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The mathematics of infectious diseases
 SIAM Review
, 2000
"... 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 R0, the contact number σ, and the replacement number R are reviewed for the classic SIR epidemic a ..."
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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 R0, the contact number σ, and the replacement number R are reviewed for the classic SIR epidemic and endemic models. Similar results with new expressions for R0 are obtained for MSEIR and SEIR endemic models with either continuous age or age groups. Values of R0 and σ 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.
Homophily and contagion are generically confounded in observational social network studies
 Sociological Methods & Research
"... We consider processes on social networks that can potentially involve three factors: homophily, or the formation of social ties due to matching individual traits; social contagion, also known as social influence; and the causal effect of an individual’s covariates on their behavior or other measura ..."
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Cited by 78 (1 self)
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We consider processes on social networks that can potentially involve three factors: homophily, or the formation of social ties due to matching individual traits; social contagion, also known as social influence; and the causal effect of an individual’s covariates on their behavior or other measurable responses. We show that, generically, all of these are confounded with each other. Distinguishing them from one another requires strong assumptions on the parametrization of the social process or on the adequacy of the covariates used (or both). In particular we demonstrate, with simple examples, that asymmetries in regression coefficients cannot identify causal effects, and that very simple models of imitation (a form of social contagion) can produce substantial correlations between an individual’s enduring traits and their choices, even when there is no intrinsic affinity between them. We also suggest some possible constructive responses to these results.
Time series analysis via mechanistic models. In review; prepublished at arxiv.org/abs/0802.0021
, 2008
"... The purpose of time series analysis via mechanistic models is to reconcile the known or hypothesized structure of a dynamical system with observations collected over time. We develop a framework for constructing nonlinear mechanistic models and carrying out inference. Our framework permits the consi ..."
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Cited by 36 (10 self)
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The purpose of time series analysis via mechanistic models is to reconcile the known or hypothesized structure of a dynamical system with observations collected over time. We develop a framework for constructing nonlinear mechanistic models and carrying out inference. Our framework permits the consideration of implicit dynamic models, meaning statistical models for stochastic dynamical systems which are specified by a simulation algorithm to generate sample paths. Inference procedures that operate on implicit models are said to have the plugandplay property. Our work builds on recently developed plugandplay inference methodology for partially observed Markov models. We introduce a class of implicitly specified Markov chains with stochastic transition rates, and we demonstrate its applicability to open problems in statistical inference for biological systems. As one example, these models are shown to give a fresh perspective on measles transmission dynamics. As a second example, we present a mechanistic analysis of cholera incidence data, involving interaction between two competing strains of the pathogen Vibrio cholerae. 1. Introduction. A
stochastic dynamics with nonlinear fractal properties
 Supplement U. S. National Report IUGG
, 1987
"... Abstract Stochastic processes with multiplicative noise have been studied independently in several different contexts over the past decades. We focus on the regime, found for a generic set of control parameters, in which stochastic processes with multiplicative noise produce intermittency of a speci ..."
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Cited by 22 (11 self)
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Abstract Stochastic processes with multiplicative noise have been studied independently in several different contexts over the past decades. We focus on the regime, found for a generic set of control parameters, in which stochastic processes with multiplicative noise produce intermittency of a special kind, characterized by a power law probability density distribution. We present a review of applications, highlight the common physical mechanism and summarize the main known results. The distribution and statistical properties of the duration of intermittent bursts are also characterized in details. 1 1
Models for spatially distributed populations: The effect of withinpatch variability
, 1981
"... This paper studies population models which have the following three ingredients: populations are divided into local subpopulations, local population dynamics are noniinear and random events occur locally in space. In this setting local stochastic phenomena have a systematic effect on average populat ..."
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Cited by 16 (3 self)
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This paper studies population models which have the following three ingredients: populations are divided into local subpopulations, local population dynamics are noniinear and random events occur locally in space. In this setting local stochastic phenomena have a systematic effect on average population density and this effect does not disappear in large populations. This result is an outcome of the interaction of the three ingredients in the models and it says that stochastic models of systems of patches can be expected to give results for average population density that differ systematically from those of deterministic models. The magnitude of these differences is related to the degree of nonlinearity of local dynamics and the magnitude of local variability. These results explain those obtained from a number of previously published models which give conclusions that differ from those of deterministic models. Results are also obtained that show how stochastic models of systems of patches may be simplified to facilitate their study. 1. INTR~OUCTI~N The chances of survival and reproduction for an individual organism
Iterated Filtering
, 2011
"... Inference for partially observed Markov process models has been a longstanding methodological challenge with many scientific and engineering applications. Iterated filtering algorithms maximize the likelihood function for partially observed Markov process models by solving a recursive sequence of fi ..."
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Cited by 14 (3 self)
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Inference for partially observed Markov process models has been a longstanding methodological challenge with many scientific and engineering applications. Iterated filtering algorithms maximize the likelihood function for partially observed Markov process models by solving a recursive sequence of filtering problems. We present new theoretical results pertaining to the convergence of iterated filtering algorithms implemented via sequential Monte Carlo filters. This theory complements the growing body of empirical evidence that iterated filtering algorithms provide an effective inference strategy for scientific models of nonlinear dynamic systems. The first step in our theory involves studying a new recursive approach for maximizing the likelihood function of a latent variable model, when this likelihood is evaluated via importance sampling. This leads to the consideration of an iterated importance sampling algorithm which serves as a simple special case of iterated filtering, and may have applicability in its own right. 1
Diffusion approximations for ecological models
 In (Ed. Fred Ghasssemi) Proceedings of the International Congress on Modelling and Simulation
, 2001
"... Abstract: Diffusion models are widely used in ecology, and in more general population biology contexts, for predicting populationsize distributions and extinction times. They are often used because they are particularly simple to analyse and give rise to explicit formulae for most of the quantities ..."
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Cited by 14 (3 self)
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Abstract: Diffusion models are widely used in ecology, and in more general population biology contexts, for predicting populationsize distributions and extinction times. They are often used because they are particularly simple to analyse and give rise to explicit formulae for most of the quantities of interest. However, whilst diffusion models are ubiquitous in the literature on population models, their use is frequently inappropriate and often leads to inaccurate predictions of critical quantities such as persistence times. This paper examines diffusion models in the context in which they most naturally arise: as approximations to discretestate Markovian models, which themselves are often more appropriate in describing the behaviour of the populations in question, yet are difficult to analyse from both an analytical and a computational point of view. We will identify a class of Markovian models (called asymptotically density dependent models) that permit a diffusion approximation through a simple limiting procedure. This procedure allows us to immediately identify the most appropriate approximating diffusion and to decide whether the diffusion approximation, and hence a diffusion model, is appropriate for describing the population in question. This will be made possible through the remarkable work of Tom Kurtz and Andrew Barbour, which is frequently cited in the applied probability literature, but is apparently not widely accessible to practitioners. Their results will be presented here in a form that most easily allows their direct application to population models. We will also present results that allow one to assess the accuracy of diffusion approximations by specifying for how long and over what ranges the underlying Markovian model is faithfully approximated. We will explain why diffusion models are not generally useful for estimating extinction times, a serious shortcoming that has been identified by other authors using empirical means.
Large graph limit for an SIR process in random network with heterogeneous connectivity
 Ann. Appl. Probab
"... We consider a SIR epidemic model propagating on a random network generated by configuration model, where the degree distribution of the vertices is given and where the edges are randomly matched. The evolution of the epidemics is summed up into three measurevalued equations that describe the degre ..."
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Cited by 12 (2 self)
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We consider a SIR epidemic model propagating on a random network generated by configuration model, where the degree distribution of the vertices is given and where the edges are randomly matched. The evolution of the epidemics is summed up into three measurevalued equations that describe the degrees of the susceptible individuals and the number of edges from an infectious or removed individual to the set of susceptibles. These three degree distributions are sufficient to describe the course of the disease. The limit in large population is investigated. As a corollary, this provides a rigorous proof of the equations obtained by
Markovian dynamics on complex reaction networks
, 2013
"... Complex networks, comprised of individual elements that interact with each other through reaction channels, are ubiquitous across many scientific and engineering disciplines. Examples include biochemical, pharmacokinetic, epidemiological, ecological, social, neural, and multiagent networks. A com ..."
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Cited by 8 (0 self)
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Complex networks, comprised of individual elements that interact with each other through reaction channels, are ubiquitous across many scientific and engineering disciplines. Examples include biochemical, pharmacokinetic, epidemiological, ecological, social, neural, and multiagent networks. A common approach to modeling such networks is by a master equation that governs the dynamic evolution of the joint probability mass function of the underling population process and naturally leads to Markovian dynamics for such process. Due however to the nonlinear nature of most reactions, the computation and analysis of the resulting stochastic population dynamics is a difficult task. This review article provides a coherent and comprehensive coverage of recently developed approaches and methods to tackle this problem. After reviewing a general framework for modeling Markovian reaction networks and giving specific examples, the authors present numerical and computational techniques capable of evaluating or approximating the solution of the master equation, discuss a recently developed approach for studying the stationary behavior of Markovian reaction networks using a potential energy landscape perspective, and provide an introduction to the emerging theory of thermodynamic analysis of such networks. Three representative problems of opinion formation, transcription regulation, and neural network dynamics are used as illustrative examples.
Truncated Importance Sampling
"... materials for this article are available online www.amstat.org/publications/JCGS ..."
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materials for this article are available online www.amstat.org/publications/JCGS