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研究生:黃義傑
研究生(外文):Yi-Jie Huang
論文名稱:具有飽和治療率的網路SIS傳染病模型之研究
論文名稱(外文):A study of a network-based SIS epidemic model with saturated treatment rate
指導教授:李俊憲李俊憲引用關係
指導教授(外文):Chun-Hsien Li
口試委員:陳振遠謝博文
口試委員(外文):Jen-Yuan ChenPo-Wen Hsieh
口試日期:2017-06-28
學位類別:碩士
校院名稱:國立高雄師範大學
系所名稱:數學系
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2017
畢業學年度:105
語文別:中文
論文頁數:31
中文關鍵詞:複雜網路傳染病模型飽和治療率平衡點穩定性
外文關鍵詞:complex networksepidemic modelsaturated treatment rateequilibriumstability
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  近年來,傳染病已經造成世界上許多人感染和死亡,因此研究傳染病的傳播行為是一個相當重要的主題。本文主要是以常微分方程組來建構模型,為了描述更符合真實的情況,我們在模型中考慮了複雜網路的結構以及飽和傳染率。此外,我們知道醫療資源是有限的,因此當某一傳染病的感染比例上升時,治療率會因有限的資源而趨近飽和。故我們也將飽和治療率加入模型中,並進一步分析此模型之平衡點的存在性與穩定性。有趣的是,我們發現加入飽和治療率會導致有多個疾病平衡點(endemic equilibrium)的存在。最後我們進行數值模擬來驗證理論分析。
  In recent years, infectious diseases have become a serious problem and have increased morbidity and mortality of individuals around the world. Thus, studying the spreading dynamics of the infectious diseases is a relevant issue. In this thesis, we will investigate an epidemic model which is composed of a system of ordinary differential equations. To fit for reality, complex network topology and saturated incidence rate are incorporated into the model. Besides, we know that the resources of treatment may be limited, and this implies that the recovery from infective individuals will reach a maximum. Therefore, we will study a network-based SIS epidemic model with saturated incidence and treatment rates. Existence and stability of the equilibria of the epidemic model will be analyzed. Interestingly, we find that incorporating the saturated treatment rate may cause the existence of two endemic equilibria. Numerical simulations will be given to demonstrate the theoretical results.
目錄

第一章 前言 1
1.1 簡介 1
1.2 研究動機 1
1.3 模型介紹 2
第二章 建構模型 4
第三章 研究分析 6
3.1 Positive invariance 6
3.2 Disease-Free Equilibrium 7
3.3 Endemic Equilibrium 11
第四章 數值模擬 19
第五章 結論 30
參考文獻 31

圖次

1. 非單調性傳染率 3
2. 飽和傳染率 3
3. 非零θ解交點示意圖 13
4. α2=6,R0<1時感染比例曲線變化 19
5. α2=6,R0<1時θ圖解 20
6. α2=0.4,R0'<1時感染比例曲線變化 20
7. α2=0.4,R0'<1時θ圖解 21
8. α2=10,R0<1<R0'感染比例曲線變化 22
9. α2=10,R0<1<R0'時θ圖解 22
10.α2=20,R0<1<R0'感染比例曲線變化 23
11.α2=20,R0<1<R0'時θ圖解 23
12.α2=40,R0<1<R0'感染比例曲線變化 24
13.α2=40,R0<1<R0'時θ圖解 24
14.α2=80,R0<1<R0'感染比例曲線變化 25
15.α2=80,R0<1<R0'時θ圖解 25
16.α2=120,200,360時θ圖解 26
17.α2=3,R0>1時感染比例曲線變化 26
18.α2=3,R0>1時θ圖解 27
19.α2=7,R0>1時感染比例曲線變化 27
20.α2=7,R0>1時θ圖解 28
21.α2=11,R0>1時感染比例曲線變化 28
22.α2=11,R0>1時θ圖解 29
23.α2=20,40,80時θ圖解 29

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[2] A. Lajmanovich and J. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci., 28 (1976), pp. 221-236.
[3]C. Li, C. Tsai, and S. Yang, Analysis of epidemic spreading of an SIRS model in complex heterogeneous networks, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), pp. 1042-1054.
[4]C. Li, Dynamics of a network-based SIS epidemic model with nonmonotone incidencerate, Physica A, 427 (2015), pp. 234-243.
[5]T. Li, Y. Wang, and Z. Guan, Spreading dynamics of a SIQRS epidemic model on scale-free networks, Commun. Nonlinear Sci. Numer.Simul., 19 (2014), pp. 686-692.
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[9]Q. Wu, X. Fu and M. Yang 2011 Epidemic thresholds in a heterogenous population with competing strains Chin. Phys. B, (2011), 046401
[10]J. Wei and J.Cui , Dynamics of SIS epidemic model with the standard incidence rate and saturated treatment function, J. Bio., 5 (2012), 1260003
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[13]G. Zhu, X. Fu, G. Chen, Spreading dynamics and global stability of a generalized epidemic model on complex heterogeneous networks, Appl. Math. Model., 36 (2012), pp. 5808-5817.

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