一类具有非线性发生率和恢复率函数的SIS传染病模型

在传染病模型建模中,采用合理的非线性发生率所得到的动力学性态与实际更加接近,并且在实际的疾病防治过程中,由于受到医院各种医疗资源的影响,染病类的恢复率也会有一定的限制.建立了具有非线性发生率和恢复率函数的SIS传染病模型并分析了其动力学性态,分析这个模型,得到了无病平衡点和地方病平衡点的存在性和稳定性的条件,以及出现Hopf分支的条件.通过数值模拟,给出系统随两个分支参数变化的分支曲线图及系统的相图.     

带有治疗函数及免疫损失率的SIRS流行病模型的动力学分析

本文研究一类带有非线性发病率、新生儿垂直传播、疫苗接种及治疗能力的SIRS传染病模型的动力学行为,该模型充分考虑了医疗资源的局限性,可用医疗资源的供应效率,当感染的数量低于容量时,治疗率与感染的数量成正比,而当感染数量达到容量时,治疗率为常数.在一定条件下,证明了该模型存在后向分支,这意味着边界平衡点与一个正平衡点共存.在这种情况下,控制基本再生数R0小于1不足以控制和根除这种疾病,需要采取额外的措施来确保其解趋近于边界平衡点.当基本再生数R0大于1时,由于治疗、疫苗接种、免疫损失和其他参数的影响,该模型可能存在多个正平衡点.本文分析了该模型平衡点的存在性和稳定性,得到了Hopf分支以及BT分支的存在性,进而发现不稳定极限环及同宿轨道的存在性,并且通过数值模拟来验证所得结果.     

一类具有非线性出生率和饱和恢复率的SEIRS传染病模型的后向分支

主要讨论一类具有非线性出生率和饱和恢复率的SEIRS传染病模型的后向分支.当R_01时,存在无病平衡点,且局部渐近稳定;考虑R_0及R_0~c的关系,得到地方病平衡点存在的条件.当R_1~*1,R_0=1时,系统出现后向分支,若R_1~*1,R_0=1,系统出现前向分支.     

一类具有非线性发生率的SIR传染病模型的全局稳定性

研究一类具有非线性发生率的SIR传染病模型.应用微分方程定性理论分别得到了该系统无病平衡点、地方病平衡点全局渐近稳定的充分条件,并进行了数值模拟.     

一类具有非线性发生率的SI传染病模型的定性分析

研究一类具有非线性发生率的SI传染病模型.应用微分方程定性理论,给出了该系统极限环的存在性、唯一性以及无病平衡点和地方病平衡点的全局渐近稳定性的充分条件.     

一类具有媒体影响的媒介传染病模型的分析北大核心CSCD

建立了一类媒体报道对媒介传染病传播影响的数学模型,研究了该传染病模型的动力学性态.通过求再生矩阵谱半径的方法得到基本再生数,并给出了地方病平衡点的存在性和局部稳定性.理论分析的结果表明,系统可能存在Hopf分支.进一步,由全局Lyapunov函数的方法得到了无病平衡点和地方病平衡点全局稳定的充分条件.     

一类具有媒体影响的媒介传染病模型的分析

建立了一类媒体报道对媒介传染病传播影响的数学模型,研究了该传染病模型的动力学性态.通过求再生矩阵谱半径的方法得到基本再生数,并给出了地方病平衡点的存在性和局部稳定性.理论分析的结果表明,系统可能存在Hopf分支.进一步,由全局Lyapunov函数的方法得到了无病平衡点和地方病平衡点全局稳定的充分条件.     

一类具有logistic增长和饱和发生率的SIRI传染病模型的全局动力学性态

研究一类具有logistic增长、潜伏期时滞、饱和发生率和疾病复发的传染病动力学模型.通过计算得到了疾病的基本再生数;通过分析相应特征方程根的分布,讨论了可行平衡点的局部稳定性和Hopf分支的存在性;通过构造Lyapunov泛函,得到了保证地方病平衡点全局渐近稳定的充分条件.     

一类依靠媒介传染的虫媒传染病模型的数学研究与分析

针对一类依靠媒介传染的虫媒传染病,建立相应的具有非线性发生率的虫媒传染病模型,定性和定量研究该类虫媒传染病的传播规律.基于此,首先根据微分方程与传染病模型的理论分析与数学推导,推出该模型的基本再生数R_0的代数表达式,并得到无病平衡点和地方病平衡点存在的充分条件;其次,利用Hurwitz判据证明了地方病平衡点的稳定性.最后将具体的结论总结如下:当R_01时,模型存在惟一渐进稳定的无病平衡点,此时疾病将随着时间的推移趋于消失;当R_0 1时,模型不存在无病平衡点,但其存在唯一渐进稳定的地方病平衡点,此时疾病将在人群和媒介中持续传播,即意味着疾病将会在某个地区或国家持续流行下去.     

具有常数迁入率和非线性传染率βI~pS~q的SI模型分析

对一种具有种群动力和非线性传染率的传染病模型进行了研究,建立了具有常数迁入率和非线性传染率βI~pS~q的SI模型.与以往的具有非线性传染率的传染病模型相比,这种模型引入了种群动力,也就是种群的总数不再为常数,因此,该类模型更精确地描述了传染病传播的规律.还讨论了模型的正不变集,运用微分方程稳定性理论分析了模型平衡点的存在性及稳定性,得出了疾病消除平衡点和地方病平衡点的全局渐进稳定的充分条件.进一步的,得出了在某些参数范围内会出现Hopf分支现象,并对上述模型进行了生物学讨论.     

Zero-Hopf bifurcation for an infection-age structured epidemic model with a nonlinear incidence rate

An infection-age structured epidemic model with a nonlinear incidence rate is investigated.We formulate the model as an abstract non-densely defined Cauchy problem and derive the condition which guarantees the existence and uniqueness for positive age-dependent equilibrium of the model.By analyzing the associated characteristic transcendental equation and applying the normal form theory presented recently for non-densely defined semilinear equations,we show that the SIR(susceptible-infected-recovered)epidemic model undergoes Zero-Hopf bifurcation at the positive equilibrium which is the main result of this paper.     

GLOBAL STABILITY OF SIRS EPIDEMIC MODELS WITH A CLASS OF NONLINEAR INCIDENCE RATES AND DISTRIBUTED DELAYS＊

In this article,we establish the global asymptotic stability of a disease-free equilibrium and an endemic equilibrium of an SIRS epidemic model with a class of nonlinear incidence rates and distributed...     

Backward bifurcation and global dynamics of an SIS epidemic model with general incidence rate and treatment

An SIS epidemic model with treatment is proposed. The incidence rate of the model, which can include the bilinear incidence rate and the standard incidence rate, is a general nonlinear incidence rate. The global dynamics of the model are studied and then we can understand the effect of the capacity for treatment. It is found that a backward bifurcation occurs and there exist bistable endemic equilibria if the capacity is low. Mathematical results suggest that decreasing the basic reproduction number is insufficient for disease eradication and improving the efficiency and capacity of treatment is important for this end.     

A class of Lyapunov functions and the global stability of some epidemic models with nonlinear incidence

In this paper, by investigating an SIR epidemic model with nonlinear incidence, we present a new technique for proving the global stability of the endemic equilibrium, which consists of introducing a variable transformation and constructing a more general Lyapunov function. For the model we obtain the following results. The disease-free equilibrium is globally stable in the feasible region as the basic reproduction number is less than or equal to unity, and the endemic equilibrium is globally stable in the feasible region as the basic reproduction number is greater than unity.The generality of the technique is illustrated by considering certain nonlinear incidences and SIS and SIRS epidemic models.     

Nonlinear dynamical analysis and control strategies of a network-based SIS epidemic model with time delay

In this paper, we establish a susceptible-infected-susceptible (SIS) epidemic model with nonlinear incidence rate and time delay on complex networks. Firstly, according to the existence of a positive equilibrium point, we work out the threshold values R₀ and λ_c of disease propagation. Secondly, we demonstrate the stabilities of the disease-free equilibrium point and the disease-spreading equilibrium point by constructing Lyapunov function and applying delay differential equations theorem. Thirdly, four different control strategies are investigated and compared, including uniform immunization control, acquaintance immunization control, active immunization control and optimal control. Finally, we perform representative numerical simulations to illustrate the theoretical results and further discover that the nonlinear incidence rate can more accurately reflect individual psychological activities when a certain disease outbreaks at a high level.     

Global analysis of an age-structured SEIR model with immigration of population and nonlinear incidence rate

Epidemic models with infection age of infectious individuals have been extensively studied, however, most of the existing works ignore the combined effects of immigration and nonlinear incidence. In this paper, we incorporate both the effects of immigration and nonlinear incidence, based on which we formulate an SEIR epidemic model. We give a rigorous mathematical analysis on some necessary technical materials. Then, by constructing a Lyapunov functional, we show that the endemic equilibrium is globally asymptotically stable. Numerical simulations of an application are given to support our theoretical results.     

一类时滞SIR传染病模型的稳定性与Hopf分岔分析

研究了一类具有时滞及非线性发生率的SIR传染病模型．首先利用特征值理论分析了地方病平衡点的稳定性,并以时滞为分岔参数,给出了Hopf分岔存在的条件．然后,应用规范型和中心流形定理给出了关于Hopf分岔周期解的稳定性及分岔方向的计算公式．最后,用Matlab软件进行了数值模拟．     

Global dynamics of an epidemic model with relapse and nonlinear incidence

On the basis of a basic SIR epidemic model, we propose and study an epidemic model with nonlinear incidence. The model also incorporates many features of the recovered such as relapse and with/without immunity. A threshold dynamics is established, which is completely determined by the basic reproduction number. The global stability of the disease‐free equilibrium is proved by means of the fluctuation lemma. To prove the global stability of the endemic equilibrium, we need some novel techniques including the transformation of variables, the construction of a new type of Lyapunov functions, and the seeking of an appropriate positively invariant set of the model.