Global Existence and Asymptotic Behavior of Solutions for Some Coupled Systems via a Lyapunov Functional

Djebara, Lamia; Abdelmalek, Salem; Bendoukha, Samir

doi:10.1007/s10473-019-0606-7

Cited by 7 publications

(8 citation statements)

References 12 publications

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“…In the present study, we study the existence of equilibria and their asymptotic stability conditions for the diffusive epidemic model considered in [10], which is an extension of that proposed in [25]. We recall that in [10], we established the global existence of solutions to problem (1)-( 3). We have now established the system model and parameter descriptions.…”

Section: Introductionmentioning

confidence: 94%

“…as our candidate Lyapunov function. The aim of this subsection is to obtain a weaker condition than that of [10].…”

Section: Global Asymptotic Stability With R 0 ≤mentioning

confidence: 99%

“…In this manuscript, we consider the reaction-diffusion epidemic phenomenon proposed in [10], which is an extended version of the SIS epidemic model with the nonlinear incidence uϕ (v). The system is described as…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Global and local asymptotic stability of an epidemic reaction-diffusion model with a nonlinear incidence

Djebara¹,

Douaifia²,

Abdelmalek³

et al. 2020

Preprint

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The aim of this paper is to study the dynamics of a reaction-diffusion SIS (susceptible-infectious-susceptible) epidemic model with a nonlinear incidence rate describing the transmission of a communicable disease between individuals. We prove that the proposed model has two steady states under one condition. By analyzing the eigenvalues and using the Routh-Hurwitz criterion and an appropriately constructed Lyapunov functional, we establish the local and global asymptotic stability of the non negative constant steady states subject to the basic reproduction number being greater than unity and of the disease-free equilibrium subject to the basic reproduction number being smaller than or equal to unity in ODE case. By applying an appropriately constructed Lyapunov functional, we identify the condition of the global stability in the PDE case. Finally, we present some numerical examples illustrating and confirming the analytical results obtained throughout the paper.

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Section: Introductionmentioning

confidence: 94%

“…as our candidate Lyapunov function. The aim of this subsection is to obtain a weaker condition than that of [10].…”

Section: Global Asymptotic Stability With R 0 ≤mentioning

confidence: 99%

See 1 more Smart Citation

Global and local asymptotic stability of an epidemic reaction-diffusion model with a nonlinear incidence

Djebara¹,

Douaifia²,

Abdelmalek³

et al. 2020

Preprint

Self Cite

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show abstract

“…In this manuscript, we consider the reaction‐diffusion epidemic phenomenon proposed in Djebara et al, 1 which is an extended version of the SIS epidemic model with the nonlinear incidence uφ ( v ). The system is described

({, \begin{matrix} center \\ array \frac{\partial u}{\partial t} - d_{1} Δ u = Λ - μ u - λ u φ (v) = : F (u, v) in (0, \infty) \times Ω, \\ array \frac{\partial v}{\partial t} - d_{2} Δ v = - σ v + λ u φ (v) = : G (u, v) in (0, \infty) \times Ω . \end{matrix})

…”

Section: Introductionmentioning

confidence: 99%

“…where 𝜃 > 0, as our candidate Lyapunov function. The aim of this subsection is to obtain a weaker condition than that of Djebara et al1 )is globally asymptotically stable, we must show that  𝜃 (t) is a Lyapunov function. The positive definiteness of  𝜃 (t) is evident.…”

mentioning

confidence: 98%

Global and local asymptotic stability of an epidemic reaction‐diffusion model with a nonlinear incidence

Djebara

Douaifia

Abdelmalek

et al. 2022

Math Methods in App Sciences

Self Cite

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The aim of this paper is to study the dynamics of a reaction‐diffusion SI (susceptible‐infectious) epidemic model with a nonlinear incidence rate describing the transmission of a communicable disease between individuals. We prove that the proposed model has two steady states under one condition. By analyzing the eigenvalues and using the linearization method and an appropriately constructed Lyapunov functional, we establish the local and global asymptotic stability of the non‐negative constant steady states subject to the basic reproduction number being greater than unity and of the disease‐free equilibrium subject to the basic reproduction number being smaller than or equal to unity in ODE case. By applying an appropriately constructed Lyapunov functional, we identify the condition of the global stability in the PDE case. Finally, we present some numerical examples illustrating and confirming the analytical results obtained throughout the paper.

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Global Existence and Asymptotic Stability for a Class of Coupled Reaction-Diffusion Systems on Growing Domains

2021

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Global Existence and Asymptotic Behavior of Solutions for Some Coupled Systems via a Lyapunov Functional

Cited by 7 publications

References 12 publications

Global and local asymptotic stability of an epidemic reaction-diffusion model with a nonlinear incidence

Global and local asymptotic stability of an epidemic reaction-diffusion model with a nonlinear incidence

Global and local asymptotic stability of an epidemic reaction‐diffusion model with a nonlinear incidence

Global Existence and Asymptotic Stability for a Class of Coupled Reaction-Diffusion Systems on Growing Domains

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