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.     

一类具有垂直传染率的SIS模型的稳定性分析

研究了一类具有垂直传染率的SIS模型,首先计算出该模型的基本再生数和平衡点,其次分析了该模型在无病平衡点处的局部渐近稳定性和全局稳定性;然后构造Lyapunov函数证明了地方病平衡点的全局稳定性;最后得到当基本再生数小于1时,传染病会逐渐消失;基本再生数大于1时,传染病将会流行并最终形成一种地方病.     

一类具有垂直传染的传染病模型的稳定性

基于动力系统的理论,讨论了一类具有垂直传染的传染病模型的稳定性.采用下一代矩阵法获得了基本再生数R₀.当R₀<1时,由Routh-Hurwitz判别法,得到了无病平衡点的局部渐近稳定性.通过构造Lyapunov函数,证明了系统在无病平衡点全局渐近稳定.当R₀> 1时,地方病平衡点存在且唯一,借助Routh判据,得出了系统在地方病平衡点局部渐近稳定的条件,并通过构造Lyapunov函数,证明了系统在地方病平衡点全局渐近稳定.最后,用数值模拟验证了结论的合理性.     

一类潜伏期与传染期均传染的SEIQR传染病模型

研究了一类潜伏期和感染期均传染的SEIQR模型的全局稳定性,找到疾病绝灭和持续生存的阈值——基本再生数R0,证明了无病平衡点和地方病平衡点的存在性和全局渐近稳定性,揭示了隔离对疾病控制的积极作用。     

一类具有垂直传染与接种的DS—I—R传染病模型研究

本文研究了-类具有垂直传染与接种的疾病在多个易感群体中传播的DS-I-R传染病模型,得到了疾病流行的阈值.运用微分方程定性与稳定性理论分析了无病平衡点的局部稳定与全局渐近稳定性及存在唯一地方病平衡点与其全局渐近稳定性.     

一类具有饱和发生率的SIVS吸毒模型的动力学分析

本文研究一类具有饱和传染率的SIVS传染病模型.首先利用Routh-Hurwitz判据和特征根方法,得到平衡点的局部渐近稳定性,其次证明系统的持久性和无病平衡点的全局渐近稳定性,并利用极限系统理论得到地方病平衡点的全局渐近稳定性.最后用数值模拟验证理论结果的正确性.     

一类具有常数输入和垂直传染的SIRI传染病模型

建立了一类易感者及染病者均有常数输入,疾病具有垂直传染以及一般形式饱和接触率的SIRI传染病模型,分别研究了p=0,0相似文献   

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．     

具有饱和传染率、免疫接种和垂直传染的SIR传染病模型的稳定性北大核心

研究了一类具有饱和传染率、免疫接种和垂直传染的SIR传染病模型,确定了疾病的基本再生数,得出当疾病的基本再生数小于1时,无病平衡点是全局指数渐近稳定的,当疾病基本再生数大于1时.地方病平衡点是全局渐近稳定的,讨论了其生物意义.     

具有饱和传染率、免疫接种和垂直传染的SIR传染病模型的稳定性

研究了一类具有饱和传染率、免疫接种和垂直传染的SIR传染病模型,确定了疾病的基本再生数,得出当疾病的基本再生数小于1时,无病平衡点是全局指数渐近稳定的,当疾病基本再生数大于1时.地方病平衡点是全局渐近稳定的,讨论了其生物意义.     

Stability and Hopf bifurcation of a delayed epidemic model with stage structure and nonlinear incidence rate

In this paper, a stage‐structured SI epidemic model with time delay and nonlinear incidence rate is investigated. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease‐free equilibrium, and the existence of Hopf bifurcations are established. By comparison arguments, it is proved that if the basic reproduction number is less than unity, the disease‐free equilibrium is globally asymptotically stable. If the basic reproduction number is greater than unity, by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium. Copyright © 2013 John Wiley & Sons, Ltd.     

具有垂直传染和接触传染的传染病模型的稳定性研究

本文研究了一类具有垂直传染和接触传染的传染病模型.利用常微分方程定性与稳定性方法,分析了该模型非负平衡点的存在性及其局部稳定性.同时,利用LaSalle不变性原理和通过构造适当的Lyapunov函数,获得了平凡平衡点、无病平衡点和地方病平衡点全局渐近稳定的充分条件.结果表明当基本再生数小于等于1时,所有种群趋于灭绝;当基本再生数大于1和病毒主导再生数小于1时,病毒很快被清除;当基本再生数大于1和病毒主导再生数大于1以及满足一定条件时,病毒持续流行并将成为一种地方病.     

基于两斑块和人口流动的SIR传染病模型的稳定性

根据传染病动力学原理,考虑人口在两斑块上流动且具有非线性传染率,建立了一类基于两斑块和人口流动的SIR传染病模型.利用常微分方程定性与稳定性方法,分析了模型永久持续性和非负平衡点的存在性,通过构造适当的Lyapunov函数和极限系统理论,获得无病平衡点和地方病平衡点全局渐近稳定的充分条件.研究结果表明:基本再生数是决定疾病流行与否的阈值,当基本再生数小于等于1时,感染者逐渐消失,病毒趋于灭绝;当基本再生数大于1并满足永久持续条件时,感染者持续存在且病毒持续流行并将成为一种地方病.     

Dynamics of an epidemic model with relapse and delay

In this paper, we consider a new epidemiological model with delay and relapse phenomena. Firstly, a basic reproduction number $R_0$ is identified, which serves as a threshold parameter for the stability of the equilibria of the model. Then, beginning with the delay-free model, the global asymptotic stability of the equilibria is obtained through the construction of suitable Lyapunov functions. For the delay model, the stability of the positive equilibrium and the existence of the local Hopf bifurcation are discussed. Furthermore, the application of the normal form theory and center manifold theorem is used to determine the direction and stability of these Hopf bifurcations. Finally, we shed light on corresponding biological implications from a numerical perspective. It turns out that time delay affects the stability of the positive equilibrium, leading to the occurrence of periodic oscillations and disease recurrence.     

Global dynamics of a delayed eco‐epidemiological model with Holling type‐III functional response

In this paper, an eco‐epidemiological model with Holling type‐III functional response and a time delay representing the gestation period of the predators is investigated. In the model, it is assumed that the predator population suffers a transmissible disease. The disease basic reproduction number is obtained. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria and the existence of Hopf bifurcations at the disease‐free equilibrium and the endemic‐coexistence equilibrium are established, respectively. By using the persistence theory on infinite dimensional systems, it is proved that if the disease basic reproduction number is greater than unity, the system is permanent. By means of Lyapunov functionals and LaSalle's invariance principle, sufficient conditions are obtained for the global stability of the endemic‐coexistence equilibrium, the disease‐free equilibrium and the predator‐extinction equilibrium of the system, respectively. Copyright © 2013 John Wiley & Sons, Ltd.     

Optimal control and stability analysis of an online game addiction model with two stages

In this paper, we establish a mathematical model of online game addiction with two stages to research the dynamic properties of it. The existence of all equilibria is obtained, and the basic reproduction number is calculated by the method of next-generation matrix. The global asymptotic stability of disease-free equilibrium (DFE) is proved by comparison principle, and the global asymptotic stability of endemic equilibrium (EE) is proved by constructing an appropriate Lyapunov function. Then we use the Pontryagin's maximum principle to find the optimal solution of the model, so that the number of infected people can be minimized. In numerical simulation, firstly, we validate the global stability of DFE and EE. Secondly, we consider three kind of control measures (treatment, isolation, and education) and divide them into four cases. The models with control and without control are solved numerically by using forward and backward sweep Runge-Kutta method. In order to achieve the best control effect, we suggest that three kind of measures should be used simultaneously according to the optimal control strategy.     

Mathematical analysis of an eco-epidemiological predator–prey model with stage-structure and latency

In this paper, an eco-epidemiological predator–prey model with stage structure for the prey and a time delay describing the latent period of the disease is investigated. By analyzing corresponding characteristic equations, the local stability of the trivial equilibrium, the predator-extinction equilibrium, the disease-free equilibrium and the endemic equilibrium is addressed. The existence of Hopf bifurcations at the endemic equilibrium is established. By using Lyapunov functionals and LaSalle’s invariance principle, sufficient conditions are obtained for the global asymptotic stability of the trivial equilibrium, the predator-extinction equilibrium, the disease-free equilibrium and the endemic equilibrium of the model.     

Global dynamics of multi‐group dengue disease model with latency distributions

In this paper, by incorporating latencies for both human beings and female mosquitoes to the mosquito‐borne diseases model, we investigate a class of multi‐group dengue disease model and study the impacts of heterogeneity and latencies on the spread of infectious disease. Dynamical properties of the multi‐group model with distributed delays are established. The results showthat the global asymptotic stability of the disease‐free equilibrium and the endemic equilibrium depends only on the basic reproduction number. Our proofs for global stability of equilibria use the classical method of Lyapunov functions and the graph‐theoretic approach for large‐scale delay systems. Copyright © 2014 John Wiley & Sons, Ltd.     

An SIRS model with a nonlinear incidence rate

The global dynamics of an SIRS model with a nonlinear incidence rate is investigated. We establish a threshold for a disease to be extinct or endemic, analyze the existence and asymptotic stability of equilibria, and verify the existence of bistable states, i.e., a stable disease free equilibrium and a stable endemic equilibrium or a stable limit cycle. In particular, we find that the model admits stability switches as a parameter changes. We also investigate the backward bifurcation, the Hopf bifurcation and Bogdanov–Takens bifurcation and obtain the Hopf bifurcation criteria and Bogdanov–Takens bifurcation curves, which are important for making strategies for controlling a disease.     

Dynamics of an age‐structured host‐vector model for malaria transmission

In this paper, we propose a host‐vector model for malaria transmission by incorporating infection age in the infected host population and nonlinear incidence for transmission from infectious vectors to susceptible hosts. One novelty of the model is that the recovered hosts only have temporary immunity and another is that successfully recovered infected hosts may become susceptible immediately. Firstly, the existence and local stability of equilibria is studied. Secondly, rigorous mathematical analyses on technical materials and necessary arguments, including asymptotic smoothness and uniform persistence of the system, are given. Thirdly, by applying the fluctuation lemma and the approach of Lyapunov functionals, the threshold dynamics of the model for a special case were established. Roughly speaking, the disease‐free equilibrium is globally asymptotically stable when the basic reproduction number is less than one and otherwise the endemic equilibrium is globally asymptotically stable when no reinfection occurs. It is shown that the infection age and nonlinear incidence not only impact on the basic reproduction number but also could affect the values of the endemic steady state. Numerical simulations were performed to support the theoretical results.