Dynamics of a nonlocal dispersal SIS epidemic model

In this paper, we are concerned with the global asymptotic stability of each equilibrium of an SIR epidemic model with nonlocal diffusion. Under the assumption of Lipschitz continuity of parameters, the eigenvalue problem associated with the linearized system around the disease-free equilibrium has a principal eigenvalue corresponding to a strictly positive eigenfunction. By setting the eigenfunction as the integral kernel of a Lyapunov function, we prove the global asymptotic stability of the disease-free equilibrium when the basic reproduction number R 0 is less than one. We also prove the uniform persistence of the system when R 0 > 1 by using the persistent theory for dynamical systems. Furthermore, in a special case where the diffusion coefficient for susceptible individuals is equal to zero, we prove the existence, uniqueness and global asymptotic stability of the endemic equilibrium when R 0 > 1 by constructing a suitable Lyapunov function.

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“…where r(•) denotes the spectral radius of an operator. For R 0 in a similar form, see, for instance, [55,57,58].…”

Section: Basic Reproduction Numbermentioning

confidence: 99%

Global dynamics of an SIR epidemic model with nonlocal diffusion

Kuniya

Wang

2018

Nonlinear Analysis: Real World Applications

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“…If and do not depend on the age variable, then model ( 1.1 ) becomes the following nonlocal SIS epidemic model: Nonlocal dispersal epidemic models (without age structure) including ( 1.5 ) have also been studied by some researchers, see Xu et al. ( 2020 ), Yang and Li ( 2017 ), and Yang et al. ( 2013 , 2019 ).…”

Section: Introductionmentioning

confidence: 99%

Mathematical analysis on an age-structured SIS epidemic model with nonlocal diffusion

Kang

Ruan

2021

J. Math. Biol.

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In this paper we propose an age-structured susceptible-infectious-susceptible epidemic model with nonlocal (convolution) diffusion to describe the geographic spread of infectious diseases via long-distance travel. We analyze the well-posedness of the model, investigate the existence and uniqueness of the nontrivial steady state corresponding to an endemic state, and study the local and global stability of this nontrivial steady state. Moreover, we discuss the asymptotic properties of the principal eigenvalue and nontrivial steady state with respect to the nonlocal diffusion rate. The analysis is carried out by using the theory of semigroups and the method of monotone and positive operators. The spectral radius of a positive linear operator associated to the solution flow of the model is identified as a threshold.

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“…Then virions arrive at location from other places at rate ∫ Ω ( − )V( ) . Based on peculiar features of the nonlocal diffusion operators, models in ecology [13][14][15], in epidemiology [16][17][18][19], and even in materials science [20,21] have been investigated.…”

Section: Introductionmentioning

confidence: 99%

Spatial and Temporal Dynamics of a Viral Infection Model with Two Nonlocal Effects

2019

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We propose and study a viral infection model with two nonlocal effects and a general incidence rate. First, the semigroup theory and the classical renewal process are adopted to compute the basic reproduction number R0 as the spectral radius of the next-generation operator. It is shown that R0 equals the principal eigenvalue of a linear operator associated with a positive eigenfunction. Then we obtain the existence of endemic steady states by Shauder fixed point theorem. A threshold dynamics is established by the approach of Lyapunov functionals. Roughly speaking, if R0<1, then the virus-free steady state is globally asymptotically stable; if R0>1, then the endemic steady state is globally attractive under some additional conditions on the incidence rate. Finally, the theoretical results are illustrated by numerical simulations based on a backward Euler method.

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Dynamics of a nonlocal dispersal SIS epidemic model

Cited by 36 publications

References 35 publications

Global dynamics of an SIR epidemic model with nonlocal diffusion

Global dynamics of an SIR epidemic model with nonlocal diffusion

Mathematical analysis on an age-structured SIS epidemic model with nonlocal diffusion

Spatial and Temporal Dynamics of a Viral Infection Model with Two Nonlocal Effects

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