具有媒体播报效应和时滞的传染病模型的研究

发布时间：2018-02-13 11:06

本文关键词： 传染病模型稳定性 Hopf分支媒体播报时滞　出处：《北京建筑大学》2015年硕士论文　论文类型：学位论文

【摘要】：在传染病爆发和流行时,各种媒体也会及时地报道传染病的相关知识,教育人们提高对传染病的防护意识,同时也教给人们正确的防护措施.另外,随着时滞微分方程的发展,时滞对传染病传播的影响也引起了很多专家学者的关注.本文将考虑三类既有时滞又有媒体播报影响的传染病模型,运用微分方程理论和时滞微分方程理论来研究平衡点的稳定性和Hopf分支发生的充分条件.第二章讨论了一类具有媒体播报效应和免疫期的SIRS模型.采用函数????1 2f I=???I/m?I来刻画媒体播报对接触率的影响,利用特征方程理论证明了平衡点的稳定性的条件,并得出了Hopf分支的相关结论,通过数值模拟,得到当时滞适当增加时,染病者的人数将会减小,说明延长免疫期有利于流行病的控制.第三章讨论了一类同时具有媒体播报效应和时滞的SISmRM模型.改进了第二章中媒体播报的刻画方式,以函数形式来刻画媒体播报的变化量,利用特征方程理论讨论了平衡点的稳定性,得到了Hopf分支存在的充分条件.理论推导表明染病者的人数随着时滞的适当增加和感染者对媒体播报的的影响的增加而减小,同时减小媒体播报的自然衰减率,也可以降低染病者的数量.第四章研究了一类媒体播报具有时间滞后性的SISmM模型.在模型中考虑了染病者康复后有一部分人仍然保持对疾病的防范意识,成为有意识的易感者,其余则成为无意识的易感者.给出了系统的基本再生数0R,利用特征方程理论和阈值理论,得到了平衡点稳定的条件和Hopf分支存在的条件.结论表明染病者康复之后成为无意识的易感者的比率越低,染病者的人数就会越小.
[Abstract]:During the outbreak and prevalence of infectious diseases, various media will also report relevant knowledge of infectious diseases in a timely manner, educate people to raise their awareness of protection against infectious diseases, and at the same time teach people the right protective measures. In addition, with the development of delay differential equations, The influence of delay on the spread of infectious diseases has also attracted the attention of many experts and scholars. In this paper, we will consider three kinds of infectious disease models, which have both delay and media effects. The stability of equilibrium points and sufficient conditions for the occurrence of Hopf bifurcation are studied by using the theory of differential equations and delay differential equations. In chapter 2, we discuss a class of SIRS models with media broadcast effect and immune period. ? ? ? 12f? ? ? I/m? I describes the influence of media broadcast on the contact rate, proves the stability condition of the equilibrium point by using the characteristic equation theory, and obtains the relevant conclusions of the Hopf bifurcation. By numerical simulation, the results show that the delay increases properly at that time. The number of infected patients will be reduced, indicating that prolonging the immunization period is beneficial to the control of the epidemic. In Chapter 3, we discuss a class of SISmRM models with both media broadcast effect and time delay, and improve the characterization of media broadcast in Chapter 2. The variation of media broadcast is described in the form of function, and the stability of equilibrium point is discussed by means of characteristic equation theory. The sufficient conditions for the existence of Hopf branches are obtained. The theoretical derivation shows that the number of infected persons decreases with the appropriate increase of time lag and the influence of infected persons on the media broadcast, and the natural attenuation rate of the media broadcast is reduced at the same time. Chapter 4th studies a class of SISmM models in which media broadcasts have a time lag. In the model, some people are still aware of the disease after their recovery and become conscious susceptible people. The others become unconscious susceptible. The basic reproduction number 0 R of the system is given, and the characteristic equation theory and threshold theory are used. The condition of stable equilibrium point and the condition of existence of Hopf branch are obtained. Conclusion the lower the proportion of patients becoming unconscious susceptible persons after recovery, the smaller the number of infected people.
【学位授予单位】：北京建筑大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175

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