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Global Attractivity of a Nonlocal Delayed Diffusive Dengue Model in a Spatially Homogeneous Environment

Xue Ren, Ran Zhang

TL;DR

This work addresses the global attractivity of the endemic equilibrium for a nonlocal-delayed dengue diffusion model in a spatially homogeneous environment. It employs a Lyapunov functional built from the logarithmic function $g(\omega)=\omega-1-\ln\omega$ and nonlocal delay terms to show that, when $\Re_0>1$, a unique constant steady state $u^*$ is globally attractive with uniform convergence on $\overline{\Omega}$. The analysis demonstrates $V'(t)\le 0$, ensuring $u(t,x,\phi)\to u^*$ as $t\to\infty$, thereby strengthening previous results by removing an extra sufficiency condition. This provides a robust approach to establishing global attractivity for nonlocal delayed epidemic PDEs and clarifies the threshold behavior around $\Re_0=1$.

Abstract

In Xu and Zhao (2015), the global attractivity of positive constant steady state is established through the application of the fluctuation method, subject to the sufficient condition that the disease will stabilized at the unique spatially-homogeneous steady state if $\Re_0>1$ exceeds a certain threshold. The focus of this study is to eliminate the need for a sufficient condition by employing a suitable Lyapunov functional and prove that the positive constant steady state is globally attractive when $\Re_0$ is exactly greater than unity, which significantly enhancing the findings outlined in Theorem 3.3 of Xu and Zhao (2015).

Global Attractivity of a Nonlocal Delayed Diffusive Dengue Model in a Spatially Homogeneous Environment

TL;DR

This work addresses the global attractivity of the endemic equilibrium for a nonlocal-delayed dengue diffusion model in a spatially homogeneous environment. It employs a Lyapunov functional built from the logarithmic function and nonlocal delay terms to show that, when , a unique constant steady state is globally attractive with uniform convergence on . The analysis demonstrates , ensuring as , thereby strengthening previous results by removing an extra sufficiency condition. This provides a robust approach to establishing global attractivity for nonlocal delayed epidemic PDEs and clarifies the threshold behavior around .

Abstract

In Xu and Zhao (2015), the global attractivity of positive constant steady state is established through the application of the fluctuation method, subject to the sufficient condition that the disease will stabilized at the unique spatially-homogeneous steady state if exceeds a certain threshold. The focus of this study is to eliminate the need for a sufficient condition by employing a suitable Lyapunov functional and prove that the positive constant steady state is globally attractive when is exactly greater than unity, which significantly enhancing the findings outlined in Theorem 3.3 of Xu and Zhao (2015).
Paper Structure (2 sections, 2 theorems, 21 equations)

This paper contains 2 sections, 2 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Main results

Key Result

Theorem 1.1

XuZhaoAMC2015 Let $\Re_0 = \sqrt{\frac{\beta_h\beta_mAH\mathrm{e}^{-\mu_h\tau_b}}{\mu_h\mu_m\rho_h}}$. If then the system (Model) admits a unique constant steady state $u^* = (u_1^*,u_2^*,u_3^*)^\mathrm{T}$ such that uniformly for $x\in \overline{\Omega}$, provided that $\phi\in C_\mathbf{M}$ with $\phi_1(0,\cdot)\not\equiv 0$ or $\phi_3(0,\cdot)\not\equiv 0$.

Theorems & Definitions (3)

  • Theorem 1.1
  • Theorem 2.1
  • proof