一类具有时滞的病菌-免疫力模型的定性研究

考虑病菌的一种信息交流机制,建立一类病菌与免疫系统竞争的时滞传染病模型.分析正平衡点的存在性、渐近稳定性、Hopf分歧的存在性.特别的,研究了无病平衡点E0 在奇异条件(R0=1)下的稳定性.数值模拟验证了所得理论结果.     

一类非连续病菌与免疫系统竞争模型的动力学分析

建立一类右端不连续的病菌与免疫系统竞争模型,讨论模型滑模的存在性,真假平衡点以及伪平衡点的存在性和全局稳定性及局部滑动边界点分歧等动力学性质.研究结果表明病菌与免疫细胞最终在伪平衡点或真平衡点处共存.最后,运用数学软件进行数值模拟,验证了所得的理论结果.     

混合随机SIR传染病模型的动力学分析

研究一类具有一般发病率函数的混合随机SIR传染病模型.首先,通过构造适当的Lyapunov函数,证明了模型全局正解的存在唯一性.然后,利用随机分析方法,建立了系统灭绝与持久的充分且几乎必要条件和遍历平稳分布的存在性.最后,通过数值模拟来验证理论结果.     

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.     

一类周期模型的周期解及次调和分歧的存在性

本文建立一类病菌与免疫系统竞争的周期模型,利用重合度理论及Lyapunov稳定性理论分别证明模型周期解的存在性及稳定性,并讨论次调和分歧的存在性.     

基于白噪声的网络传染病模型动力学分析北大核心CSCD

该文基于确定性网络传染病模型,建立了白噪声影响下的随机网络传染病模型,证明了模型全局解的存在唯一性,利用随机微分方程理论得到了传染病随机灭绝和随机持久的充分条件.结果表明,白噪声对网络传染病传播动力学有很大的影响,白噪声能有效抑制传染病的传播,大的白噪声甚至能让原本持久的传染病变得灭绝.最后,通过数值模拟验证了理论结果.     

一类具有标准发生率和双垂直传播的媒介传染病模型分析

建立了一类具有标准发生率和双垂直传播的媒介传染病模型,通过构造Lyapunov函数,利用LaSalle不变集原理等理论,证明了无病平衡点和地方病平衡点的存在性和稳定性,并对其进行数值模拟.得出通过采取降低人群与媒介之间接触率或者提高医疗水平等措施,能够控制疾病的蔓延.     

具脉冲预防接种的SIRS传染病模型中地方病周期解的存在性

对具脉冲预防接种的SIRS传染病模型进行分析,利用分支理论得到了系统中地方病周期解的存在性,并利用数值模拟的方法验证了所得结论的正确性,完善了对该系统的讨论结果     

一类含有脉冲免疫和治疗的SIR流行病的分析

在考虑脉冲接种和脉冲治疗的基础上,本文提出了一类新的含有两个脉冲过程和治疗的SIR传染病模型.利用频闪映射和Floquet理论,研究了无病周期解的存在性与稳定性,这意味着疫情最终可能灭绝.此外,研究了该流行病持久流行的条件,获得了决定疫情是否发生的基本再生数.最后,通过数值模拟分析,说明了脉冲接种和脉冲治疗对疾病控制的影响.     

基于随机与异质网络共存的SEIRS传染病模型

讨论了随机与异质网络共存的SEIRS传染病模型,通过正平衡点的存在性给出基本再生数R_0=((1-η)Aλ+ηβ)/μ.结果表明,当R_01时,无病平衡点(1,0,0,0)局部稳定;当R_01时,无病平衡点(1,0,0,0)不稳定,此时系统存在唯一的地方病平衡点,并且一致持续存在.最后通过数值仿真,验证了理论结果的正确性.     

Traveling Epizootic Waves of a Fox Rabies Model with Small Spatial Diffusion

In this paper, to describe the spread of fox rabies, a degenerate SEI epidemic model with small spatial diffusion equipped by infectious foxes due to rabies is investigated. In particular, the existence of traveling waves is established by the geometric singular perturbation theory for the larger speeds, while the non-existence of traveling wave is still derived for the smaller speeds. Moreover, some numerical simulations are implemented to illustrate the propagation dynamics driven by traveling waves.     

The threshold of a stochastic Susceptible–Infective epidemic model under regime switching

In this paper, we consider a stochastic Susceptible–Infective (SI) epidemic model under regime switching. Firstly, by constructing suitable Lyapunov functions, we establish sufficient criteria for the existence and uniqueness of an ergodic stationary distribution. Then we obtain the threshold which guarantees the extinction and the existence of the stationary distribution of the epidemic. Finally, some numerical simulations are introduced to illustrate our main results.     

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.     

Effects of awareness program and delay in the epidemic outbreak

In the present paper, an epidemic model has been proposed and analyzed to investigate the impact of awareness program and reporting delay in the epidemic outbreak. Awareness programs induce behavioral changes within the population, and divide the susceptible class into two subclasses, aware susceptible and unaware susceptible. The existence and the stability criteria of the equilibrium points are obtained in terms of the basic reproduction number. Considering time delay as the bifurcating parameter, the Hopf bifurcation analysis has been performed around the endemic equilibrium. The direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are investigated by using the normal form theory and central manifold theorem. To verify the analytical results, comprehensive numerical simulations are carried out. Copyright © 2016 John Wiley & Sons, Ltd.     

Stability analysis in a class of discrete SIRS epidemic models

In this paper, the dynamical behaviors of a class of discrete-time SIRS epidemic models are discussed. The conditions for the existence and local stability of the disease-free equilibrium and endemic equilibrium are obtained. The numerical simulations not only illustrate the validity of our results, but also exhibit more complex dynamical behaviors, such as flip bifurcation, Hopf bifurcation and chaos phenomenon. These results reveal far richer dynamical behaviors of the discrete epidemic model compared with the continuous epidemic models.     

ANALYSIS OF A SUSCEPTIBLE-EXPOSED-INFECTED EPIDEMIC MODEL WITH RANDOM PERTURBATION AND VARYING POPULATION SIZE

In this paper, we study a type of susceptible-exposed-infected (SEI) epidemic model with varying population size and introduce the random perturbation of the constant contact rate into the SEI epidemic model due to the universal existence of fluctuations. Under some moderate conditions, the density of the exposed and the infected individuals exponentially approaches zero almost surely are derived. Furthermore, the stochastic SEI epidemic model admits a stationary distribution around the endemic equilibrium, and the solution is ergodic. Some numerical simulations are carried out to demonstrate the efficiency of the main results.     

Traveling Waves of a Reaction-Diffusion SIRQ Epidemic Model with Relapse

This paper is concerned with the traveling waves of a reaction-diffusion SIRQ epidemic model with relapse. We find that the existence and nonexistence of traveling waves are determined by the basic reproduction number of the system and the minimal wave speed. This threshold dynamics is proved by Schauder''s fixed-point theorem combining Lyapunov functional with the theory of asymptotic spreading. Moreover, the numerical simulations are provided to illustrate our analytical results and the effect of the relapse is also discussed.     

考虑污染和种内关系的种群扩散最优控制模型

针对污染和种内关系均影响细菌种群扩散这一管理生态学问题,本文建立了基于非线性拟抛物方程的最优控制模型,将外界环境向细菌种群输入的毒素率作为控制变量,运用控制理论和方法探讨污染和种内关系双重影响下种群扩散系统的最优控制问题。利用Schauder不动点定理证明了该种群扩散系统的适定性;同时,通过建立新的Carleman型估计,给出了容许控制和最优控制的存在性。最后,通过数值算例分析了理论推导的结果,在算例中都找到一对时间最优控制,验证了种群扩散系统最优控制模型的有效性。该研究结果对现代传染病预防具有借鉴意义,也为有效控制瘟疫的爆发和流行提供理论参考。     

Hopf bifurcation direction of a delayed bacteria-immunity system with quorum sensing mechanism

Considering the mechanism of quorum sensing, we formulate a bacteria-immunity model to describe the competition between bacteria and immune cells on the basis of Zhang’s model (see [9] for more details). A time delay is introduced to characterize the time in which bacteria receive signal molecules and then combat with immune cells. In the sequel, the length of delay which preserves the stability of the positive equilibrium is estimated, and the existence of Hopf bifurcation when the delay crosses through a critical value is investigated. Further, by using the normal form theory and center manifold theory, the explicit formulae are calculated which determine the stability, the direction and the period of bifurcating periodical solutions. Finally, numerical simulations are employed to verify the mathematical conclusions.