On the asymptotic behavior of a diffusive epidemic model (AIDS)

Hamaya, Yoshihiro

doi:10.1016/s0362-546x(98)00118-7

Cited by 9 publications

(11 citation statements)

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“…The considered system ()–() may describe the transmission of a communicable disease between individuals such as HIV/AIDS. The population assumed in the model is divided into two classes, susceptible and infective 2 . Functions u ( t , x ) and v ( t , x ) denote the non‐dimensional population densities of the susceptible and infective individuals at time t and location x , respectively.…”

Section: Introductionmentioning

confidence: 99%

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

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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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Section: Introductionmentioning

confidence: 99%

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

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show abstract

“…The considered system (1)-(3) may describe the transmission of a communicable disease between individuals such as HIV/AIDS. The population assumed in the model is divided into two classes, susceptible and infective [13]. Functions u(x, t) and v(x, t) denote the non dimensional population densities of the susceptible and infective individuals at location x and time t, respectively.…”

Section: Introductionmentioning

confidence: 99%

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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“…In the rest of this paper, we will report results only for system (3). Before proofs of Theorem 1, we prepare lemmas.…”

Section: Introductionmentioning

confidence: 99%

“…Then there arises a contradiction by the strong maximum principle (cf. [3], [4], [8], [12]). Indeed, if x 2 ∈ Ω, then k∆w − ∂w/∂t − dw must be negative at (t 2 , x 2 ).…”

Section: Introductionmentioning

confidence: 99%