具有饱和发生率的离散SIR模型的分支

本文研究了一类具有饱和发生率的离散SIR传染病模型的动力学性态.我们利用再生矩阵的方法定义了模型的基本再生数;直接计算得到了平衡点的存在性;利用线性化矩阵和Jury判据讨论了平衡点的稳定性;利用中心流形定理讨论了平衡点处可能发生的分支,包括flip分支和Hopf分支.最后,通过数值模拟展示了所得到的理论结果和模型的复杂动力学性态.     

考虑媒体作用的传染病模型的分析与控制?

针对媒体宣传教育对人们行为方式和生活习惯的影响,本文考虑了由于媒体影响而导致易感性不同的一个SEI传染病模型。分析了模型可能出现的后项分支及其平衡点的稳定性和持久性。结果表明,当基本再生数小于1时,模型的无病平衡点全局稳定;当基本再生数大于1时,地方病平衡点一致持久。同时,利用控制理论,本文也研究了媒体的宣传作用对易感者进行影响和教育的最优控制措施,给出了使目标函数值最小的最优控制,并用数值模拟显示了模型解的动力学性态及控制措施对防止疾病蔓延所起的作用。     

一类具有阶段结构和饱和发生率的生态流行病模型的稳定性

本文研究一类食饵具有阶段结构且捕食者染病的具有饱和发生率的捕食者-食饵模型的稳定性及其Hopf分支,讨论了由疾病的潜伏期引起的时滞对种群动力学性态的影响．通过分析特征方程,运用Hurwitz判定定理,讨论了该模型边界平衡点和正平衡点的局部稳定性,并得到了Hopf分支存在的充分条件;通过构造适当的Lyapunov泛函,运用LaSall不变集原理,讨论了该模型边界平衡点和正平衡点的全局稳定性,从而得到了疾病流行而最终形成地方病及消灭的充分条件．     

具有一般形式非线性饱和传染率染病年龄结构SIS模型渐近分析

：研究一类具有一般形式非线性饱和传染率染病年龄结构SIS流行病传播数学模型动力学性态,得到疾病绝灭和持续生存的阈值条件——基本再生数。当基本再生数小于或等于1时,仅存在无病平衡点,且在其小于1的情况下,无病平衡点全局渐遗稳定,疾病将逐渐消除;当基本再生数大于1时,存在不稳定的无病平衡点和唯一的局部渐近稳定的地方病平衡点,疾病将持续存在。已有的两类模型可视为本模型的特例,其相关结论可作为本文的推论。     

一类具有垂直传播的HIV模型的稳定性分析

根据艾滋病的传播规律,本文建立了一类传染病模型.在模型中,HIV携带者分为幼年和成年两类,HIV可垂直传染,艾滋病患者有额外死亡.我们用再生矩阵求出了模型的基本再生数,并得出当基本再生数小于1时,模型只有无病平衡点,而当基本再生数大于1时,模型还有地方病平衡点.最后,应用第二加性复合矩阵等理论,文中证明了各平衡点全局渐近稳定性.     

一类具有Beverton-Holt出生函数的阶段结构传染病模型的全局分析

针对一些疾病仅在成年个体间传播和成年个体的成长受到密度制约等因素,建立了一类具有幼年和成年两个阶段且疾病仅在成年个体间传播的传染病模型,其中以具有饱和性质的Beverton-Holt函数作为幼年出生函数．通过构造恰当的Lyapunov函数和定性分析,得到了模型的全局动力学性态,并确定了决定模型动力学性态的种群存活的基本再生数和疾病传播的基本再生数．所得结果表明：当种群的基本再生数不大于 1 时,种群灭绝;当种群的基本再生数大于 1 而疾病传播的基本再生数不大于 1 时,种群持续生存而疾病灭绝;当疾病传播的基本再生数大于 1 时,种群持续存活且疾病会发展成地方病．     

具有吸毒年龄和治疗年龄的海洛因模型的全局稳定性（英）

本文建立了一个吸毒人群具有吸毒年龄,治疗人群具有治疗年龄的海洛因传播模型．得到了基本再生数．通过波动引理和李雅普诺夫泛函,证明了当基本再生数小于1时无海洛因吸食平衡点是全局渐近稳定的,当基本再生数大于1时,海洛因传播平衡点是全局渐近稳定的．     

具有年龄阶段的离散SCIRS模型在我国流脑中的应用

本文主要研究了具有三个年龄阶段的离散SCIRS模型的动力学性态.首先,利用再生矩阵的方法定义了模型的基本再生数R0,证明了当R01时,模型存在唯一的无病平衡点并且是全局渐近稳定的,当R01时,除了无病平衡点,模型还存在唯一的地方病平衡点.其次,利用法定传染病报告的流脑数据,把模型应用到我国流脑的流行传播中.针对模型中很多参数的不确定性,对基本再生数中的参数进行了敏感性分析.最后,在模型的基础上考虑流脑发病的季节因素对模型加以改进,预测分析了我国流脑的发病情况,数值模拟的结果显示季节因素对疾病进展率的影响程度大于对疾病传染率的影响,为控制流脑在我国的流行传播提供建议.     

具有年龄结构的流行病模型的全局稳定性北大核心CSCD

基于重新感染情形,建立了一个具有接种、潜伏和染病年龄结构的流行病模型,目的在于讨论疫苗接种年龄、潜伏年龄和感染年龄对模型全局动力学的影响,得到了模型的全局动力学由基本再生数决定。首先,利用偏微分方程沿特征线积分理论,给出了模型解的存在唯一性、连续有界性和渐近光滑性;其次,利用微分方程解的理论,得到模型的平衡点和基本再生数。再次,结合引入的基本再生数和构造的Lyapunov函数,应用LaSalle不变性原理得到结论:若基本再生数小于1,则无病平衡点全局渐近稳定;若基本再生数大于1,则无病平衡点不稳定。最后,数值模拟验证了所讨论模型的解收敛于无病平衡点。     

具有吸毒年龄和治疗年龄的海洛因模型的全局稳定性（英文）

本文建立了一个吸毒人群具有吸毒年龄,治疗人群具有治疗年龄的海洛因传播模型.得到了基本再生数.通过波动引理和李雅普诺夫泛函,证明了当基本再生数小于1时无海洛因吸食平衡点是全局渐近稳定的,当基本再生数大于1时,海洛因传播平衡点是全局渐近稳定的.     

预防接种情况下非线性饱和接触率SIR流行病模型动力学性态研究

研究了一类预防接种情况下具有一般非线性饱和接触率SIR流行病模型动力学性态。得到决定疾病灭绝和持续生存的基本再生数。当基本再生数小于等于1时，仅存在无病平衡态：当基本再生数大于1时，除存在无病平衡态外，还存在惟一的地方病平衡态。利用Hurwitz判据、Liapunov-Lasalle不变集原理得到各个平衡态局部渐近稳定及无病平衡态全局渐近稳定的条件。特别地。当传染率为双线性时，无病平衡态及地方病平衡态全局渐近稳定。     

Global stability of infection‐free state and endemic infection state of a modified human immunodeficiency virus infection model

This study proposes a modified human immunodeficiency virus (HIV) infection differential equation model with a saturated infection rate. This model has an infection‐free equilibrium point and an endemic infection equilibrium point. Using Lyapunov functions and LaSalle’s invariance principle shows that if the model’s basic reproductive number R ₀ < 1, the infection‐free equilibrium point is globally asymptotically stable, otherwise the endemic infection equilibrium point is globally asymptotically stable. It is shown that a forward bifurcation will occur when R ₀ = 1. The basic reproductive number R ₀ of the modified model is independent of plasma total CD4⁺ T cell counts and thus the modified model is more reasonable than the original model proposed by Buonomo and Vargas‐De‐León. Based on the clinical data from HIV drug resistance database of Stanford University, using the proposed model simulates the dynamics of two group patients’ anti‐HIV infection treatments. The simulation results have shown that the first 4 weeks’ treatments made the two group patients’ R′ ₀ < 1, respectively. After the period, drug resistance made the two group patients’ R′ ₀ > 1. The results explain why the two group patients’ mean CD4⁺ T cell counts raised and mean HIV RNA levels declined in the first period, but contrary in the following weeks.Inspec keywords: microorganisms, cellular biophysics, differential equations, Lyapunov methods, blood, drugs, patient treatment, RNAOther keywords: global stability, infection‐free state, endemic infection state, modified human immunodeficiency virus infection model, HIV, differential equation model, saturated infection rate, infection‐free equilibrium point, endemic infection equilibrium point, Lyapunov functions, LaSalle invariance principle, forward bifurcation, plasma total CD4⁺ T cell counts, HIV drug resistance database, mean HIV RNA levels     

Analysis and simulation of an Adefovir anti‐hepatitis B virus infection therapy immune model with alanine aminotransferase

Hepatitis B virus (HBV) infection models and anti‐HBV infection therapy models have been set up to understand and explain clinical phenomena. Many of these models have been proposed based on Zeuzem et al. and Nowak et al.''s basic virus infection model (BVIM). Some references have pointed out that the basic infection reproductive number of the BVIM is biologically questionable and gave the modified models with standard mass action incidences. This study describes one anti‐HBV therapy immune model with alanine aminotransferase (ALT) based on standard mass action incidences. There are two basic infection reproductive numbers R ₀ and R ₁ in the model. It is proved that if R ₀ < 1 and R ₁ < 1, the disease free equilibrium is locally and globally asymptotically stable, respectively. For the endemic equilibrium, simulation shows that if R ₁ > 1, it may be also globally asymptotically stable. Simulations based on clinical data of HBV DNA and ALT can explain some clinical phenomena. Simulations of the correlation between liver cells, HBV DNA, cytotoxic T lymphocytes and ALT are also given.Inspec keywords: blood, cellular biophysics, diseases, DNA, enzymes, liver, microorganisms, molecular biophysics, patient treatmentOther keywords: Adefovir antihepatitis B virus infection therapy immune model analysis, Adefovir antihepatitis B virus infection therapy immune model simulation, alanine aminotransferase, clinical phenomena, basic infection reproductive number, standard mass action incidences, disease free equilibrium, asymptotic stability, endemic equilibrium, HBV DNA, ALT, liver cells, cytotoxic T lymphocytes     

Global dynamics of a time delayed SIR model with varying population size

A disease transmission model of susceptible-infective-recovered type with a constant latent period is analysed. The global dynamics of the disease-free equilibrium is investigated. If the basic reproduction number is greater than unity, a unique endemic equilibrium exists. Using Lyapunov functional approach, this endemic equilibrium is globally stable in the feasible region. The disease will persist (and is permanent) at the endemic equilibrium if it is initially present. The effects of loss of immunity on the dynamics of the model are analysed, and the parameters that drive the disease dynamics are obtained. Numerical simulations support our analytical results and illustrate possible behavioural scenarios of the model.     

具有时滞和旅途过程中有传染的两斑块SIS模型的全局动力学

本文建立了一个具有时滞的SI S模型,研究了旅途过程中疾病的传染.得到了基本再生数.通过线性化方法和比较原理,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,疾病绝灭.当基本再生数大于1时,系统存在唯一的全局吸引的地方病平衡点,且疾病持续生存.数值模拟验证了扩散率对疾病传播的影响.分析了基本再生数对扩散率的依赖性.     

On infinite period bifurcations with an application to roll waves

Summary By considering a model equation we are able to derive conditions under which a limit cycle, created (at small amplitude) by a Hopf bifurcation, can be destroyed (at finite amplitude) by an infinite period bifurcation, this latter appearing out of a homoclinic orbit formed by the separatrices of a saddle-point equilibrium state. Further, we are able to extend the methods used for showing the existence of an infinite period bifurcation to calculate the amplitude of the limit cycle over its whole range of existence. These ideas are then applied to an equation arising in the theory of roll waves down an open inclined channel, extending previous work to include the case when the Reynolds number is large with the Froude number close to its critical value for the temporal instability of the uniform flow. Here the governing equation reduces to one similar in form to the model equation.With 3 Figures