本文關(guān)鍵詞： 傳染病模型 穩(wěn)定性 Hopf分支 媒體播報(bào) 時(shí)滯　出處：《北京建筑大學(xué)》2015年碩士論文　論文類型：學(xué)位論文

【摘要】：在傳染病爆發(fā)和流行時(shí),各種媒體也會(huì)及時(shí)地報(bào)道傳染病的相關(guān)知識(shí),教育人們提高對(duì)傳染病的防護(hù)意識(shí),同時(shí)也教給人們正確的防護(hù)措施.另外,隨著時(shí)滯微分方程的發(fā)展,時(shí)滯對(duì)傳染病傳播的影響也引起了很多專家學(xué)者的關(guān)注.本文將考慮三類既有時(shí)滯又有媒體播報(bào)影響的傳染病模型,運(yùn)用微分方程理論和時(shí)滯微分方程理論來研究平衡點(diǎn)的穩(wěn)定性和Hopf分支發(fā)生的充分條件.第二章討論了一類具有媒體播報(bào)效應(yīng)和免疫期的SIRS模型.采用函數(shù)????1 2f I=???I/m?I來刻畫媒體播報(bào)對(duì)接觸率的影響,利用特征方程理論證明了平衡點(diǎn)的穩(wěn)定性的條件,并得出了Hopf分支的相關(guān)結(jié)論,通過數(shù)值模擬,得到當(dāng)時(shí)滯適當(dāng)增加時(shí),染病者的人數(shù)將會(huì)減小,說明延長(zhǎng)免疫期有利于流行病的控制.第三章討論了一類同時(shí)具有媒體播報(bào)效應(yīng)和時(shí)滯的SISmRM模型.改進(jìn)了第二章中媒體播報(bào)的刻畫方式,以函數(shù)形式來刻畫媒體播報(bào)的變化量,利用特征方程理論討論了平衡點(diǎn)的穩(wěn)定性,得到了Hopf分支存在的充分條件.理論推導(dǎo)表明染病者的人數(shù)隨著時(shí)滯的適當(dāng)增加和感染者對(duì)媒體播報(bào)的的影響的增加而減小,同時(shí)減小媒體播報(bào)的自然衰減率,也可以降低染病者的數(shù)量.第四章研究了一類媒體播報(bào)具有時(shí)間滯后性的SISmM模型.在模型中考慮了染病者康復(fù)后有一部分人仍然保持對(duì)疾病的防范意識(shí),成為有意識(shí)的易感者,其余則成為無意識(shí)的易感者.給出了系統(tǒng)的基本再生數(shù)0R,利用特征方程理論和閾值理論,得到了平衡點(diǎn)穩(wěn)定的條件和Hopf分支存在的條件.結(jié)論表明染病者康復(fù)之后成為無意識(shí)的易感者的比率越低,染病者的人數(shù)就會(huì)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.
【學(xué)位授予單位】：北京建筑大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：O175

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本文編號(hào)：1508015