A simplified SIS model is proposed and investigated to understand the impact of spatial heterogeneity of environment and advection on the persistence and eradication of an infectious disease. The free boundary is introduced to model the spreading front of the disease. The basic reproduction number associated with the diseases in the spatial setting is introduced. Sufficient conditions for the disease to be eradicated or to spread are given. Our result shows that if the spreading domain is high-risk at some time, the disease will continue to spread till the whole area is infected; while if the spreading domain is low-risk, the disease may be vanishing or keep spreading depending on the expanding capability and the initial number of the infective individuals. The spreading speeds are also given when spreading happens, numerical simulations are presented to illustrate the impacts of the advection and the expanding capability on the spreading fronts.
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
In this paper, we consider a semilinear heat equation u t = u + c(x, t)u p for (x, t) ∈ Ω × (0, ∞) with nonlinear and nonlocal boundary condition u| ∂Ω×(0,∞) = Ω k(x, y, t)u l dy and nonnegative initial data where p > 0 and l > 0. We prove global existence theorem for max(p, l) 1. Some criteria on this problem which determine whether the solutions blow up in a finite time for sufficiently large or for all nontrivial initial data or the solutions exist for all time with sufficiently small or with any initial data are also given.
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