit almost every population.
Abstract: (1) One of the most striking features in the study of epidemics is the difficulty of finding a causal factor which appears to be adequate to account for the magnitude of the frequent epidemics of disease which visit almost every population. It was with a view to obtaining more insight regarding the effects of the various factors which govern the spread of contagious epidemics that the present investigation was undertaken. Reference may here be made to the work of Ross and Hudson (1915-17) in which the same problem is attacked. The problem is here carried to a further stage, and it is considered from a point of view which is in one sense more general. The problem may be summarised as follows: One (or more) infected person is introduced into a community of individuals, more or less susceptible to the disease in question. The disease spreads from the affected to the unaffected by contact infection. Each infected person runs through the course of his sickness, and finally is removed from the number of those who are sick, by recovery or by death. The chances of recovery or death vary from day to day during the course of his illness. The chances that the affected may convey infection to the unaffected are likewise dependent upon the stage of the sickness. As the epidemic spreads, the number of unaffected members of the community becomes reduced. Since the course of an epidemic is short compared with the life of an individual, the population may be considered as remaining constant, except in as far as it is modified by deaths due to the epidemic disease itself. In the course of time the epidemic may come to an end. One of the most important probems in epidemiology is to ascertain whether this termination occurs only when no susceptible individuals are left, or whether the interplay of the various factors of infectivity, recovery and mortality, may result in termination, whilst many susceptible individuals are still present in the unaffected population. It is difficult to treat this problem in its most general aspect. In the present communication discussion will be limited to the case in which all members of the community are initially equally susceptible to the disease, and it will be further assumed that complete immunity is conferred by a single infection.
8,238 citations
01 Jan 1927
TL;DR: The present communication discussion will be limited to the case in which all members of the community are initially equally susceptible to the disease, and it will be further assumed that complete immunity is conferred by a single infection.
Abstract: (1) One of the most striking features in the study of epidemics is the difficulty of finding a causal factor which appears to be adequate to account for the magnitude of the frequent epidemics of disease which visit almost every population. It was with a view to obtaining more insight regarding the effects of the various factors which govern the spread of contagious epidemics that the present investigation was undertaken. Reference may here be made to the work of Ross and Hudson (1915-17) in which the same problem is attacked. The problem is here carried to a further stage, and it is considered from a point of view which is in one sense more general. The problem may be summarised as follows: One (or more) infected person is introduced into a community of individuals, more or less susceptible to the disease in question. The disease spreads from the affected to the unaffected by contact infection. Each infected person runs through the course of his sickness, and finally is removed from the number of those who are sick, by recovery or by death. The chances of recovery or death vary from day to day during the course of his illness. The chances that the affected may convey infection to the unaffected are likewise dependent upon the stage of the sickness. As the epidemic spreads, the number of unaffected members of the community becomes reduced. Since the course of an epidemic is short compared with the life of an individual, the population may be considered as remaining constant, except in as far as it is modified by deaths due to the epidemic disease itself. In the course of time the epidemic may come to an end. One of the most important probems in epidemiology is to ascertain whether this termination occurs only when no susceptible individuals are left, or whether the interplay of the various factors of infectivity, recovery and mortality, may result in termination, whilst many susceptible individuals are still present in the unaffected population. It is difficult to treat this problem in its most general aspect. In the present communication discussion will be limited to the case in which all members of the community are initially equally susceptible to the disease, and it will be further assumed that complete immunity is conferred by a single infection.
7,769 citations
"An empirically adjusted approach to..." refers background in this paper
...Background Compartmental models have a long history in the epidemic modeling literature, beginning with the SIR technique specified by Kermack and McKendrick (1927)....
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TL;DR: Threshold theorems involving the basic reproduction number, the contact number, and the replacement number $R$ are reviewed for classic SIR epidemic and endemic models and results with new expressions for $R_{0}$ are obtained for MSEIR and SEIR endemic models with either continuous age or age groups.
Abstract: Many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases. Threshold theorems involving the basic reproduction number $R_{0}$, the contact number $\sigma$, and the replacement number $R$ are reviewed for the classic SIR epidemic and endemic models. Similar results with new expressions for $R_{0}$ are obtained for MSEIR and SEIR endemic models with either continuous age or age groups. Values of $R_{0}$ and $\sigma$ are estimated for various diseases including measles in Niger and pertussis in the United States. Previous models with age structure, heterogeneity, and spatial structure are surveyed.
5,915 citations
"An empirically adjusted approach to..." refers background in this paper
...Hethcote (2000) defines a similarly motivated quantity known as the replacement number for deterministic versions of the models discussed here....
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