發(fā)布時(shí)間：2018-10-22 08:53

【摘要】：本文研究了幾類生態(tài)系統(tǒng)中的傳染病模型.全文共分為四章:第一章,緒論,介紹了本文的研究背景和主要工作.以及所用到的預(yù)備知識(shí).第二章.考慮了一類具有飛沫和直接接觸感染的傳染病.建立了具有非線性接觸率和非線性治愈率的脈沖時(shí)滯SIRS傳染病模型.首先,運(yùn)用脈沖微分方程理論討論了該系統(tǒng)無(wú)病周期解的全局吸引性.其次,通過(guò)構(gòu)造適當(dāng)?shù)腣函數(shù)證明了該系統(tǒng)的持久性.最后,通過(guò)數(shù)值模擬驗(yàn)證本章的相關(guān)結(jié)論.第三章,在第二章的基礎(chǔ)上研究了具有兩個(gè)時(shí)滯和一般非線性發(fā)生率的傳染病模型.將與飛沫感染的個(gè)體接觸的發(fā)病率和與直接接觸感染的個(gè)體的發(fā)病率分別用非線性發(fā)生率函數(shù)f(P),f(I)來(lái)表示,使得該系統(tǒng)更具有一般性.首先,證明了該系統(tǒng)解的正有界性及平衡點(diǎn)的存在性.其次,構(gòu)造Lyapunov函數(shù)證明了無(wú)病平衡點(diǎn)的全局漸近穩(wěn)定性.本文還利用開(kāi)關(guān)原理證明了地方病平衡點(diǎn)的局部漸近穩(wěn)定性并分析了 Hopf分支存在的條件.最后通過(guò)數(shù)值模擬驗(yàn)證本章的主要理論結(jié)果.第四章.研究了在社會(huì)網(wǎng)絡(luò)中,具有潛伏期和常數(shù)補(bǔ)充的謠言傳播模型.首先,本章證明了無(wú)謠言平衡點(diǎn)和謠言平衡點(diǎn)的存在性.其次,運(yùn)用特征值的方法證明了無(wú)謠言平衡點(diǎn)的局部漸近穩(wěn)定性并且利用左右特征向量的方法研究了超臨界分支的存在性,利用M矩陣的方法證明了無(wú)謠言平衡點(diǎn)的全局漸近穩(wěn)定性.在一定條件下,該系統(tǒng)存在唯一的謠言平衡點(diǎn).利用Routh-Hurwitz判據(jù)證明了謠言平衡點(diǎn)的局部漸近穩(wěn)定性,而且該平衡點(diǎn)是全局穩(wěn)定的.最后.給出簡(jiǎn)單的結(jié)論以及一些理論性結(jié)果的數(shù)值模擬.
[Abstract]:The infectious disease models in several ecosystems are studied in this paper. The paper is divided into four chapters: the first chapter, introduction, introduces the research background and main work of this paper. And the preparatory knowledge used. Chapter 2 A class of infectious diseases with droplet and direct contact infection was considered. A impulsive time-delay SIRS epidemic model with nonlinear contact rate and nonlinear cure rate is established. Firstly, the global attractivity of the disease-free periodic solution of the system is discussed by using the theory of impulsive differential equations. Secondly, the persistence of the system is proved by constructing appropriate V function. Finally, the relevant conclusions of this chapter are verified by numerical simulation. In chapter 3, based on the second chapter, the infectious disease model with two delays and general nonlinear incidence is studied. The incidence rate of individual contact with droplet infection and the incidence rate with direct contact infection are represented by nonlinear incidence function (f (P), f (I), which makes the system more general. Firstly, the positive boundedness and the existence of equilibrium point are proved. Secondly, Lyapunov function is constructed to prove the global asymptotic stability of disease-free equilibrium. This paper also proves the local asymptotic stability of endemic equilibrium by using the switching principle and analyzes the conditions for the existence of Hopf bifurcation. Finally, the main theoretical results of this chapter are verified by numerical simulation. Chapter IV A rumour propagation model with latent period and constant supplement in social network is studied. Firstly, this chapter proves the existence of non-rumor equilibrium and rumor equilibrium. Secondly, the eigenvalue method is used to prove the local asymptotic stability of the non-rumor equilibrium point, and the existence of supercritical bifurcation is studied by using the left and right eigenvector method. The global asymptotic stability of the rumor free equilibrium is proved by using the method of M matrix. Under certain conditions, the system has a unique equilibrium of rumors. By using Routh-Hurwitz criterion, the local asymptotic stability of the rumor equilibrium is proved, and the equilibrium is globally stable. Last A simple conclusion and numerical simulation of some theoretical results are given.
【學(xué)位授予單位】：山西師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175

【參考文獻(xiàn)】

相關(guān)期刊論文 前1條

1 Yoichi Enatsu;Yukihiko Nakata;Yoshiaki Muroya;;GLOBAL STABILITY OF SIRS EPIDEMIC MODELS WITH A CLASS OF NONLINEAR INCIDENCE RATES AND DISTRIBUTED DELAYS[J];Acta Mathematica Scientia;2012年03期

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本文編號(hào)：2286699