本文選題：傳染病模型 + 輸入輸出��； 參考：《蘭州交通大學(xué)》2015年碩士論文

【摘要】：傳染病作為人類生命的第一殺手,一直以來是世界各國人民高度關(guān)注的問題.近年來,由于各方面的原因各種各樣的新型傳染病相用而現(xiàn),肆虐的吞噬者人類的生命.嚴(yán)重危害著人類的健康,影響著人們的生產(chǎn)生活和經(jīng)濟的發(fā)展.因此,揭示傳染病的發(fā)展規(guī)律將有助于預(yù)測傳染病的發(fā)展趨勢,為有關(guān)部門制定傳染病的預(yù)防與控制策略提供有效的科學(xué)依據(jù),使人們遠(yuǎn)離疾病的困擾.目前,利用動力學(xué)方法建立傳染病的相關(guān)數(shù)學(xué)模型,并對其模型從定性和定量的兩方面進行動力學(xué)的分析與研究,從而來揭示傳染病的流行規(guī)律,已經(jīng)成為了一種趨勢,備受國內(nèi)為眾多學(xué)者的廣泛關(guān)注,在這方面研究的豐碩成果已大量存在.本論文針對幾類新型傳染病建立相應(yīng)的數(shù)學(xué)模型,分析并研究了其動力學(xué)行為即自治系統(tǒng)平衡點的局部和全局的漸進穩(wěn)定性,非自治系統(tǒng)解的正性和周期解的存在性和穩(wěn)定性以及疾病的持久性、滅絕性等主要內(nèi)容如下：(1)討論了一類具有垂直傳染的sIQR傳染病模型,得到了傳染病流行與否的閾值條件R01*,當(dāng)R01*≤1時,無病平衡點是全局穩(wěn)定的；當(dāng)R01*1時,無病平衡點E0是不穩(wěn)定的；當(dāng)R*01=min{R*01,R*02,R*02,1}時,全局漸進穩(wěn)定的是地方病平衡點E1和E2.(2)討論了帶有人口輸入和人口輸出平均分配的SEIR傳染病的數(shù)學(xué)模型,通過分析求得基本再生數(shù)R0,當(dāng)R01時,無病平衡點E0是全局穩(wěn)定的,即無論疾病初始狀態(tài)如何,最終會走向滅亡.當(dāng)R01時,地方病平衡點唯一存在且是全局漸進穩(wěn)定的,即疾病在人群中持續(xù)不斷的流行,最終形成地方性傳染病.(3)討論了帶有人口輸入和人口輸出的非自治情形下的SEIR傳染病的數(shù)學(xué)模型,解的正性用反正法證得,同時疾病的持久性及滅絕性的充要條件被求得,以及當(dāng)模型中的系數(shù)函數(shù)均為時間t的周期函數(shù)時,通過構(gòu)造Lyapunov函數(shù)并對其求右導(dǎo)數(shù)的方法,得到了周期系統(tǒng)對應(yīng)的周期解的存在性及穩(wěn)定性的充分條件.
[Abstract]:Infectious disease, as the first killer of human life, has always been a high concern of people all over the world. In recent years, various kinds of new infectious diseases have been used for various reasons, which engulfs human life. Serious harm to human health, affecting people's production and life and economic development. Therefore, revealing the development law of infectious diseases will help to predict the development trend of infectious diseases, provide effective scientific basis for the departments concerned to formulate the prevention and control strategies of infectious diseases, and make people away from the troubles of diseases. At present, it has become a trend to establish relevant mathematical models of infectious diseases by using kinetic methods, and to analyze and study the dynamics of infectious diseases from both qualitative and quantitative aspects, so as to reveal the epidemic law of infectious diseases. It has been paid much attention by many scholars in our country, and there have been a lot of achievements in this field. In this paper, the corresponding mathematical models of several new infectious diseases are established, and the dynamic behavior of these models is analyzed and studied, that is, the local and global asymptotic stability of the equilibrium points of autonomous systems. The main contents of the positive and periodic solutions of nonautonomous systems, as well as the persistence and extinction of diseases, are as follows: (1) A class of sIQR infectious disease models with vertical transmission are discussed. The threshold condition of epidemic or not is obtained. When R01 * 鈮，

本文編號：2088072