本文選題：SIR + 倉室模型��； 參考：《西安科技大學》2012年碩士論文

【摘要】：本文以經典的SIR倉室模型為基礎，考慮疾病具有潛伏期、人員在不同區(qū)域流動等因素，建立了幾類甲型H1N1流感傳播的SEIRS傳染病模型.利用矩陣譜半徑定義基本再生數并得到了幾類模型的基本再生數，證明了基本再生數決定的模型的動力學性態(tài)，如平衡點的存在性和穩(wěn)定性等.所得結果能夠為疾病的預防和控制提供理論依據和數量基礎. 首先，建立了一類具有兩個彼此獨立斑塊的SEIRS模型，利用矩陣譜半徑來定義模型的基本再生數，得到了該模型的基本再生數R0的表達式，并證明了當R01時無病平衡點0的不穩(wěn)定性及當R01時無病平衡點是全局漸近穩(wěn)定性.通過分析基本再生數的表達式發(fā)現當參數c（國家和政府采取的保護措施）的值越大，基本再生數越小，疾病越容易控制.最后通過數值模擬也驗證了該理論結果. 其次，建立了一類具有兩個斑塊且只有一個斑塊遷移的SEIRS模型.得到了決定疾病消亡與否的基本再生數R0.證明了當R01時無病平衡點0的不穩(wěn)定性以及當R01時無病平衡點0的全局漸近穩(wěn)定性.數值模擬結果顯示：當R01時，第二個斑塊的染病者曲線很快地收斂于零，而第一個斑塊的染病者曲線則是先逐漸上升達到某一峰值然后再逐漸下降最終趨于零.接著考慮兩個斑塊具有相同或者不同遷移率的SEIRS模型并得到基本再生數的理論表達式.數值模擬結果顯示：當R01時，兩個斑塊的染病者人數均趨近于一個常數，表明該疾病將會在此地流行而成為地方�。划擱01時，兩個斑塊的染病者人數最終均趨近于零，，表明該疾病將會逐漸消亡. 最后，推廣兩個斑塊上的SEIRS模型到n個斑塊，利用矩陣譜半徑的方法定義并得到了具有n個斑塊的SEIRS模型基本再生數的一般表達式，討論了該模型的平衡點存在性和穩(wěn)定性.
[Abstract]:Based on the classical SIR chamber model and considering the latent period of the disease and the movement of personnel in different regions, several SEIRS infectious disease models of A / H1N1 influenza transmission were established in this paper.The fundamental reproducing number is defined by using the matrix spectral radius and the basic reproducing numbers of several kinds of models are obtained. The dynamical behavior of the model determined by the basic reproduction number is proved, such as the existence and stability of the equilibrium point, etc.The results can provide theoretical basis and quantitative basis for disease prevention and control.Firstly, a class of SEIRS model with two independent patches is established. The basic reproduction number of the model is defined by using the matrix spectral radius, and the expression of the basic reproduction number R _ 0 of the model is obtained.The instability of the disease-free equilibrium 0 at R01 and the global asymptotic stability of the disease-free equilibrium at R01 are proved.By analyzing the expression of basic reproduction number, it is found that when the value of parameter c (protection measures taken by state and government) is larger, the number of basic regeneration is smaller, and the disease is easier to control.Finally, the theoretical results are verified by numerical simulation.Secondly, a SEIRS model with two patches and only one patch migration is established.The basic regeneration number R0.The instability of the disease-free equilibrium 0 at R01 and the global asymptotic stability of the disease-free equilibrium 0 at R01 are proved.The numerical simulation results show that when R01, the second patch curve converges to zero quickly, while the first patch curve rises to a certain peak at first and then decreases gradually to zero.Then the SEIRS model of two patches with the same or different mobility is considered and the theoretical expression of the basic regeneration number is obtained.The numerical simulation results show that when R01, the number of people infected with both plaques approaches a constant, indicating that the disease will become endemic in this area, and when R01, the number of infected people of both plaques eventually approaches zero.This suggests that the disease will die out.Finally, we generalize the SEIRS model on two patches to n patches, define and obtain the general expression of the basic regeneration number of the SEIRS model with n patches by using the method of matrix spectral radius, and discuss the existence and stability of the equilibrium point of the model.
【學位授予單位】：西安科技大學
【學位級別】：碩士
【學位授予年份】：2012
【分類號】：O175;R311

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