An SIR epidemic model with free boundary

Abstract

An SIR epidemic model with free boundary is investigated.This model describes the transmission of diseases. The behavior of positive solutions to a reaction-diffusion system in a radially symmetric domain is investigated. The existence and uniqueness of the global solution are given by the contraction mapping theorem. Sufficient conditions for the disease vanishing or spreading are given. Our result shows that the disease will not spread to the whole area if the basic reproduction number R 0 < 1 or the initial infected radius h 0 is sufficiently small even that R 0 > 1. Moreover, we prove that the disease will spread to the whole area if R 0 > 1 and the initial infected radius h 0 is suitably large. IntroductionRecently epidemic model has been received a great attention in mathematical ecology. To describe the development of an infectious disease, compartmental models have been given to separate a population into various classes based on the stages of infection [2]. The classical SIR model is described by partitioning the population into susceptible, infectious and recovered individuals, denoted by S, I and R, respectively. Assume that the disease incubation period is negligible so that each susceptible individual becomes infectious and later recovers with a permanently or temporarily acquired immunity, then the SIR model is governed by the following system of differential equations:( 1.1) where the total population size has been normalized to one and the influx of the susceptible comes from a constant recruitment rate b. The death rate for the S, I and R class is, respectively, given by µ 1 , µ 2 and µ 3 . Biologically, it is natural to assume that µ 1 < min{µ 2 , µ 3 }. The standard incidence of disease is denoted by βSI, where β is the constant effective contact rate, which is the average number of contacts of the infectious per unit time. The recovery rate of the infectious is

Dynamics of a nonlocal SIS epidemic model with free boundary

“…Then [26], due to the effect of the nonlocal term, we find that our result shows that the disease will die out eventually with a smaller h 0 but with no restriction on the relation between βσ µ(µ+γ) and 1. Proof.…”

“…The organization of this paper is as follows. In Section 2, we prove the general existence and uniqueness result, which implies in particular that problem (6)- (7) has a unique positive solution defined for all t > 0, the method is inspired by [10,8,12,26]. In Section 3, we firstly analyze an eigenvalue problem and discuss the property of its principal eigenvalue λ 1 .…”

“…We are going to construct a suitable upper solution to (6) and then apply Lemma 3.3 to prove this theorem. As in [10,26], we define…”

“…Recalling the fact that there exists T 1 > 0 such that 0 < N (x, t) ≤ σ µ in [g(t), h(t)] × [T 1 , +∞), then as in [10,26], we define…”

A SIS reaction–diffusion–advection model in a low-risk and high-risk domain

“…For example, it was used to describe the melting of ice in contact with water [32], the oxygen in muscles in [10], the wound healing in [9], the spreading of the invasion species in [11][12][13][14]18,20,24,26,36,37,39]. Recently, it was used to study an SIR epidemic model in a homogeneous environment in [21].…”