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1.
A model with differential susceptibility, differential infectivity (DS–DI), and age of infection is formulated in this paper. The susceptibles are divided into n groups according to their susceptibilities. The infectives are divided into m groups according to their infectivities. The total population size is assumed constant. Formula for the reproductive number is derived so that if the reproduction number is less than one, the infection-free equilibrium is locally stable, and unstable otherwise. Furthermore, if the reproductive number is less than one, the infection-free equilibrium is globally asymptotically stable. If the reproductive number is greater than one, it is shown that there exists a unique endemic equilibrium which is globally asymptotically stable. This result is obtained through a Lyapunov function.  相似文献   

2.
We derive and analyze a general class of difference equation models for the dynamics of hierarchically organized populations. Different forms of intra-specific competition give rise to different types of nonlinearities. For our models, we prove that contest competition results asymptotically in only equilibrium dynamics. Scramble competition, on the other hand, can result in more complex asymptotic dynamics. We study both the case when the limiting resource is a constant and when it is dynamically modeled. We prove, in all cases, that the population persists if the inherent net reproductive number of the population is greater than one.  相似文献   

3.
In this article we study the dynamical behaviour of a intracellular delayed viral infection with immune impairment model and general non-linear incidence rate. Several techniques, including a non-linear stability analysis by means of the Lyapunov theory and sensitivity analysis, have been used to reveal features of the model dynamics. The classical threshold for the basic reproductive number is obtained: if the basic reproductive number of the virus is less than one, the infection-free equilibrium is globally asymptotically stable and the infected equilibrium is globally asymptotically stable if the basic reproductive number is higher than one.  相似文献   

4.
In this paper, a virus dynamics model with intracellular delay and Crowley–Martin functional response is discussed. By constructing suitable Lyapunov functions and using LaSalles invariance principle for delay differential equations, we established the global stability of uninfected equilibrium and infected equilibrium; it is proved that if the basic reproductive number is less than or equal to one, the uninfected equilibrium is globally asymptotically stable; if the basic reproductive number is more than one, the infected equilibrium is globally asymptotically stable. We also discuss the effects of intracellular delay on global dynamical properties by comparing the results with the stability conditions for the model without delay. Copyright © 2013 John Wiley & Sons, Ltd.  相似文献   

5.
具有饱和治疗函数与密度制约的SIS传染病模型的后向分支   总被引:2,自引:0,他引:2  
讨论了一个具有饱和治疗函数以及出生率和死亡率均具有密度制约的SIS传染病模型,其中总人口的变化满足Logistic方程,治疗项采用一个连续可微的函数,描述在医疗条件有限的情况下患病者的治疗被耽误的影响.研究发现当患病者的治疗被耽误的影响较强时,模型将出现后向分支,因此基本再生数R_0=1不再是疾病是否消亡的阈值.另外还得到无病平衡点和地方平衡点全局稳定的充分条件.  相似文献   

6.
In this paper, the asymptotic behavior of solutions of an autonomous SEIRS epidemic model with the saturation incidence is studied. Using the method of Liapunov–LaSalle invariance principle, we obtain the disease-free equilibrium is globally stable if the basic reproduction number is not greater than one. Moreover, we show that the disease is permanent if the basic reproduction number is greater than one. Furthermore, the sufficient conditions of locally and globally asymptotically stable convergence to an endemic equilibrium are obtained base on the permanence.  相似文献   

7.
In this paper, the global stability of a virus dynamics model with intracellular delay, Crowley–Martin functional response of the infection rate, and CTL immune response is studied. By constructing suitable Lyapunov functions and using LaSalles invariance principle, the global dynamics is established; it is proved that if the basic reproductive number, R0, is less than or equal to one, the infection‐free equilibrium is globally asymptotically stable; if R0 is more than one, and if immune response reproductive number, R0, is less than one, the immune‐free equilibrium is globally asymptotically stable, and if R0 is more than one, the endemic equilibrium is globally asymptotically stable. Copyright © 2014 John Wiley & Sons, Ltd.  相似文献   

8.
研究一类具有非线性染病年龄结构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--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,无病平衡点全局渐近稳定,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.  相似文献   

9.
A cholera epidemic model with periodic transmission rate is presented. The basic reproduction number is defined. It is shown that the disease-free equilibrium is globally asymptotically stable and the cholera eventually disappears if the basic reproduction number is less than one. And if the basic reproduction number is greater than one, there exists a positive periodic solution which is globally asymptotically stable. Numerical simulations are provided to illustrate analytical results.  相似文献   

10.
A deterministic model for the transmission dynamics of measles in a population with fraction of vaccinated individuals is designed and rigorously analyzed. The model with standard incidence exhibits the phenomenon of backward bifurcation, where a stable disease‐free equilibrium coexists with a stable endemic equilibrium whenever the associated reproduction number is less than unity. This phenomenon can be removed if either measles vaccine is assumed to be perfect or disease related mortality rates are negligible. In the latter case, the disease‐free equilibrium is shown to be globally asymptotically stable whenever the associated reproduction number is less than unity. Furthermore, the model has a unique endemic equilibrium whenever the reproduction threshold exceeds unity. This equilibrium is shown, using a nonlinear Lyapunov function of Goh‐Volterra type, to be globally asymptotically stable for a special case.  相似文献   

11.
研究一类具有预防接种免疫力的双线性传染率 SIR流行病模型全局稳定性 ,找到了决定疾病灭绝和持续生存的阈值——基本再生数 R0 .当 R0 ≤ 1时 ,仅存在无病平衡态 E0 ;当 R0 >1时 ,存在唯一的地方病平衡态 E* 和无病平衡态 E0 .利用 Hurwitz判据及 Liapunov-Lasalle不变集原理可以得知 :当 R0 <1时 ,无病平衡态 E0 全局渐近稳定 ;当 R0 >1时 ,地方病平衡态 E*全局渐近稳定 ,无病平衡态 E0 不稳定 ;当 R0 =1时 ,计算机数值模拟结果显示 ,无病平衡态 E0 有可能是稳定的  相似文献   

12.
针对HIV/AIDS传播的具有常数移民和指数出生的SI型模型,为了更加符合实际意义,对具有双线性传染率的模型进行局部改进,并对改进后的动力学模型进行了简化.对于改进后的模型,证明了平衡点的存在与局部稳定性,并证明了传染病毒的灭绝与持续性,得到了传染病毒的基本再生数.结果表明:当单位时间内从外界迁入人口中染病者的比例系数c近似等于零时,基本再生数小于1时,传染病毒最终灭绝;当基本再生数大于1时,模型存在唯一的正平衡点,且是局部渐近稳定的,说明传染病毒一致持续存在.  相似文献   

13.
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.  相似文献   

14.
Global behavior and permanence of SIRS epidemic model with time delay   总被引:1,自引:0,他引:1  
In this paper an autonomous SIRS epidemic model with time delay is studied. The basic reproductive number R0 is obtained which determines whether the disease is extinct or not. When the basic reproductive number is greater than 1, it is proved that the disease is permanent in the population, and explicit formula are obtained by which the eventual lower bound of the fraction of infectious individuals can be computed. Throughout the total paper, we mainly use the technique of Lyapunov functional to establish the global stability of the infection-free equilibrium and the local stability of the endemic equilibrium but need another sufficient condition.  相似文献   

15.
In this paper, we study a model for the dynamics of dengue fever when only one type of virus is present. For this model, Lyapunov functions are used to show that when the basic reproduction ratio is less than or equal to one, the disease-free equilibrium is globally asymptotically stable, and when it is greater than one there is an endemic equilibrium which is also globally asymptotically stable.  相似文献   

16.
两类带有确定潜伏期的SEIS传染病模型的分析   总被引:2,自引:0,他引:2  
通过研究两类带有确定潜伏期的SEIS传染病模型,发现对种群的常数输入和指数输入会使疾病的传播过程产生本质的差异.对于带有常数输入的情形,找到了地方病平衡点存在及局部渐近稳定的阈值,证明了地方病平衡点存在时一定局部渐近稳定,并且疾病一致持续存在.对于带有指数输入的情形,发现地方病平衡点当潜伏期充分小时是局部渐近稳定的,当潜伏期充分大时是不稳定的.  相似文献   

17.
A neglected disease with a nearly forgotten name is making a comeback following a global control programme that almost eradicated it more than forty years ago. Until the 1970s the prevalence of non-venereal treponematosis, including yaws, was greatly reduced after worldwide mass treatment. In 2005, cases were again reported in the Democratic Republic of the Congo. A deterministic model is formulated to investigate the impact of poverty on yaws eradication. Threshold parameters are determined and stabilities analysed. The reproductive number was also used to assess the impact of birth rate in resource-constrained families on the dynamics of yaws. The model was shown to be globally stable whenever the associated reproductive number is less than a unity. Using the Lyapunov function it was proved that whenever the associated reproductive number is greater than a unity an endemic equilibrium exists and is globally asymptotically stable. Results from this theoretical study suggests that if the population of children in the community is dominated by those from resource-constrained families, then yaws eradication will remain difficulty to attain. Thus, more needs to be done in addressing issues such as high fertility rate, overcrowding, poor sanitation, etc. and poverty in general so that yaws epidemic which was successfully controlled several decades ago will cease to reemerge and can easily be eradicated.  相似文献   

18.
In this paper, the single species modelled by (asymptotically) periodic Gompertz equation is investigated. It is shown that the (asymptotically) periodic system has a unique (asymptotically) periodic solution which is globally asymptotically stable for the positive solution. When the nonautonomous Gompertz equation is subject to harvesting, we study the optimal harvesting policy for the periodic system and obtain the corresponding optimal population level and the maximum sustainable yield. Further, when the functions in the exploited Gompertz system are stably bounded functions, we study the ultimately optimal harvesting policy. By choosing the average limiting maximum sustainable yield as management objective, the corresponding optimal population level is determined.  相似文献   

19.
This paper deals with global dynamics of an SIRS epidemic model for infections with non permanent acquired immunity. The SIRS model studied here incorporates a preventive vaccination and generalized non-linear incidence rate as well as the disease-related death. Lyapunov functions are used to show that the disease-free equilibrium state is globally asymptotically stable when the basic reproduction number is less than or equal to one, and that there is an endemic equilibrium state which is globally asymptotically stable when it is greater than one.  相似文献   

20.
具有Logistic增长和年龄结构的SIS模型   总被引:3,自引:2,他引:1  
讨论具有Logistic增长和年龄结构的SIS流行病模型.运用微分、积分方程理论,得到了当再生数R0<1时,无病平衡点E0是全局渐近稳定的;当R0>1时,地方病平衡点E*是局部渐近稳定的.  相似文献   

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