本文選題：SIS傳染病模型　切入點：基本再生數(shù)　出處：《陜西科技大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

【摘要】：隨著傳染性疾病的大肆流行,人們關(guān)于傳染病的研究越來越多。傳染病模型主要分為連續(xù)模型和離散模型兩大類。因為傳染病模型的數(shù)據(jù)大多采用離散時間,因此離散傳染病模型的描述更為合理。離散傳染病模型的求解問題是傳染病模型研究過程中的重中之重。關(guān)于離散傳染病模型的研究主要集中于模型的平衡性,持久性,分支理論。在大量傳染病模型研究的基礎(chǔ)上,本文構(gòu)造出一類具有指數(shù)型發(fā)生率的離散SIS傳染病模型。這類指數(shù)型發(fā)生率的離散SIS傳染病模型是在倉室理論基礎(chǔ)上建立的。所研究的人群分為易感者和染病者。通過一定的控制措施,疾病的傳播速度降低,疾病的發(fā)生率可調(diào)控為指數(shù)型發(fā)生率。因為易感者是通過上一刻的染病者被傳染的,因此疾病的發(fā)生率也與上一時刻的染病者數(shù)量存在密不可分的關(guān)系。根據(jù)離散SIS傳染病模型的主要的研究內(nèi)容和研究方向,本文主要的工作和內(nèi)容如下:首先,在精確分析和合理假設(shè)的情況下,建立了一個具有指數(shù)型發(fā)生率的離散SIS傳染病模型。討論了這個具有指數(shù)型發(fā)生率的離散SIS傳染病模型的平衡點的穩(wěn)定性,分析了該模型的持久性,通過穩(wěn)定性理論,得到了模型無病平衡點的全局漸近穩(wěn)定性,以及有病平衡點的局部漸近穩(wěn)定性,運用數(shù)值模擬的方法驗證了理論成果,并展示了模型動力學(xué)性態(tài)的復(fù)雜性。其次,在已構(gòu)造具有指數(shù)型發(fā)生率的模型的基礎(chǔ)上,建立了一個具有時滯的離散傳染病模型。討論了這個時滯模型的平衡點的存在性和穩(wěn)定性,經(jīng)過分析可知,這個時滯模型在有病正平衡點不穩(wěn)定情況下出現(xiàn)了flip分支,通過中心流形理論證明了該模型的flip分支是2周期穩(wěn)定的,最后運用數(shù)值模擬驗證了該flip分支是2周期穩(wěn)定的。再次,分析了一個具有飽和恢復(fù)率和具有指數(shù)型發(fā)生率的離散SIS傳染病模型。證明了該模型存在一個無病平衡點和兩個有病平衡點,并且該模型的無病平衡點是全局漸近穩(wěn)定的,因為計算的復(fù)雜性,采取數(shù)值模擬的方法證明了有病平衡點的穩(wěn)定性,結(jié)果顯示這個模型在其有病平衡點不穩(wěn)定時會出現(xiàn)后向分支。最后,改進(jìn)了一個具有指數(shù)型發(fā)生率的離散SIS傳染病模型,討論了該模型的無病平衡點和有病平衡點的穩(wěn)定性。結(jié)果顯示,其無病平衡點是全局漸近穩(wěn)定的,其有病平衡點是局部漸近穩(wěn)定的,當(dāng)其有病平衡點不穩(wěn)定時,該模型在有病平衡點處可產(chǎn)生neimark-sacker分支,并通過數(shù)值模擬的方法,驗證了研究結(jié)果。
[Abstract]:With the prevalence of infectious diseases, more and more people are studying infectious diseases. Infectious disease models are divided into two categories: continuous model and discrete model. Therefore, the description of discrete infectious disease model is more reasonable. The solution of discrete infectious disease model is the most important in the process of studying infectious disease model. The research on discrete infectious disease model mainly focuses on the balance and persistence of the model. Branch theory. Based on a large number of infectious disease models, In this paper, a discrete SIS infectious disease model with exponential incidence is constructed. The discrete SIS infectious disease model with exponential incidence is established on the basis of storeroom theory. Through certain controls, The rate of spread of the disease is reduced, and the incidence of the disease can be regulated as an exponential incidence, because the susceptible person is transmitted through the infected person of the last moment. Therefore, the incidence of disease is closely related to the number of infected people at the last moment. According to the main research content and research direction of discrete SIS infectious disease model, the main work and contents of this paper are as follows: first, Under the condition of accurate analysis and reasonable assumption, a discrete SIS infectious disease model with exponential incidence is established. The stability of the equilibrium point of the discrete SIS infectious disease model with exponential incidence is discussed. The persistence of the model is analyzed. The global asymptotic stability of the disease-free equilibrium and the local asymptotic stability of the disease-free equilibrium are obtained by using the stability theory. It also shows the complexity of the dynamic behavior of the model. Secondly, on the basis of the established model with exponential incidence, In this paper, a discrete epidemic model with time delay is established. The existence and stability of the equilibrium point of the model are discussed. The results show that the flip bifurcation of the time-delay model appears under the unstable condition of the diseased positive equilibrium. The flip bifurcation of the model is proved to be 2-period stable by the center manifold theory. Finally, the numerical simulation is used to verify that the flip bifurcation is 2-period stable. A discrete SIS infectious disease model with saturation recovery rate and exponential incidence is analyzed. It is proved that the model has one disease-free equilibrium point and two disease-free equilibrium points, and the disease-free equilibrium point of the model is globally asymptotically stable. Because of the complexity of the calculation, the stability of the diseased equilibrium point is proved by numerical simulation. The results show that the model has backward bifurcation when the ill equilibrium point is unstable. Finally, A discrete SIS infectious disease model with exponential incidence is improved, and the stability of disease-free equilibrium and disease-free equilibrium is discussed. The results show that the disease-free equilibrium is globally asymptotically stable. The diseased equilibrium is locally asymptotically stable. When the diseased equilibrium is unstable, the model can produce neimark-sacker bifurcation at the diseased equilibrium. The results are verified by numerical simulation.
【學(xué)位授予單位】：陜西科技大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O175

【參考文獻(xiàn)】

相關(guān)期刊論文 前10條

1 傅金波;陳蘭蓀;;基于兩斑塊和人口流動的SIR傳染病模型的穩(wěn)定性[J];應(yīng)用數(shù)學(xué)和力學(xué);2017年04期

2 楊彩虹;胡志興;;一類具有飽和發(fā)生率和治愈率的SEIR模型研究[J];鄭州大學(xué)學(xué)報(理學(xué)版);2017年02期

3 王振國;劉桂榮;;具有非線性傳染率的SIS網(wǎng)絡(luò)傳染病模型的穩(wěn)定性和分支分析[J];西南師范大學(xué)學(xué)報(自然科學(xué)版);2017年03期

4 董宜靜;李桂花;;醫(yī)療資源影響下具有潛伏期的一類傳染病模型建立及性態(tài)分析[J];數(shù)學(xué)的實踐與認(rèn)識;2017年05期

5 杜燕飛;肖鵬;曹慧;;具有階段結(jié)構(gòu)的周期SEIR傳染病模型的動力學(xué)性態(tài)[J];四川師范大學(xué)學(xué)報(自然科學(xué)版);2017年01期

6 王婷;王輝;胡志興;;一類非線性SEIRS傳染病傳播數(shù)學(xué)模型[J];河南科技大學(xué)學(xué)報(自然科學(xué)版);2017年02期

7 謝英超;程燕;賀天宇;;一類具有非線性發(fā)生率的時滯傳染病模型的全局穩(wěn)定性[J];應(yīng)用數(shù)學(xué)和力學(xué);2015年10期

8 李冬梅;徐亞靜;韓春晶;;一類具有非線性接觸率的SEIR傳染病模型[J];哈爾濱理工大學(xué)學(xué)報;2015年02期

9 張輝;徐文雄;李應(yīng)岐;;一類非線性SEIR傳染病模型的全局穩(wěn)定性[J];數(shù)學(xué)的實踐與認(rèn)識;2015年04期

10 郭金生;張生成;唐玉玲;;一類傳染病模型的穩(wěn)定性分析[J];貴州大學(xué)學(xué)報(自然科學(xué)版);2014年06期

相關(guān)碩士學(xué)位論文 前3條

1 湯倩;具有非線性發(fā)生率的連續(xù)SIRS傳染病動力學(xué)模型的穩(wěn)定性研究[D];新疆大學(xué);2014年

2 牛文英;幾類高階差分方程的動力學(xué)行為[D];山西大學(xué);2008年

3 趙柱;幾類非線性差分方程的動力學(xué)行為[D];蘭州大學(xué);2006年

，

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