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19
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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Cited by 200 (1 self)
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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.
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 19 (10 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
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 13 (5 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
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 8 (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
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 7 (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.
Bringing consistency to simulation of population models  Poisson Simulation as a bridge between micro and macro simulation
, 2007
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MODELING, ANALYSIS AND DISCRETIZATION OF STOCHASTIC LOGISTIC EQUATIONS
, 2007
"... The well–known logistic model has been extensively investigated in deterministic theory. There are numerous case studies where such type of nonlinearities occur in Ecology, Biology and Environmental Sciences. Due to the presence of environmental fluctuations and a lack of precision of measurements, ..."
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Cited by 2 (0 self)
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The well–known logistic model has been extensively investigated in deterministic theory. There are numerous case studies where such type of nonlinearities occur in Ecology, Biology and Environmental Sciences. Due to the presence of environmental fluctuations and a lack of precision of measurements, one has to deal with effects of randomness on such models. As a more realistic modeling, we suggest nonlinear stochastic differential equations (SDEs) dX(t) = [(ρ + λX(t))(K − X(t)) − µX(t)]dt + σX(t) α K − X(t)  β dW (t) of Itôtype to model the growth of populations or innovations X, driven by a Wiener process W and positive real constants ρ, λ, K, µ, α, β ≥ 0. discuss well–posedness, regularity (boundedness) and uniqueness of their solutions. However, explicit expressions for analytical solution of such random logistic equations are rarely known. Therefore one has to resort to numerical solution of SDEs for studying various aspects like the time–evolution of growth patterns, exit frequencies, mean passage times and impact of fluctuating growth parameters. We present some basic aspects of adequate numerical analysis of these random extensions of these models such as numerical regularity and mean square convergence. The problem of keeping reasonable boundaries for analytic solutions under discretization plays an essential role for practically meaningful models, in particular the preservation of intervals with reflecting or absorbing barriers. A discretization of the continuous state space can be circumvented by appropriate methods. Balanced implicit methods (see Schurz, IJNAM 2 (2), p. 197220, 2005) are used to construct strongly converging approximations with the desired monotone properties. Numerical studies can bring out salient features of the stochastic logistic models (e.g. We almost sure monotonicity, almost sure uniform boundedness, delayed initial evolution or earlier points of inflection compared to deterministic model).
Fluctuations and oscillations in a simple epidemic model
, 810
"... We show that the simplest stochastic epidemiological models with spatial correlations exhibit two types of oscillatory behaviour in the endemic phase. In a large parameter range, the oscillations are due to resonant amplification of stochastic fluctuations, a general mechanism first reported for pre ..."
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Cited by 1 (0 self)
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We show that the simplest stochastic epidemiological models with spatial correlations exhibit two types of oscillatory behaviour in the endemic phase. In a large parameter range, the oscillations are due to resonant amplification of stochastic fluctuations, a general mechanism first reported for predatorprey dynamics. In a narrow range of parameters that includes many infectious diseases which confer long lasting immunity the oscillations persist for infinite populations. This effect is apparent in simulations of the stochastic process in systems of variable size, and can be understood from the phase diagram of the deterministic pair approximation equations. The two mechanisms combined play a central role in explaining the ubiquity of oscillatory behaviour in real data and in simulation results of epidemic and other related models. PACS numbers: 87.10.Mn; 87.19.ln; 05.10.Gg Cycles are a very striking behaviour of preypredator systems also seen in a variety of other hostenemy systems — a case in point is the pattern of recurrent epidemics of many endemic infectious diseases [1]. The controversy in the literature over the driving mechanisms
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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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
DEDICATION.................................
, 2010
"... Time series analysis for nonlinear dynamical systems with applications to modeling of infectious diseases by ..."
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Time series analysis for nonlinear dynamical systems with applications to modeling of infectious diseases by