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## Susceptible-infected-recovered and susceptible-exposed-infected models

Tome, Tania; Oliveira, Mario Jose de
Fonte: IOP PUBLISHING LTD Publicador: IOP PUBLISHING LTD
Tipo: Artigo de Revista Científica
ENG
Relevância na Pesquisa
46%
Two stochastic epidemic lattice models, the susceptible-infected-recovered and the susceptible-exposed-infected models, are studied on a Cayley tree of coordination number k. The spreading of the disease in the former is found to occur when the infection probability b is larger than b(c) = k/2(k - 1). In the latter, which is equivalent to a dynamic site percolation model, the spreading occurs when the infection probability p is greater than p(c) = 1/(k - 1). We set up and solve the time evolution equations for both models and determine the final and time-dependent properties, including the epidemic curve. We show that the two models are closely related by revealing that their relevant properties are exactly mapped into each other when p = b/[k - (k - 1) b]. These include the cluster size distribution and the density of individuals of each type, quantities that have been determined in closed forms.; Brazilian agency CNPq; Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq); INCT/CNPq of Complex Fluids; Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

## Stochastic lattice gas model describing the dynamics of the SIRS epidemic process

Souza, David Rodrigues de; Tome, Tania
Fonte: ELSEVIER SCIENCE BV Publicador: ELSEVIER SCIENCE BV
Tipo: Artigo de Revista Científica
ENG
Relevância na Pesquisa
46.2%
We study a stochastic process describing the onset of spreading dynamics of an epidemic in a population composed of individuals of three classes: susceptible (S), infected (I), and recovered (R). The stochastic process is defined by local rules and involves the following cyclic process: S -> I -> R -> S (SIRS). The open process S -> I -> R (SIR) is studied as a particular case of the SIRS process. The epidemic process is analyzed at different levels of description: by a stochastic lattice gas model and by a birth and death process. By means of Monte Carlo simulations and dynamical mean-field approximations we show that the SIRS stochastic lattice gas model exhibit a line of critical points separating the two phases: an absorbing phase where the lattice is completely full of S individuals and an active phase where S, I and R individuals coexist, which may or may not present population cycles. The critical line, that corresponds to the onset of epidemic spreading, is shown to belong in the directed percolation universality class. By considering the birth and death process we analyze the role of noise in stabilizing the oscillations. (C) 2009 Elsevier B.V. All rights reserved.; Brazilian agency CNPq; Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq); INCT of Complex Fluids (CNPq and FAPESP); Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP); Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

## Modelagem de problemas da dinâmica de populações por meio da dinâmica estocástica; Modeling problems of population by the stochastic dynamics

Souza, David Rodrigues de
Fonte: Biblioteca Digitais de Teses e Dissertações da USP Publicador: Biblioteca Digitais de Teses e Dissertações da USP
Tipo: Dissertação de Mestrado Formato: application/pdf
Publicado em 29/09/2009 PT
Relevância na Pesquisa
46.03%
Apresentamos o estudo de três modelos estocásticos governadas por equações mestras que descrevem a propagação de epidemias em uma comunidade de indivíduos;; We present a study of three stochastic models, governed by master equations, that decribe the epidemic spreding in a community of individuals;

## Teoremas limiares para o modelo SIR estocástico de epidemia; Threshold theorems for the SIR stochastic epidemic model

Mario Andrés Estrada López
Fonte: Biblioteca Digital da Unicamp Publicador: Biblioteca Digital da Unicamp
Tipo: Dissertação de Mestrado Formato: application/pdf
Publicado em 25/02/2015 PT
Relevância na Pesquisa
66.21%
Este trabalho tem como objetivo estudar o modelo SIR (suscetível-infectado-removido) de epidemia nas versões determinística e estocástica. Nosso objetivo é encontrar limitantes para a probabilidade de que o tamanho da epidemia não sobrepasse certa proporção do número inicial de suscetíveis. Iniciamos apresentando as definições e a dinâmica do processo de epidemia determinístico. Obtemos um valor limiar para o número inicial de suscetíveis para que a epidemia exploda ou não. Consideramos o modelo de epidemia estocástico SIR assumindo que não há período latente, isto é, que um infectado pode transmitir a infecção ao instante de ser contagiado. O modelo é considerado com uma configuração inicial de suscetíveis e infectados e é feita especial ênfases no estudo da variável aleatória 'tamanho da epidemia', que é definida como a diferença entre o número de suscetíveis ao começar e ao terminar a propagação da doença. Como na parte determinística, obtemos teoremas limiares para o modelo de epidemia estocástico. Os métodos usados para encontrar os limitantes são os de análise da cadeia de Markov imersa e de comparação estocástica.; This work has as objective to study the SIR (susceptible-infected-removed) epidemic model in the deterministic and stochastic version. Our objective is to find bounds for the probability that the size of the epidemic does not exceed certain proportion of the initial number of susceptible individuals. We begin presenting the definitions and the dynamics for the deterministic model for a general epidemic. We obtain a threshold value for the initial number of susceptible individuals for the epidemic to build up or not. As fundamental part of this work...

## Network-based analysis of stochastic SIR epidemic models with random and proportionate mixing

Kenah, Eben; Robins, James M.
Fonte: PubMed Publicador: PubMed
Tipo: Artigo de Revista Científica
EN
Relevância na Pesquisa
46.33%
In this paper, we outline the theory of epidemic percolation networks and their use in the analysis of stochastic SIR epidemic models on undirected contact networks. We then show how the same theory can be used to analyze stochastic SIR models with random and proportionate mixing. The epidemic percolation networks for these models are purely directed because undirected edges disappear in the limit of a large population. In a series of simulations, we show that epidemic percolation networks accurately predict the mean outbreak size and probability and final size of an epidemic for a variety of epidemic models in homogeneous and heterogeneous populations. Finally, we show that epidemic percolation networks can be used to re-derive classical results from several different areas of infectious disease epidemiology. In an appendix, we show that an epidemic percolation network can be defined for any time-homogeneous stochastic SIR model in a closed population and prove that the distribution of outbreak sizes given the infection of any given node in the SIR model is identical to the distribution of its out-component sizes in the corresponding probability space of epidemic percolation networks. We conclude that the theory of percolation on semi-directed networks provides a very general framework for the analysis of stochastic SIR models in closed populations.

## Maximal Sensitive Dependence and the Optimal Path to Epidemic Extinction

Forgoston, Eric; Bianco, Simone; Shaw, Leah B.; Schwartz, Ira B.
Fonte: PubMed Publicador: PubMed
Tipo: Artigo de Revista Científica
EN
Relevância na Pesquisa
45.98%
Extinction of an epidemic or a species is a rare event that occurs due to a large, rare stochastic fluctuation. Although the extinction process is dynamically unstable, it follows an optimal path that maximizes the probability of extinction. We show that the optimal path is also directly related to the finite-time Lyapunov exponents of the underlying dynamical system in that the optimal path displays maximum sensitivity to initial conditions. We consider several stochastic epidemic models, and examine the extinction process in a dynamical systems framework. Using the dynamics of the finite-time Lyapunov exponents as a constructive tool, we demonstrate that the dynamical systems viewpoint of extinction evolves naturally toward the optimal path.

## Improved Estimation of the Initial Number of Susceptible Individuals in the General Stochastic Epidemic Model Using Penalized Likelihood

Oh, Changhyuck
Fonte: Hindawi Publishing Corporation Publicador: Hindawi Publishing Corporation
Tipo: Artigo de Revista Científica
EN
Relevância na Pesquisa
46.01%
The initial size of a completely susceptible population in a group of individuals plays a key role in drawing inferences for epidemic models. However, this can be difficult to obtain in practice because, in any population, there might be individuals who may not transmit the disease during the epidemic. This short note describes how to improve the maximum likelihood estimators of the infection rate and the initial number of susceptible individuals and provides their approximate Hessian matrix for the general stochastic epidemic model by using the concept of the penalized likelihood function. The simulations of major epidemics show significant improvements in performance in averages and coverage ratios for the suggested estimator of the initial number in comparison to existing methods. We applied the proposed method to the Abakaliki smallpox data.

## Bayesian spatio-temporal epidemic models with applications to sheep pox

Malesios, C.; Demiris, N.; Kalogeropoulos, K.; Ntzoufras, I.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46%
Epidemic data often possess certain characteristics, such as the presence of many zeros, the spatial nature of the disease spread mechanism or environmental noise. This paper addresses these issues via suitable Bayesian modelling. In doing so we utilise stochastic regression models appropriate for spatio-temporal count data with an excess number of zeros. The developed regression framework can incorporate serial correlation and time varying covariates through an Ornstein Uhlenbeck process formulation. In addition, we explore the effect of different priors, including default options and techniques based upon variations of mixtures of $g$-priors. The effect of different distance kernels for the epidemic model component is investigated. We proceed by developing branching process-based methods for testing scenarios for disease control, thus linking traditional spatio-temporal models with epidemic processes, useful in policy-focused decision making. The approach is illustrated with an application to a sheep pox dataset from the Evros region, Greece.

## Statistical inference for stochastic epidemic models with three levels of mixing

Britton, Tom; Kypraios, Theodore; O'Neill, Philip
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
55.98%
A stochastic epidemic model is defined in which each individual belongs to a household, a secondary grouping (typically school or workplace) and also the community as a whole. Moreover, infectious contacts take place in these three settings according to potentially different rates. For this model we consider how different kinds of data can be used to estimate the infection rate parameters with a view to understanding what can and cannot be inferred, and with what precision. Among other things we find that temporal data can be of considerable inferential benefit compared to final size data, that the degree of heterogeneity in the data can have a considerable effect on inference for non-household transmission, and that inferences can be materially different from those obtained from a model with two levels of mixing. Keywords: Basic reproduction number, Bayesian inference, Epidemic model, Infectious disease data, Markov chain Monte Carlo, Networks.

## Sequential Bayesian Inference in Hidden Markov Stochastic Kinetic Models with Application to Detection and Response to Seasonal Epidemics

Lin, Junjing; Ludkovski, Michael
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.01%
We study sequential Bayesian inference in stochastic kinetic models with latent factors. Assuming continuous observation of all the reactions, our focus is on joint inference of the unknown reaction rates and the dynamic latent states, modeled as a hidden Markov factor. Using insights from nonlinear filtering of continuous-time jump Markov processes we develop a novel sequential Monte Carlo algorithm for this purpose. Our approach applies the ideas of particle learning to minimize particle degeneracy and exploit the analytical jump Markov structure. A motivating application of our methods is modeling of seasonal infectious disease outbreaks represented through a compartmental epidemic model. We demonstrate inference in such models with several numerical illustrations and also discuss predictive analysis of epidemic countermeasures using sequential Bayes estimates.; Comment: 26 pages, 7 figures

## Dynamical Monte Carlo method for stochastic epidemic models

Aiello, O. E.; da Silva, M. A. A.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46%
In this work we introduce a new approach to Dynamical Monte Carlo methods to simulate markovian processes. We apply this approach to formulate and study an epidemic generalized SIRS model. The results are in excellent agreement with the fourth order Runge-Kutta method in a region of deterministic solution. Introducing local stochastic interactions, the Runge-Kutta method is no longer applicable. Thus, we solve the system described by a set of stochastic differential equations by a Dynamical Monte Carlo technique and check the solutions self-consistently with a stochastic version of the Euler method. We also analyzed the results under the herd-immunity concept.; Comment: 18 pages, 4 figures in ps format, regular article, Latex, written with Scientific WorkPlace 3.51

## A note on fractional linear pure birth and pure death processes in epidemic models

Garra, Roberto; Polito, Federico
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.03%
In this note we highlight the role of fractional linear birth and linear death processes recently studied in \citet{sakhno} and \citet{pol}, in relation to epidemic models with empirical power law distribution of the events. Taking inspiration from a formal analogy between the equation of self consistency of the epidemic type aftershock sequences (ETAS) model, and the fractional differential equation describing the mean value of fractional linear growth processes, we show some interesting applications of fractional modelling to study \textit{ab initio} epidemic processes without the assumption of any empirical distribution. We also show that, in the frame of fractional modelling, subcritical regimes can be linked to linear fractional death processes and supercritical regimes to linear fractional birth processes. Moreover we discuss a simple toy model to underline the possible application of these stochastic growth models to more general epidemic phenomena such as tumoral growth.

## Network-based analysis of stochastic SIR epidemic models with random and proportionate mixing

Kenah, Eben; Robins, James M.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.33%
In this paper, we outline the theory of epidemic percolation networks and their use in the analysis of stochastic SIR epidemic models on undirected contact networks. We then show how the same theory can be used to analyze stochastic SIR models with random and proportionate mixing. The epidemic percolation networks for these models are purely directed because undirected edges disappear in the limit of a large population. In a series of simulations, we show that epidemic percolation networks accurately predict the mean outbreak size and probability and final size of an epidemic for a variety of epidemic models in homogeneous and heterogeneous populations. Finally, we show that epidemic percolation networks can be used to re-derive classical results from several different areas of infectious disease epidemiology. In an appendix, we show that an epidemic percolation network can be defined for any time-homogeneous stochastic SIR model in a closed population and prove that the distribution of outbreak sizes given the infection of any given node in the SIR model is identical to the distribution of its out-component sizes in the corresponding probability space of epidemic percolation networks. We conclude that the theory of percolation on semi-directed networks provides a very general framework for the analysis of stochastic SIR models in closed populations.; Comment: 40 pages...

## Stochastic epidemic models featuring contact tracing with delays

Ball, Frank G; Knock, Edward S; O'Neill, Philip D
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.04%
This paper is concerned with a stochastic model for the spread of an SEIR (susceptible -> exposed (=latent) -> infective -> removed) epidemic with a contact tracing scheme, in which removed individuals may name some of their infectious contacts, who are then removed if they have not been already after some tracing delay. The epidemic is analysed via an approximating, modified birth-death process, for which a type-reproduction number is derived in terms of unnamed individuals, that is shown to be infinite when the contact rate is sufficiently large. We obtain explicit results under the assumption of either constant or exponentially distributed infectious periods, including the epidemic extinction probability in the former case. Numerical illustrations show that, while the distributions of latent periods and delays have an effect on the spread of the epidemic, the assumption of whether the delays experienced by individuals infected by the same individual are of the same or independent length makes little difference.; Comment: Mathematical Biosciences 2015

## Second Quantization Approach to Stochastic Epidemic Models

Mondaini, Leonardo
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
66.08%
We show how the standard field theoretical language based on creation and annihilation operators may be used for a straightforward derivation of closed master equations describing the population dynamics of multivariate stochastic epidemic models. In order to do that, we introduce an SIR-inspired stochastic model for hepatitis C virus epidemic, from which we obtain the time evolution of the mean number of susceptible, infected, recovered and chronically infected individuals in a population whose total size is allowed to change.; Comment: To appear in Biomedical Sciences Today, 6 pages

## Critical scaling of stochastic epidemic models

Lalley, Steven P.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
55.99%
In the simple mean-field SIS and SIR epidemic models, infection is transmitted from infectious to susceptible members of a finite population by independent $p-$coin tosses. Spatial variants of these models are proposed, in which finite populations of size $N$ are situated at the sites of a lattice and infectious contacts are limited to individuals at neighboring sites. Scaling laws for both the mean-field and spatial models are given when the infection parameter $p$ is such that the epidemics are critical. It is shown that in all cases there is a critical threshold for the numbers initially infected: below the threshold, the epidemic evolves in essentially the same manner as its branching envelope, but at the threshold evolves like a branching process with a size-dependent drift.; Comment: Published at http://dx.doi.org/10.1214/074921707000000346 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)

## Information geometry and entropy in a stochastic epidemic rate process

Dodson, C. T. J.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.04%
Epidemic models with inhomogeneous populations have been used to study major outbreaks and recently Britton and Lindenstrand \cite{BL} described the case when latency and infectivity have independent gamma distributions. They found that variability in these random variables had opposite effects on the epidemic growth rate. That rate increased with greater variability in latency but decreased with greater variability in infectivity. Here we extend their result by using the McKay bivariate gamma distribution for the joint distribution of latency and infectivity, recovering the above effects of variability but allowing possible correlation. We use methods of stochastic rate processes to obtain explicit solutions for the growth of the epidemic and the evolution of the inhomogeneity and information entropy. We obtain a closed analytic solution to the evolution of the distribution of the number of uninfected individuals as the epidemic proceeds, and a concomitant expression for the decay of entropy. The family of McKay bivariate gamma distributions has a tractable information geometry which provides a framework in which the evolution of distributions can be studied as the outbreak grows, with a natural distance structure for quantitative tracking of progress.; Comment: 10 pages...

## Stochastic epidemic models: a survey

Britton, Tom
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
66.15%
This paper is a survey paper on stochastic epidemic models. A simple stochastic epidemic model is defined and exact and asymptotic model properties (relying on a large community) are presented. The purpose of modelling is illustrated by studying effects of vaccination and also in terms of inference procedures for important parameters, such as the basic reproduction number and the critical vaccination coverage. Several generalizations towards realism, e.g. multitype and household epidemic models, are also presented, as is a model for endemic diseases.; Comment: 26 pages, 4 figures

## On the advancement of optimal experimental design with applications to infectious diseases.

Price, David James
Tipo: Tese de Doutorado