Stability analysis for an epidemic model with stage structure

An epidemic model with stage structure is formulated. The period of infection is partitioned into the early and later stages according to the developing process of infection, and the infectious individuals in the different stages have the different ability of transmitting disease. The constant recruitment rate and exponential natural death, as well as the disease-related death, are incorporated into the model. The basic reproduction number of this model is determined by the method of next generation matrix. The global stability of the disease-free equilibrium and the local stability of the endemic equilibrium are obtained; the global stability of the endemic equilibrium is got under the case that the infection is not fatal.     

具有扩散病毒感染群体动力学模型的稳定性与Hopf分歧

讨论了一个具有诺依曼边界条件扩散病毒感染群体动力学模型.证明了模型正常数平衡点的稳定性和扩散引起的Hopf分歧的存在性.     

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.     

一类人体内寄生虫传染的数学模型

研究了一类人体内寄生虫传染的数学模型,分析了人体内健康细胞、受传染细胞和寄生虫的变化规律,对模型进行了定性和稳定性分析,讨论了模型无病平衡点和地方病平衡点的存在条件,得到了各个平衡点渐近稳定的充分条件.结合实际钩虫病感染数据,对模型进行数值模拟,并绘制出模型的变化趋势图.     

STABILITY ANALYSIS OF A TRITROPHIC FOOD CHAIN MODEL WITH AN ADAPTIVE PARAMETER FOR THE PREDATOR

Abstract The study of three‐species communities have become the focus of considerable attention, and because the studies of ecological communities start with their food web, we consider a tritrophic food chain model comprised of the prey, the predator, and the super‐predator. The classical assumption of the domino effect is supplemented with an adaptive parameter for the predator (in the absence of prey). Thus, the model exhibits an equilibrium with the predator‐top‐predator steady state, which is a saddle point. Dynamical behaviors such as boundedness, existence of periodic orbits, persistence, as well as stability are analyzed. The long‐term coexistence of the three interacting species is addressed, and the stability analysis of the model shows that the biologically most relevant equilibrium point is globally asymptotically stable whenever it satisfies a certain criterion. Practical implications are explored and related to real populations.     

A compartmental model for global spread dynamics of malware under mutation

Malware mutation is pervasive among networks. Modeling and understanding its propagation characteristics have been of great importance. In this study, a new compartmental model that extends the present model by incorporating mutated malware into the modeling process as a separate dynamic variable is proposed and theoretically analyzed to deepen the understanding of the spreading mechanisms of mutated malware. The model involves two equilibria, namely, malware‐free equilibrium and malware equilibrium, wherein both have proven to be locally and globally asymptotically stable through the Routh‐Hurwitz criterion and Lyapunov functional approach, respectively. An epidemic threshold is obtained that clearly forms the boundary among the comprehensive dynamics of the model between two distinct ramifications: one with mutation infection prevalence and the other without any mutation infection. Both are incarnated via the existence and stability of the equilibria admitted by the model. Further analyses show that the mutation is related not only to the epidemic threshold, but also to the malware prevalence level. The numerical simulations based on the analytic results demonstrate that the diffusion of mutated malware can fall away or can be maintained at a suitable level.     

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.     

一个淋病扩散模型的定性分析

本文利用奇点理论与Ｂｅｎｄｉｘｓｏｎ定理对淋病扩散模型进行了定性分析,给出了可行平衡点附近,特别是高阶平衡点附近轨线的定性结构,研究了可行平衡点的全局渐近稳定性,得到了完整的结果。     

Game analysis of technology innovation alliance stability based on knowledge transfer

Under incomplete information, a game model is used to investigate the influence of ownership level and learning ability on the stability of technology innovation alliance from the perspective of knowledge transfer. The decision-making processes of involved parties are divided into two stages in the model. In the first stage, the firm possessing advanced technology decides on the level of knowledge it transfers to its alliance partner. In the second stage, the decision of the parties on whether to maintain or terminate the alliance is based on two factors: the level of knowledge learned and profits gained. The outcomes of the Cournot–Nash equilibrium in the model can reveal when the parties decide to maintain or terminate the alliance. The model explores the status of alliance stability under different ownership levels and learning abilities to provide theoretical support for the selection of optimal dynamic competitive-cooperative relationship and managerial flexibility.     

A Mathematical Model with Delays for Schistosomiasis Japonicum Transmission

A dynamic model of schistosoma japonicum transmission is presented that incorporates effects of the prepatent periods of the different stages of schistosoma into Baxbour＇s model. The model consists of four delay differential equations. Stability of the disease free equilibrium and the existence of an endemic equilibrium for this model are stated in terms of a key threshold parameter. The study of dynamics for the model shows that the endemic equilibrium is globally stable in an open region if it exists and there is no delays, and for some nonzero delays the endemic equilibrium undergoes Hopf bifurcation and a periodic orbit emerges. Some numerical results are provided to support the theoretic results in this paper. These results suggest that prepatent periods in infection affect the prevalence of schistosomiasis, and it is an effective strategy on schistosomiasis control to lengthen in prepatent period on infected definitive hosts by drug treatment （or lengthen in prepatent period on infected intermediate snails by lower water temperature）.     

Stability and Hopf bifurcation of a delayed viral infection model with logistic growth and saturated immune impairment

In this paper, the stability and Hopf bifurcation of a delayed viral infection model with logistic growth and saturated immune impairment is studied. It is shown that there exist 3 equilibria. The sufficient conditions for local asymptotic stability of the infection‐free equilibrium and no‐immune equilibrium are given. We also discussed the local stability of positive equilibrium and the existence of Hopf bifurcation. Moreover, the direction and stability of Hopf bifurcation is obtained by using standard form theory and the center manifold theorem. Finally, numerical simulations are performed to verify the theoretical conclusions.     

ASYMPTOTICAL ANALYSIS OF A REACTIONDIFFUSION EQUATIONS D-SIS EPIDEMIC MODEL

By monotone methods and invariant region theory,a reaction-diffusion equa- tions D-SIS epidemic model with bilinear rate is studied.The existence and uniqueness of the solution of the model are proved.The basic reproductive number which determines whether the disease is extinct or not is found.The globally asymptotical stability of the disease-free equilibrium and the endemic equilibrium are obtained.Some results of the ordinary differential equations model are extended to the present partial differential equations model.     

阻滞增长模型支持下对益虫和害虫数量关系及农药影响的分析

研究了在自然状态和喷洒农药情况下益虫和害虫数量的发展趋势.基于logistic模型并加以改进建模,通过分析微分方程平衡点的稳定性,分别得出了自然状态下益虫和害虫的数量发展规律.在此基础上,进而建立了农药影响的模型并进行了计算机仿真,仿真结果证明了新模型的合理性和适用性,解决了Lotka—Volterra模型存在的问题.另外,该仿真程序本身可以作为应用程序用于实际的农田管理.     

Global stability and periodic oscillations for an SIV infection model with immune response and intracellular delays

In this paper, we consider the combined effects of cytotoxic T lymphocyte (CTL) responses on the competition dynamics of two Simian immunodeficiency virus (SIV) strains model. One of strains concerns a relatively slowly replicating and mildly cytopathic virus in the early infection (SIVMneCL8), the other is faster replicating and more cytopathic virus at later stages of the infection (SIVMne170). It is shown that the global dynamics of the ordinary differential equations can be determined by several threshold parameters, and we prove the global stability of the equilibria by rigorous mathematical analysis. To account for a series of infection mechanism leading to viral production, we incorporate time delays in the infection term. Using the methods of constructing suitable Lyapunov functionals and LaSalle’s invariance principle, we obtain the sufficient conditions for the global attractiveness of infection-free equilibrium with both virus strains going extinct, single-infection equilibrium with one of two virus strains out-competing the other one and the two strains coexisting infection equilibrium. We establish that the intracellular delays can destabilize the single-infection equilibrium leading to Hopf bifurcation and periodic oscillations. We show that introduction of immune responses is responsible for the coexistence of two virus strains and the intracellular delays may alter the two-strain competition results. Numerical simulations are presented to illustrate the theoretical conclusions.     

一类具有Beddington-DeAngelis型功能反应函数的HIV病毒动力学系统模型的稳定性

基于Nowak等于1996年提出的一类经典的HIV病毒动力学模型,考虑了一类具有Beddington-DeAngelis功能反映函数的HIV病毒动力学模型,并研究了无病毒平衡点的全局稳定性与感染平衡点的局部稳定性等.     

A MODEL OF THE ENZOOTIOLOGY OF LYME DISEASE IN THE ATLANTIC NORTHEAST OF THE UNITED STATES

A mathematical model is presented for the dynamics of the rate of infection of the Lyme disease vector tick Ixodes dammini (Acari: Ixodidae) by the spirochete Borrelia burgdorferi, in the Atlantic Northeast of the United States. According to this model, moderate reductions in the abundance of white-tailed deer Odocoileus virginianus may either decrease or increase the spirochete infection rate in ticks, provided the deer are not reservoir hosts for Lyme disease. Expressions for the basic reproductive rate of the disease are computed analytically for special cases, and it is shown that as the basic reproductive rate increases, a proportional reduction in the tick population produces a smaller proportional reduction in the infection rate, so that vector control is less effective far above the threshold. The model also shows that control of the mouse reservoir hosts Peromyscus leucopus could reduce the infection rate if the survivorship of juvenile stages of ticks were reduced as a consequence. If the survivorship of juvenile stages does not decline as the rodent population is reduced, then rodent reduction can increase the spirochete infection rate in the ticks.     

具有媒体饱和效应影响的时滞SIS模型研究

媒体报道对疾病的预防和控制有着重要的作用,其可以减少人们感染疾病的机会.通过建立具有媒体饱和的传染病时滞模型来刻画媒体报道对感染率的影响,首先计算出无病平衡点和当R_01时存在唯一的地方病平衡点;其次,分析了平衡点的稳定性,并得到当参数满足一定条件时,时滞τ超过临界值τ_0,地方病平衡点处会出现Hopf分支;最后,通过数值模拟来验证理论分析.     

Global asymptotical properties for a diffused HBV infection model with CTL immune response and nonlinear incidence

This article proposes a diffused hepatitis B virus (HBV) model with CTL immune response and nonlinear incidence for the control of viral infections. By means of different Lyapunov functions, the global asymptotical properties of the viral-free equilibrium and immune-free equilibrium of the model are obtained. Global stability of the positive equilibrium of the model is also considered. The results show that the free diffusion of the virus has no effect on the global stability of such HBV infection problem with Neumann homogeneous boundary conditions.     

A Unified theory of tokamaks and stellarators

Computational methods are described to study the equilibrium, stability, and transport of a fusion plasma confined by a magnetic field with toroidal geometry. Misconceptions in conventional theories of transport are analyzed and new results more consistent with experimental observations are presented. Specifications are given for an advanced stellarator designed to compete with tokamaks as a fusion reactor. © 1994 John Wiley & Sons, Inc.