Piecewise‐deterministic Markov Processes for Spatio‐temporal Population Dynamics

Abboud, Candy; Senoussi, Rachid; Soubeyrand, Samuel

doi:10.1002/9781119507338.ch7

Cited by 3 publications

(5 citation statements)

References 34 publications

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“…1995) or a piecewise deterministic Markov process (Abboud et al. 2018). These approaches, allowing a finer quantification of local and long distance dispersal, are generally expected to yield better predictions (Higgins and Richardson 1999; Nathan et al.…”

Section: Discussionmentioning

confidence: 99%

“…Abboud et al. (2018) precisely discuss a PDMP embedding multiple introductions without implementing it in practice. This is one of the most attractive perspectives for furthering our work.…”

Section: Discussionmentioning

confidence: 99%

“…An intermediate modeling approach for gaining in realism but conserving parsimony, could be to couple the conciseness of our reaction-diffusion-absorption equation with a stochastic jump process in the settings of Piecewise Deterministic Markov Process (PDMP; Azaïs and Bouguet, 2018). In this framework, each jump could be localized in space and time and would represent a new introduction of the pathogen, as proposed by Abboud et al (2018). A strong limitation of the number of jumps would be needed to keep the model and the estimation of its parameters simple.…”

Section: Discussionmentioning

confidence: 99%

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Dating and localizing an invasion from post-introduction data and a coupled reaction–diffusion–absorption model

et al. 2019

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Invasion of new territories by alien organisms is of primary concern for environmental and health agencies and has been a core topic in mathematical modeling, in particular in the intents of reconstructing the past dynamics of the alien organisms and predicting their future spatial extents. Partial differential equations offer a rich and flexible modeling framework that has been applied to a large number of invasions. In this article, we are specifically interested in dating and localizing the introduction that led to an invasion using mathematical modeling, post-introduction data and an adequate statistical inference procedure. We adopt a mechanistic-statistical approach grounded on a coupled reaction–diffusion–absorption model representing the dynamics of an organism in an heterogeneous domain with respect to growth. Initial conditions (including the date and site of the introduction) and model parameters related to diffusion, reproduction and mortality are jointly estimated in the Bayesian framework by using an adaptive importance sampling algorithm. This framework is applied to the invasion of Xylella fastidiosa , a phytopathogenic bacterium detected in South Corsica in 2015, France. Electronic supplementary material The online version of this article (10.1007/s00285-019-01376-x) contains supplementary material, which is available to authorized users.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

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Dating and localizing an invasion from post-introduction data and a coupled reaction–diffusion–absorption model

et al. 2019

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“…Stochastic metapopulation models offer precise mathematical descriptions but, to the best of our knowledge, have not yet been investigated in the context of epidemic modeling. Piecewise-deterministic metapopulation models have been largely applied to model pathogen spreading [25] , [26] due to (1) their smaller computational complexity compared to ABMs; (2) better spatial resolution as opposed to ODE models, which are typically used. Despite the enormous recent advancement in adapting these models to realistic epidemic scenarios, this topic is still at the infancy of its development.…”

Section: Modeling Covid-19 Epidemic Spreadingmentioning

confidence: 99%

“…There exist several approaches to describe epidemic (or other types of) population dynamics by semi-deterministic processes [25] , [26] . A piecewise-deterministic metapopulation model (PDMM), where deterministic dynamics within the subpopulations are combined with stochastic transitions between the subpopulations, is formulated in [27] for modeling a cattle trade network.…”

Section: Introductionmentioning

confidence: 99%

Mathematical modeling of spatio-temporal population dynamics and application to epidemic spreading

Winkelmann

Zonker

Schütte

et al. 2021

Mathematical Biosciences

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Agent based models (ABMs) are a useful tool for modeling spatio-temporal population dynamics, where many details can be included in the model description. Their computational cost though is very high and for stochastic ABMs a lot of individual simulations are required to sample quantities of interest. Especially, large numbers of agents render the sampling infeasible. Model reduction to a metapopulation model leads to a significant gain in computational efficiency, while preserving important dynamical properties. Based on a precise mathematical description of spatio-temporal ABMs, we present two different metapopulation approaches (stochastic and piecewise deterministic) and discuss the approximation steps between the different models within this framework. Especially, we show how the stochastic metapopulation model results from a Galerkin projection of the underlying ABM onto a finite-dimensional ansatz space. Finally, we utilize our modeling framework to provide a conceptual model for the spreading of COVID-19 that can be scaled to real-world scenarios.

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Filling the gap between Continuous and Discrete Time Dynamics of Autoregressive Processes

Girardin

Senoussi

2019

Journal Time Series Analysis

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Discretization of continuous time autoregressive (AR) processes driven by a Brownian motion and embedding of discrete time AR sequences driven by a Gaussian white noise are classical issues. The article aims at establishing and using such discretization and embedding formulae between extended AR continuous time processes and discrete time sequences. The continuous‐time processes are driven by either Brownian or jump processes, and may have random coefficients depending on time; Lévy‐driven processes are also considered. The innovation of the discrete time processes may be of many types – including Gaussian. In one way, observing the continuous time AR process at discrete times leads the AR dynamics of the discretized process to be characterized. The other way round, AR sequences can be embedded, in the almost sure sense, into continuous time AR processes with the same dynamics. Illustration is provided through many examples and simulation.

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Piecewise‐deterministic Markov Processes for Spatio‐temporal Population Dynamics

Cited by 3 publications

References 34 publications

Dating and localizing an invasion from post-introduction data and a coupled reaction–diffusion–absorption model

Dating and localizing an invasion from post-introduction data and a coupled reaction–diffusion–absorption model

Mathematical modeling of spatio-temporal population dynamics and application to epidemic spreading

Filling the gap between Continuous and Discrete Time Dynamics of Autoregressive Processes

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