传染病动力学模型性态分析
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摘要
本文主要研究了几个寄生虫宿主数学模型的动力学性态。我们首先在第二部分研究了一类具有一般传染率寄生虫宿主模型的全局性态,此传染率是在双线性与标准传染率之间连续变化的。我们利用微分方程定性理论证明了只有在标准传染率时,宿主和寄生虫可能绝灭;我们还得到了宿主和寄生虫共存即系统正平衡点存在的条件,证明了正平衡点只要存在一定是全局稳定的。
本文第三部分对一类具有S形函数非线性传染率的传染病模型进行了分支分析,我们得到了两个正平衡点存在的阈值条件,通过分析平衡点的稳定性我们给出系统存在Allee效应的条件,我们证明了在一定条件下,系统经历超临界Hopf分支和次临界Hopf分支;还证明了在一定条件下系统经历Bogdanov-Takens分支,通过数值模拟分析,我们发现,系统可能出现两个极限环并且两个极限环可能由不同情况引起:一种是一个极限环从Hopf分支出现,而另一个极限环由同宿轨分支引起;第二种是首先从Hopf分支分支出第一个极限环,接着从退化的极限点出现第二个极限环,并且发现从退化的极限点出现极限环有三种方式,通过数值模拟表明S形函数的倾角如果有微小的变化,可能会导致系统完全不同的分支结构。
本文第四部分分析了第二部分的模型加空间效应后系统的斑图动力学性态,我们利用线性稳定性分析得到具有双线性传染率的反应扩散系统的均匀定态解是稳定的;具有标准传染率的反应扩散系统在一定条件下是稳定的,在一定条件下会出现图灵斑图,我们通过分析系统的振幅方程,给出图灵斑图出现点状和条状及其稳定的条件。进一步,利用中心流形法确定振幅方程的系数,给出在固定参数值时系统出现图灵斑图及图灵斑图形状变化的数值分析。
本文第五部分研究了具有Allee效应的反应扩散系统的斑图动力学性态,首先,我们利用线性稳定性理论给出图灵不稳定和Hopf分支发生的条件且分析了图灵斑图的稳定性,进一步,我们对照不加扩散的系统与加扩散的系统的性态,进行了数值模拟,通过数值模拟,我们发现系统可能出现图灵斑图、缺口状斑图、迷宫斑图、螺旋波斑图、静态斑图及混沌斑图。若寄生虫剂量作为分支参数,发现随着寄生虫剂量的增加,有两种情况发生:一种是当易感者宿主的扩散率超过临界值时系统依次出现图灵斑图、缺口状斑图和迷宫斑图,意味着染病者宿主的分布会由疏到密接着再变疏;另一种是当易感者宿主的扩散率低于临界值时系统依次出现螺旋波斑图、静态斑图和迷宫斑图。若取易感者宿主的扩散率作为分支参数,发现易感者宿主的扩散率大于或小于染病者宿主的扩散率都可能发生稳定的螺旋波,并且随着染病者宿主扩散率的增加,系统螺旋波缺陷数增加导致混沌发生。
In the paper, the dynamical behaviors of several epidemic models for host-parasite are investigated. In the second part of this paper, we consider the global dynamics of a microparasite model with more general incidences. The incidence characterize continuous transitions from the bilinear incidence to the standard incidence. Firstly, the extinction of both host population and parasite population occurs only under standard incidence by the methods of differential qualitative theory. Furthermore, we obtain the existence conditions of the endemic equilibrium and prove it is global whenever it exists.
In the third part of this paper, we consider an epidemic model with the nonlinear incidence of a sigmoidal function. Firstly, a threshold condition of the endemic equilibria is obtained. The dynamical analysis of the model shows that the non-linearity of the incidence rate may lead to Allee effect. Secondly, it is shown that the epidemic model undergoes a supercritical Hopf bifurcation a subcritical Hopf bifurcation under the some conditions. Furthermore, it is shown that the model undergoes a Bogdanov-Takens bifurcation which means that it exhibits a saddle-node bifurcation, a Hopf bifurcation and a homoclinic bifurcation. Thirdly, by numerical simulations, We find two approaches that the model admits two limit cycles. First, a limit cycle emerges from a Hopf bifurcation, while another limit cycle occurs from a homoclinic bifurcation. Secondly, a branch of limit cycles is born from a Hopf bifurcation at first. Then there exists a second bifurcation from a degenerate limit cycle such that two limit cycles emerge. We find also three patterns of limit cycle bifurcations from a degenerate limit cycle. Our analysis shows that a little change of parameter shape of the incidence could lead to quite different bifurcation structures.
In the fourth part of this paper, pattern dynamical behaviors of the model with space effect in the second part are analyzed. Applying the standard linear stability analysis, we" obtain the given steady state is stable in the reaction-diffusion system with bilinear incidence. However, the stability of the given steady state with standard incidence is obtained under the some conditions and Turing instability is given under the other conditions in the reaction-diffusion system. Furthermore, analyzing the behaviors of the amplitude equations of the model, we get the conditions of Turing pattern occurrence in the fixed parameter values.
In the firth part of this paper, we analyze the spatiotemporal patterns in an epidemic model with the Allee effect. Using the linear stability theory, we obtain the conditions of existence for Turing instability and Hopf bifurcation. By means of analytic analysis and numerical simulations, we show that the model exhibits Turing peaks, Turing holes, labyrinths, spiral waves, standing waves and chaotic pattern. There are two cases if we increase the infectious dose. One case is that Turing peaks, holes and labyrinths occur if diffusion rate is greater than a certain value, and the patterns undergo transitions from spots, stripes to holes and labyrinths, which indicates that the spatial distribution of infected individuals changes from sparse to dense, then to sparse. The other case is that spiral waves, standing waves and labyrinths happen if diffusion rate is less than the value, and the patterns range from traveling wave to static wave and labyrinths. If the diffusion rate of susceptible individuals is taken as bifurcation parameter, we find that stable spiral waves occur whenever the diffusion rate is more than or less than the diffusion rate of infectious individuals and the increase of number of defects for spiral waves leads to chaotic pattern.
本文第三部分对一类具有S形函数非线性传染率的传染病模型进行了分支分析,我们得到了两个正平衡点存在的阈值条件,通过分析平衡点的稳定性我们给出系统存在Allee效应的条件,我们证明了在一定条件下,系统经历超临界Hopf分支和次临界Hopf分支;还证明了在一定条件下系统经历Bogdanov-Takens分支,通过数值模拟分析,我们发现,系统可能出现两个极限环并且两个极限环可能由不同情况引起:一种是一个极限环从Hopf分支出现,而另一个极限环由同宿轨分支引起;第二种是首先从Hopf分支分支出第一个极限环,接着从退化的极限点出现第二个极限环,并且发现从退化的极限点出现极限环有三种方式,通过数值模拟表明S形函数的倾角如果有微小的变化,可能会导致系统完全不同的分支结构。
本文第四部分分析了第二部分的模型加空间效应后系统的斑图动力学性态,我们利用线性稳定性分析得到具有双线性传染率的反应扩散系统的均匀定态解是稳定的;具有标准传染率的反应扩散系统在一定条件下是稳定的,在一定条件下会出现图灵斑图,我们通过分析系统的振幅方程,给出图灵斑图出现点状和条状及其稳定的条件。进一步,利用中心流形法确定振幅方程的系数,给出在固定参数值时系统出现图灵斑图及图灵斑图形状变化的数值分析。
本文第五部分研究了具有Allee效应的反应扩散系统的斑图动力学性态,首先,我们利用线性稳定性理论给出图灵不稳定和Hopf分支发生的条件且分析了图灵斑图的稳定性,进一步,我们对照不加扩散的系统与加扩散的系统的性态,进行了数值模拟,通过数值模拟,我们发现系统可能出现图灵斑图、缺口状斑图、迷宫斑图、螺旋波斑图、静态斑图及混沌斑图。若寄生虫剂量作为分支参数,发现随着寄生虫剂量的增加,有两种情况发生:一种是当易感者宿主的扩散率超过临界值时系统依次出现图灵斑图、缺口状斑图和迷宫斑图,意味着染病者宿主的分布会由疏到密接着再变疏;另一种是当易感者宿主的扩散率低于临界值时系统依次出现螺旋波斑图、静态斑图和迷宫斑图。若取易感者宿主的扩散率作为分支参数,发现易感者宿主的扩散率大于或小于染病者宿主的扩散率都可能发生稳定的螺旋波,并且随着染病者宿主扩散率的增加,系统螺旋波缺陷数增加导致混沌发生。
In the paper, the dynamical behaviors of several epidemic models for host-parasite are investigated. In the second part of this paper, we consider the global dynamics of a microparasite model with more general incidences. The incidence characterize continuous transitions from the bilinear incidence to the standard incidence. Firstly, the extinction of both host population and parasite population occurs only under standard incidence by the methods of differential qualitative theory. Furthermore, we obtain the existence conditions of the endemic equilibrium and prove it is global whenever it exists.
In the third part of this paper, we consider an epidemic model with the nonlinear incidence of a sigmoidal function. Firstly, a threshold condition of the endemic equilibria is obtained. The dynamical analysis of the model shows that the non-linearity of the incidence rate may lead to Allee effect. Secondly, it is shown that the epidemic model undergoes a supercritical Hopf bifurcation a subcritical Hopf bifurcation under the some conditions. Furthermore, it is shown that the model undergoes a Bogdanov-Takens bifurcation which means that it exhibits a saddle-node bifurcation, a Hopf bifurcation and a homoclinic bifurcation. Thirdly, by numerical simulations, We find two approaches that the model admits two limit cycles. First, a limit cycle emerges from a Hopf bifurcation, while another limit cycle occurs from a homoclinic bifurcation. Secondly, a branch of limit cycles is born from a Hopf bifurcation at first. Then there exists a second bifurcation from a degenerate limit cycle such that two limit cycles emerge. We find also three patterns of limit cycle bifurcations from a degenerate limit cycle. Our analysis shows that a little change of parameter shape of the incidence could lead to quite different bifurcation structures.
In the fourth part of this paper, pattern dynamical behaviors of the model with space effect in the second part are analyzed. Applying the standard linear stability analysis, we" obtain the given steady state is stable in the reaction-diffusion system with bilinear incidence. However, the stability of the given steady state with standard incidence is obtained under the some conditions and Turing instability is given under the other conditions in the reaction-diffusion system. Furthermore, analyzing the behaviors of the amplitude equations of the model, we get the conditions of Turing pattern occurrence in the fixed parameter values.
In the firth part of this paper, we analyze the spatiotemporal patterns in an epidemic model with the Allee effect. Using the linear stability theory, we obtain the conditions of existence for Turing instability and Hopf bifurcation. By means of analytic analysis and numerical simulations, we show that the model exhibits Turing peaks, Turing holes, labyrinths, spiral waves, standing waves and chaotic pattern. There are two cases if we increase the infectious dose. One case is that Turing peaks, holes and labyrinths occur if diffusion rate is greater than a certain value, and the patterns undergo transitions from spots, stripes to holes and labyrinths, which indicates that the spatial distribution of infected individuals changes from sparse to dense, then to sparse. The other case is that spiral waves, standing waves and labyrinths happen if diffusion rate is less than the value, and the patterns range from traveling wave to static wave and labyrinths. If the diffusion rate of susceptible individuals is taken as bifurcation parameter, we find that stable spiral waves occur whenever the diffusion rate is more than or less than the diffusion rate of infectious individuals and the increase of number of defects for spiral waves leads to chaotic pattern.
引文
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