具有因病死亡且输人为Berverton-Holt函数的离散SIS传染病模型分析

引入相应的概率建立了考虑因病死亡且输入为Berverton-Holt的离散SIS传染病模型,确定了决定其动力性态的阈值,在阈值之下模型仅存在无病平衡点,且无病平衡点是全局渐近稳定的;在阈值之上模型是一致持续的,有唯一的地方病平衡点存在,且可以猜想地方病平衡点是全局渐近稳定的.     

染病者有常数输入的传染病模型

讨论在隔离措施下易感者和染病者都有常数移民的传染病模型.给出了模型的地方病平衡点,证明了地方病平衡点的稳定性.     

具有阶段结构的SIR传染病模型

研究了一类具有阶段结构的SIR传染病模型，在模型中假设种群分幼年和成年两个阶段，且只有成年种群染病，并且采用与成年易感者数量有关的一般非线性传染率，得到了系统解的有界性及无病平衡点和地方病平衡点存在的条件．通过对平衡点对应的特征方程的讨论得到了平衡点局部渐近稳定的条件，同时证明了平衡点的全局渐近稳定性，并对结论进行了数值模拟．     

潜伏期和染病期均具有传染性的媒介传染病模型的全局稳定性分析

讨论潜伏期和染病期均具有传染性的媒介传染病模型.得到模型基本再生数的表达式,证明了当基本再生数小于1时,无病平衡点是全局渐近稳定的,此时疾病消亡;当基本再生数大于1时,无病平衡点是不稳定的,系统存在全局渐近稳定的地方病平衡点,此时,疾病将在人群中持续存在,数值模拟验证了理论结果.     

一类具有非线性发生率的SIR传染病模型的全局稳定性

研究一类具有非线性发生率的SIR传染病模型.应用微分方程定性理论分别得到了该系统无病平衡点、地方病平衡点全局渐近稳定的充分条件,并进行了数值模拟.     

一类具有不同感染率的SIR模型的稳定性分析

建立了一类具有不同感染率且出生和死亡具有密度制约的SIR传染病模型,应用极限系统理论以及Liapunov稳定性定理得到该系统平衡点的稳定性.     

具有常数输入的SEIR模型的稳定性分析

讨论了易感者类和潜伏者类均为常数输入,潜伏期、染病期和恢复期均具有传染力,且传染率为一般传染率的SEIR传染病模型.利用Hurwitz判据证明了地方病平衡点的局部渐近稳定性,进一步利用复合矩阵理论得到了地方病平衡点全局渐近稳定的充分条件.     

一类潜伏期和染病期均具有传染性的手足口病模型

根据手足口病的病理特性及传播特点,建立一类描述其传播的数学模型并对模型的动力学性态进行分析.首先利用再生矩阵的方法定义了模型的基本再生数R_0,同时通过构造Lyapunov函数和Routh-Hurwitz判据证明了当R_0≤1时无病平衡点E_0的金局渐近稳定性,R_0>1时地方病平衡点E_*的局部渐近稳定性,并进一步证明了在一定条件下地方病平衡点的全局渐近稳定性.     

具有免疫接种且总人口规模变化的SIR传染病模型的稳定性

讨论一类具有预防免疫接种且有效接触率依赖于总人口的SIR传染病模型,给出了决定疾病灭绝和持续生存的基本再生数σ的表达式,在一定条件下证明了疾病消除平衡点的全局稳定性,得到了唯一地方病平衡点的存在性和局部渐近稳定性条件.最后研究了具有双线性传染率和标准传染率的两个具体模型,并证明了当σ>1时该模型地方病平衡点的全局渐近稳定性.     

离散的SI和SIS传染病模型的研究

为了描述个体的死亡、染病者的恢复以及疾病的传染,引入了相应的概率.基于总种群中个体数量为常数的假设,根据染病者能否恢复分别建立了具有生命动力学的离散SI和SIS传染病模型.所得到的结果显示:它们具有与相应连续模型相同的动力学性态,并确定了各自的阈值.在它们的阈值之下,传染病最终将灭绝;在它们的阈值之上,传染病将会发展成为地方病,染病者的数量将趋向于一确定的正常数.     

Allee effects in a discrete-time SIS epidemic model with infected newborns

We extend the framework of Rios-Soto et al. (Contemporary Mathematics, 2006, 410, 297) to include both compensatory (contest competition) and overcompensatory (scramble competition) population dynamics with and without the Allee effect. We compute the basic reproductive number ?₀, and use it to predict the (uniform) persistence or extinction of the infective population, where the population dynamics are compensatory and the Allee effect is either present or absent. We also explore the relationship between the demographic equation and the epidemic process, where the total population dynamics are overcompensatory. In particular, we show that the demographic dynamics drive both the susceptible and infective dynamics. This is in contrast to the recent observations of Franke and Yakubu, that the demographic dynamics can be chaotic while the infective dynamics are oscillatory and non-chaotic in periodically-forced SIS epidemic models (Mathematical Biosciences, 2006, 204, 68).     

Two-group SIR epidemic model with stochastic perturbation

A stochastic two-group SIR model is presented in this paper.The existence and uniqueness of its nonnegative solution is obtained,and the solution belongs to a positively invariant set.Furthermore,the globally asymptotical stability of the disease-free equilibrium is deduced by the stochastic Lyapunov functional method if R0 ≤ 1,which means the disease will die out.While if R0 1,we show that the solution is fluctuating around a point which is the endemic equilibrium of the deterministic model in time average.In addition,the intensity of the fluctuation is proportional to the intensity of the white noise.When the white noise is small,we consider the disease will prevail.At last,we illustrate the dynamic behavior of the model and their approximations via a range of numerical experiments.     

Dynamic behaviour of a discrete-time SIR model with information dependent vaccine uptake

This paper proposes and analyzes a discrete-time deterministic SIR model with information dependent immunization behaviour, where vaccination coverage at birth during any period of time is a general phenomenological function of the risk of infection that is perceived at the beginning of the period. Results on existence of equilibria, their local stability, and system persistence are proved. Then, by considering the noteworthy subcase of a piecewise linear ‘prevalence-dependent’ coverage function, the local stability of the endemic state is proved and conditions for its global asymptotic stability are given. Some insight on both Neimarck-Sacher and period-doubling bifurcations are provided. Overall we show that prevalence-dependent coverage is an essentially stabilising force. However period-doubling bifurcations are possible though under stressed parameter constellations.     

Numerical modelling of an SIR epidemic model with diffusion

A spatial SIR reaction-diffusion model for the transmission disease such as whooping cough is studied. The behaviour of positive solutions to a reaction-diffusion system with homogeneous Neumann boundary conditions are investigated. Sufficient conditions for the local and global asymptotical stability are given by linearization and by using Lyapunov functional. Our result shows that the disease-free equilibrium is globally asymptotically stable if the contact rate is small. These results are verified numerically by constructing, and then simulating, a robust implicit finite-difference method. Furthermore, the new implicit finite-difference method will be seen to be more competitive (in terms of numerical stability) than the standard finite-difference method.     

Persistence in a discrete-time,stage-structured epidemic model

Discrete-time SI and SIR epidemic models, formulated by Emmert and Allen [J. Differ. Equ. Appl., 10 (2004), pp. 1177–1199] for the spread of a fungal disease in a structured amphibian host population, are analysed. Criteria for persistence of the population as well as for persistence of the disease are established. Global stability results for host extinction and for the disease-free equilibrium are presented.     

Traveling waves in a nonlocal dispersal SIR epidemic model

This paper is concerned with traveling wave solutions of a nonlocal dispersal SIR epidemic model. The existence and nonexistence of traveling wave solutions are determined by the basic reproduction number and the minimal wave speed. This threshold dynamics are proved by Schauder’s fixed point theorem and the Laplace transform. The main difficulties are that the semiflow generated by the model does not have the order-preserving property and the solutions lack of regularity.     

Permanence of a delayed SIR epidemic model with general nonlinear incidence rate

In this paper, a SIR model with two delays and general nonlinear incidence rate is considered. The local and global asymptotical stabilities of the disease‐free equilibrium are given. The local asymptotical stability and the existence of Hopf bifurcations at the endemic equilibrium are also established by analyzing the distribution of the characteristic values. Furthermore, the sufficient conditions for the permanence of the system are given. Some numerical simulations to support the analytical conclusions are carried out. At last, some conclusions are given. Copyright © 2014 John Wiley & Sons, Ltd.     

Damped oscillating traveling waves of a diffusive SIR epidemic model

This paper is concerned with the damped oscillating traveling waves of a diffusive SIR model with vital dynamics. Our result implies that the vital dynamics can induce damped spatio-temporal oscillations. Biologically, this shows that the vital dynamics can cause multiple outbreaks of infection with decreasing maximum outbreak sizes.