Longtime behavior of a class of stochastic tumor-immune systems

Tuong, Tran Dinh; Nguyen, Nhu N.; Yin, George

doi:10.1016/j.sysconle.2020.104806

Cited by 17 publications

(7 citation statements)

References 18 publications

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“…By now, Kolmogorov stochastic population systems (using stochastic differential equations or difference equations) together with their longtime behavior have been relatively well understood; see [5,54,56] for Kolmogorov stochastic systems in compact domains and [4,25] for certain general Kolmogorov systems in non-compact domains. Variants of Kolmogorov systems such as epidemic models [12,14,16,47], tumor-immune systems [61] and chemostat models [44] etc. have also been studied.…”

Section: Introductionmentioning

confidence: 99%

Stochastic Functional Kolmogorov Equations (I): Persistence

Nguyen¹,

Nguyen²,

Yin³

2021

Preprint

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This work (Part (I)) together with its companion (Part (II) [45]) develops a new framework for stochastic functional Kolmogorov equations, which are nonlinear stochastic differential equations depending on the current as well as the past states. Because of the complexity of the results, it seems to be instructive to divide our contributions to two parts. In contrast to the existing literature, our effort is to advance the knowledge by allowing delay and past dependence, yielding essential utility to a wide range of applications. A long-standing question of fundamental importance pertaining to biology and ecology is: What are the minimal necessary and sufficient conditions for long-term persistence and extinction (or for long-term coexistence of interacting species) of a population? Regardless of the particular applications encountered, persistence and extinction are properties shared by Kolmogorov systems. While there are many excellent treaties of stochastic-differential-equation-based Kolmogorov equations, the work on stochastic Kolmogorov equations with past dependence is still scarce. Our aim here is to answer the aforementioned basic question. This work, Part (I), is devoted to characterization of persistence, whereas its companion, Part (II) [45], is devoted to extinction. The main techniques used in this paper include the newly developed functional Itô formula and asymptotic coupling and Harris-like theory for infinite dimensional systems specialized to functional equations. General theorems for stochastic functional Kolmogorov equations are developed first. Then a number of applications are examined to obtain new results substantially covering, improving, and extending the existing literature. Furthermore, these conditions reduce to that of Kolmogorov systems when there is no past dependence.

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Section: Introductionmentioning

confidence: 99%

Stochastic Functional Kolmogorov Equations (I): Persistence

Nguyen¹,

Nguyen²,

Yin³

2021

Preprint

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“…Why is the "growth rate method" in SDEs no longer works? The growth rate idea is the most effective to characterize the persistence and extinction of a stochastic population modeled by SDEs; see [3,7,15,26,40,49,54,55] and the reference therein. The main idea is to define the growth rate of a species using its Lyapunov exponent.…”

Section: Discussionmentioning

confidence: 99%

Stochastic Lotka-Volterra Competitive Reaction-Diffusion Systems Perturbed by Space-Time White Noise: Modeling and Analysis

Nguyen

Yin

2020

Preprint

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Motivated by the traditional Lotka-Volterra competitive models, this paper proposes and analyzes a class of stochastic reaction-diffusion partial differential equations. In contrast to the models in the literature, the new formulation enables spatial dependence of the species. In addition, the noise process is allowed to be space-time white noise. In this work, wellposedness, regularity of solutions, existence of density, and existence of an invariant measure for stochastic reaction-diffusion systems with non-Lipschitz and non-linear growth coefficients and multiplicative noise are considered. By combining the random field approach and infinite integration theory approach in SPDEs for mild solutions, analysis is carried out. Then this paper develops a Lotka-Volterra competitive system under general setting; longtime properties are studied with the help of newly developed tools in stochastic calculus.

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“…It simplifies the conditions required for the research results, and can obtain almost consistent threshold conditions to distinguish extinction and persistence. This method has been used in some models, see [ 20 , 25 , 26 ]. However, these models are mainly presented in the form of two-dimensional, and there is few research on the model of three-dimensional with regime-switching.…”

Section: Introductionmentioning

confidence: 99%

Dynamical behavior of a stochastic SIQS model via isolation with regime-switching

Wang

Liu

2022

J. Appl. Math. Comput.

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A stochastic SIQS model via isolation with regime-switching is studied in this paper. The range of positive solution of the model is presented. Threshold to determine extinction and invariant measure is obtained by a new technique, which can be seen as the sufficient and almost necessary condition. Meantime, a value to judge the existence of stationary distribution is acquired by constructing the suitable hybrid Lyapunov function. Two values are proved to be consistent. Several examples are enumerated to test the theoretical results.

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Longtime behavior of a class of stochastic tumor-immune systems

Cited by 17 publications

References 18 publications

Stochastic Functional Kolmogorov Equations (I): Persistence

Stochastic Functional Kolmogorov Equations (I): Persistence

Stochastic Lotka-Volterra Competitive Reaction-Diffusion Systems Perturbed by Space-Time White Noise: Modeling and Analysis

Dynamical behavior of a stochastic SIQS model via isolation with regime-switching

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