Global dynamics of an SIR epidemic model with nonlocal diffusion

Abstract

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.

Dynamics in a reaction-diffusion epidemic model via environmental driven infection in heterogenous space

“…Let λ 0 (η) be the principal eigenvalue of problem (23). There exists an enough small constant η * ∈ (0, 1), so λ 0 (η * ) > 0, and S 0 (x) − η * > 0 for all x ∈ .…”

“…Although convenient transportation can bring great convenience to people's travel, it also makes the epidemic to spread to other areas quickly. After analysing the number of patients with a disease in different cities, many researchers found that the number of patients was closely related to the mobility of population, and proposed different types of reactiondiffusion epidemic models on this phenomenon (see, for example [5,23,25,44,47,48]). Especially, Yang et al [48] studied a seasonal brucellosis SIV epidemic model with nonlocal transmissions and spatial diffusions.…”

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

“…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.…”

Bifurcation analysis of a general activator-inhibitor model with nonlocal dispersal

“…is the rate at which they leave position x to reach any other position. For example, in water-biomass models the kernel function J(x−y) means the probability per unit length of seeds originating at the point y being dispersed to point x [1,12,30,31]; in epidemic models the kernel function J(x−y) weights the contributions of the susceptible/infective/recovered individuals at location y to the susceptible/infective/recovered individuals at location x [4,26,37,38]. In recent years, the nonlocal dispersal has attracted the attention of a great number of investigators [2,3,7,8,9,10,14,20,21,36] and the references therein for more details.…”

“…Thus it is difficult to analyze the global dynamics of these nonlocal models. Thanks to [26,37,38], by Lyapunov functional and LaSalle's invariance principle, some results about the existence, uniqueness and stability of steady states of nonlocal SIS epidemic models have been obtained if the dispersal spread is held fixed, i.e. d…”