Boundedness and Large-Time Behavior Results for a Diffusive Epidemic Model

Melkemi, Lamine; Mokrane, Ahmed Zerrouk; Youkana, Amar

doi:10.1155/2007/17930

Cited by 6 publications

(4 citation statements)

References 13 publications

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“…The idea behind the Lyapunov functional stems from Zelenyak's article [23], which has also been used by Crandall et al [5] for other purposes. The case Π > 0, α > 0, σ > 0 L. Melkemi et al [19] established the existence of global solutions, when…”

Section: Introductionmentioning

confidence: 99%

Global solution for a diffusive epidemic model (HIV/AIDS) with an exponential behavior of source

Daddiouaissa

2023

Stud. Univ. Babes-Bolyai Math.

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Section: Introductionmentioning

confidence: 99%

Global solution for a diffusive epidemic model (HIV/AIDS) with an exponential behavior of source

Daddiouaissa

2023

Stud. Univ. Babes-Bolyai Math.

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“…In particular, as spatially heterogeneous epidemic models, reaction-diffusion equations have been widely studied by many authors. We refer the reader to [2][3][4][5][6][7][8][9][10][11][12][13][14] and the references cited therein.…”

Section: Introductionmentioning

confidence: 99%

“…However, for space-structured epidemic models as reaction-diffusion equations, such a construction method has not been sufficiently established and the global stability results are restricted to some special cases that the coefficients are constant (see, for instance, Capone et al [3], Chinviriyasit and Chinviriyasit [4], Fitzgibbon et al [7], Lotfi et al [24], Melkemi et al [12], and Webb [13]). For diffusive epidemic models, the existence of traveling wave solutions has attracted sustained attention of researchers and has been one of the most interesting topics (see, for instance, Ducrot [5], Ducrot and Magal [6], Hosono and Ilyas [8], and Zhang and Wang [14]).…”

Section: Introductionmentioning

confidence: 99%

Lyapunov functions and global stability for a spatially diffusive SIR epidemic model

Kuniya

Wang

2016

Applicable Analysis

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This paper deals with the problem of global asymptotic stability for equilibria of a spatially diffusive SIR epidemic model with homogeneous Neumann boundary condition. By discretizing the model with respect to the space variable, we first construct Lyapunov functions for the corresponding ODEs model, and then broaden the construction method into the PDEs model in which either susceptible or infective populations are diffusive. In both cases, we obtain the standard threshold dynamical behaviors, that is, if 0 ≤ 1, then the disease-free equilibrium is globally asymptotically stable and if 0 > 1, then the (strictly positive) endemic equilibrium is so. Numerical examples are given to illustrate the effectiveness of the theoretical results. ARTICLE HISTORY

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“…results for the continuous RD system have been obtained using duality arguments, Sobolev embedding theorems, and Lyapunov-type structures [17,12,16,7]. However, to the best of our knowledge, our work is the first that derives conditions to guarantee the discretized RD system is uniformly bounded for all time.…”

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confidence: 99%

Boundedness of a class of discretized reaction-diffusion systems

Wentz¹,

Bortz²

2019

Preprint

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Although the spatially continuous version of the reaction-diffusion equation has been well studied, in some instances a spatially-discretized representation provides a more realistic approximation of biological processes. Indeed, mathematically the discretized and continuous systems can lead to different predictions of biological dynamics. It is well known in the continuous case that the incorporation of diffusion can cause diffusion-driven blow-up with respect to the L ∞ norm. However, this does not imply diffusion-driven blow-up will occur in the discretized version of the system. For example, in a continuous reaction-diffusion system with Dirichlet boundary conditions and nonnegative solutions, diffusion-driven blow up occurs even when the total species concentration is non-increasing. For systems that instead have homogeneous Neumann boundary conditions, it is currently unknown whether this deviation between the continuous and discretized system can occur. Therefore, it is worth examining the discretized system independently of the continuous system. Since no criteria exist for the boundedness of the discretized system, the focus of this paper is to determine sufficient conditions to guarantee the system with diffusion remains bounded for all time. We consider reaction-diffusion systems on a 1D domain with homogeneous Neumann boundary conditions and non-negative initial data and solutions. We define a Lyapunov-like function and show that its existence guarantees that the discretized reaction-diffusion system is bounded. These results are considered in the context of three example systems for which Lyapunov-like functions can and cannot be found.

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Boundedness and Large-Time Behavior Results for a Diffusive Epidemic Model

Cited by 6 publications

References 13 publications

Global solution for a diffusive epidemic model (HIV/AIDS) with an exponential behavior of source

Global solution for a diffusive epidemic model (HIV/AIDS) with an exponential behavior of source

Lyapunov functions and global stability for a spatially diffusive SIR epidemic model

Boundedness of a class of discretized reaction-diffusion systems

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