具有染病者输入的离散SIR传染病模型分析

引入相应的概率建立了具有染病者输入的离散SIR传染病模型,确定了决定其动力学性态的阂值.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,且地方病平衡点是局部渐近稳定的.     

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.     

一类带有一般出生率的SIS传染病模型的全局分析

将一般出生率系数引入S IS传染病模型,得到了种群灭绝和疾病灭绝的阈值条件.分别借助S tokes定理和D u lac函数对染病者的数量模型和染病者在种群中所占比例的模型进行了讨论,得到了相应模型的全局动力学行为.     

一类带有阶段结构的传染病模型的定性分析

通过假设被感染者恢复后不具有免疫力,但易感性不同于未被感染过的易感者,建立了一类带有双线性传染率的传染病模型,发现该模型对一定参数会发生后向分支,找到了相应的阈值,完整分析了该模型的动力学性态.     

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.     

一类带有一般接触率和常数输入的流行病模型的全局分析

借助极限系统理论和构造适当的Liapunov函数,对带有一般接触率和常数输入的SIR型和SIRS型传染病模型进行讨论．当无染病者输入时,地方病平衡点存在的阈值被找到A·D2对相应的SIR模型,关于无病平衡点和地方病平衡点的全局渐近稳定性均得到充要条件;对相应的SIRS模型,得到无病平衡点和地方病平衡点全局渐近稳定的充分条件．当有染病者输入时,模型不存在无病平衡点．对相应的SIR模型,地方病平衡点是全局渐近稳定的;对相应的SIRS模型,得到地方病平衡点全局渐近稳定的充分条件．     

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.     

一类基于集合种群网络的传染病模型研究

研究了一类基于集合种群网络的传染病模型.针对在疾病传播过程中,随着染病者数量的增加,被感染的人数会达到饱和,研究了带有饱和发生率的传染病模型,建立了不同集合种群之间扩散模式,并分析了模型动力学的性态,给出了无病平衡点及其稳定性和正平衡点的存在性.最后用数值模拟验证了理论结果的正确性.     

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   

一类具有常数输入的随机SIR流行病模型的定性分析

研究了一类易感者和恢复者具有常数输入的随机SIR传染病模型.利用停时理论及Lyapunov分析方法,证明了该随机模型正解的全局存在唯一性和有界性,讨论了随机模型的解在相应确定模型的无病平衡点和地方病平衡点附近的振荡行为以及得到了随机模型的解的平均持久和疾病灭绝的充分条件.最后,通过数值模拟验证和理论推导的一致性.     

Backward bifurcation of an epidemic model with standard incidence rate and treatment rate

An epidemic model with standard incidence rate and treatment rate of infectious individuals is proposed to understand the effect of the capacity for treatment of infectives on the disease spread. It is assumed that treatment rate is proportional to the numbers of infectives below the capacity and is a constant when the number of infectives is greater than the capacity. It is proved that the existence and stability of equilibria for the model is not only related to the basic reproduction number but also the capacity for treatment of infectives. It is found that a backward bifurcation occurs if the capacity is small. It is also found that there exist bistable endemic equilibria if the capacity is low.     

Epidemic outbreaks on networks with effective contacts

In real-world networks of disease transmission, the incidence of infection among individuals conforms to a certain fixed probability of effective contact between them, which must meet some necessary conditions for the disease to continue to spread. Based on susceptible/infective/removed (SIR) models in homogeneous or heterogeneous networks, we find that these models evolve dynamically just like in networks without connectivity fluctuations if all the susceptible individuals are supposed to have the same effective contact. This means that effectively heterogeneous contacts play a striking role in epidemic dynamics. To go a step further, we introduce the effective contact function (ECF) into models and present an analytical and numerical study for the threshold and dynamical behaviors of epidemic incidence. The power-law and proportional ECFs are considered, and, we demonstrate analytically that the epidemic incidence is generally a monotone decreasing function of the epidemic threshold and increasing function of the number of effective contacts. Certain exceptional cases are also discussed. This tells us that we cannot always focus on the threshold to evaluate the extent of epidemic outbreaks.     

一类带有非线性传染率的SEIR传染病模型的全局分析

通过假设被传染的易感者一部分经过一段潜伏期后才具有传染性,而另一部分被感染的易感者直接成为传染者,建立了一类带有非线性传染率的SEIR传染病模型,得到了确定疾病是否成为地方病的基本再生数以及无病平衡点和地方病平衡点的全局稳定性.     

Bifurcations in an epidemic model with constant removal rate of the infectives

An epidemic model with a constant removal rate of infective individuals is proposed to understand the effect of limited resources for treatment of infectives on the disease spread. It is found that it is unnecessary to take such a large treatment capacity that endemic equilibria disappear to eradicate the disease. It is shown that the outcome of disease spread may depend on the position of the initial states for certain range of parameters. It is also shown that the model undergoes a sequence of bifurcations including saddle-node bifurcation, subcritical Hopf bifurcation, and homoclinic bifurcation.     

Stability and bifurcation of an SIS epidemic model with treatment

An SIS epidemic model with a limited resource for treatment is introduced and analyzed. It is assumed that treatment rate is proportional to the number of infectives below the capacity and is a constant when the number of infectives is greater than the capacity. It is found that a backward bifurcation occurs if the capacity is small. It is also found that there exist bistable endemic equilibria if the capacity is low.     

Bifurcations analysis and tracking control of an epidemic model with nonlinear incidence rate

In this paper, a discrete epidemic model with nonlinear incidence rate obtained by the forward Euler method is investigated. The conditions for existence of codimension-1 bifurcations (fold bifurcation, flip bifurcation and Neimark-Sacker bifurcation) are derived by using the center manifold theorem and bifurcation theory. Furthermore, the condition for the occurrence of codimension-2 bifurcation (fold-flip bifurcation) is presented. In order to eliminate the chaos or Neimark-Sacker bifurcation of the discrete epidemic model, a tracking controller is designed. The number of the infectives tends to zero when the number of iterations is gradually increasing, that is, the disease disappears gradually. Finally, numerical simulations not only illustrate the validity of the proposed results, but also display the interesting and complex dynamical behaviors.     

Modelling the effect of tuberculosis on the spread of HIV infection in a population with density-dependent birth and death rate

A nonlinear mathematical model is proposed to study the effect of tuberculosis on the spread of HIV infection in a logistically growing human population. The host population is divided into four sub classes of susceptibles, TB infectives, HIV infectives (with or without TB) and that of AIDS patients. The model exhibits four equilibria namely, a disease free, HIV free, TB free and an endemic equilibrium. The model has been studied qualitatively using stability theory of nonlinear differential equations and computer simulation. We have found a threshold parameter R₀ which is if less than one, the disease free equilibrium is locally asymptotically stable otherwise for R₀>1, at least one of the infections will be present in the population. It is shown that the positive endemic equilibrium is always locally stable but it may become globally stable under certain conditions showing that the disease becomes endemic. It is found that as the number of TB infectives decreases due to recovery, the number of HIV infectives also decreases and endemic equilibrium tends to TB free equilibrium. It is also observed that number of AIDS individuals decreases if TB is not associated with HIV infection. A numerical study of the model is also performed to investigate the influence of certain key parameters on the spread of the disease.     

Threshold dynamics for compartmental epidemic models with impulses

The basic reproductive number and its calculation for general impulsive compartmental epidemic models, with pulses on both the infected and the uninfected compartments, are established. Theoretical results show that the basic reproductive number serves as a threshold parameter: the disease dies out if the basic reproductive number is smaller than unity, and breaks out if it is larger than unity. The global dynamics of a viral dynamical model with impulsive immune response is analyzed to study how the vaccination strength and the vaccination interval affect the basic reproductive number and virus progression.     

A nonlinear HIV/AIDS model with contact tracing

A nonlinear mathematical model is proposed and analyzed to study the effect of contact tracing on reducing the spread of HIV/AIDS in a homogeneous population with constant immigration of susceptibles. In modeling the dynamics, the population is divided into four subclasses of HIV negatives but susceptibles, HIV positives or infectives that do not know they are infected, HIV positives that know they are infected and that of AIDS patients. Susceptibles are assumed to become infected via sexual contacts with (both types of) infectives and all infectives move with constant rates to develop AIDS. The model is analyzed using the stability theory of differential equations and numerical simulation. The model analysis shows that contact tracing may be of immense help in reducing the spread of AIDS epidemic in a population. It is also found that the endemicity of infection is reduced when infectives after becoming aware of their infection do not take part in sexual interaction.