Disease emergence in deterministic and stochastic models for host and pathogen

The spatial component of ecological interactions plays an important role in shaping ecological communities. A crucial ecological question is how does habitat disturbance and fragmentation affect species persistence and diversity? In this paper, we develop a deterministic metapopulation model that takes into account a time-dependent patchy environment, thus our model and analysis take into account environmental changes. We investigate the effects that spatial variations have on persistence and coexistence of two competing species. In particular, we study the local behaviour of the model, and we provide a rigorous proof for the global analysis of our model. Also, we compare the results of the deterministic model with simulations of a stochastic version of the model.

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“…From Equation (19) and the third equation of (18) we know that z * satisfies the following quadratic equation…”

Section: Global Analysis Of the Coexistence Equilibriummentioning

confidence: 99%

“…Many epidemiological and ecological models with demographic or environmental stochasticity have been studied in the past few decades (see, e.g. [1,[17][18][19]25,32]). In this paper, we incorporate such stochastic factors in our model and examine how these factors may influence the persistence and coexistence of two competing species.…”

Section: Introductionmentioning

confidence: 99%

Coexistence of competitors in deterministic and stochastic patchy environments

Feng

Liu

Qiu

et al. 2011

Journal of Biological Dynamics

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“…Populations of different species may interact, for example, as competitors or as predator and prey. More details about modeling the stochastic dynamics of populations using the procedure described in this section can be found, for example, in references [4,7,8,11,12,39,68,86].…”

Section: General Model Of Two Interacting Populationsmentioning

confidence: 99%

“…The discrete stochastic model then leads to a stochastic differential equation model as ∆t → 0. This model development procedure has been applied to several interesting biological and physical phenomena [4,7,8,11,12,13,39,55,68,86,105]. Furthermore, as ∆t → 0, a continuous-time stochastic model is obtained from the discrete stochastic model and it appears that similarities between continuous-time Markov chain models and stochastic differential equation models were first noted by Kurtz in 1971 [78].…”

Section: Introductionmentioning

confidence: 99%

Modeling with Itô Stochastic Differential Equations

Allen¹

2007

Mathematical Modelling: Theory and Applications

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This series is aimed at publishing work dealing with the definition, development and application of fundamental theory and methodology, computational and algorithmic implementations and comprehensive empirical studies in mathematical modelling. Work on new mathematics inspired by the construction of mathematical models, combining theory and experiment and furthering the understanding of the systems being modelled are particularly welcomed.Manuscripts to be considered for publication lie within the following, nonexhaustive list of areas: mathematical modelling in engineering, industrial mathematics, control theory, operations research, decision theory, economic modelling, mathematical programming, mathematical system theory, geophysical sciences, climate modelling, environmental processes, mathematical modelling in psychology, political science, sociology and behavioural sciences, mathematical biology, mathematical ecology, image processing, computer vision, artificial intelligence, fuzzy systems, and approximate reasoning, genetic algorithms, neural networks, expert systems, pattern recognition, clustering, chaos and fractals.Original monographs, comprehensive surveys as well as edited collections will be considered for publication.

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“…Tremendous crop losses and global threat to food security have been caused by plant diseases [1,2]. In recent years, plant diseases have attracted the interest of many mathematical modeling researchers and epidemiologists [3][4][5][6][7].…”

Section: Introductionmentioning

confidence: 99%

Stability and Hopf Bifurcation Analysis of a Plant Virus Propagation Model with Two Delays

Zhang

2018

Discrete Dynamics in Nature and Society

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To understand the interaction between the insects and the plants, a system of delay differential equations is proposed and studied. We prove that if 0 ≤ 1, the disease-free equilibrium is globally asymptotically stable for any length of time delays by constructing a Lyapunov functional, and the system admits a unique endemic equilibrium if 0 > 1. We establish the sufficient conditions for the stability of the endemic equilibrium and existence of Hopf bifurcation. Using the normal form theory and center manifold theorem, the explicit formulae which determine the stability, direction, and other properties of bifurcating periodic solutions are derived. Some numerical simulations are given to confirm our analytic results.

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Disease emergence in deterministic and stochastic models for host and pathogen

Cited by 12 publications

References 15 publications

Coexistence of competitors in deterministic and stochastic patchy environments

Coexistence of competitors in deterministic and stochastic patchy environments

Modeling with Itô Stochastic Differential Equations

Stability and Hopf Bifurcation Analysis of a Plant Virus Propagation Model with Two Delays

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