【作者】 董霖；

【导师】 李学鹏；

【作者基本信息】 福建师范大学 ， 基础数学， 2008， 硕士

【摘要】 本文研究了具有非线性传染率的四类传染病模型:首先,研究了一类易感者、潜伏者和染病者均有常数输入,且传染率是非线性传染率βf(S)I的SEIR传染病模型.研究表明此时系统不存在无病平衡点,只存在唯一一个地方病平衡点.利用Hurwitz判别法证明了地方病平衡点的局部稳定性,进一步利用Li和Muldowney所发展的几何方法证明了地方病平衡点的全局稳定性.其次,研究了两类SIQR传染病模型,第一类为各仓室均有常数输入(除了隔离仓室),且传染率为一般形式非线性饱和传染率的SIQR模型,第二类为具有强非线性传染率的SIQR模型.对第一个模型,当不考虑隔离者的因病死亡时,引入变量代换将四维模型转化为二维渐近自治系统,而后利用Dulac函数和极限方程理论证明了地方病平衡点的全局稳定性.对第二个模型,运用Hurwitz判别法分析了各平衡点的局部稳定性,发现了在一定的条件下,该模型会发生Hopf分支产生周期解,进一步我们应用Dulac函数和极限方程理论证明了当0＜p≤1时地方病平衡点的全局稳定性.最后,研究了一类易感者和染病者均有常数输入,疾病具有垂直传染,且传染率是一般形式非线性饱和传染率的SIRI传染病模型.结果表明此时系统不存在无病平衡点,只存在唯一一个地方病平衡点.利用Hurwitz判别法证明了地方病平衡点的局部稳定性.当传染率为双线性传染率和标准传染率时,利用广义BendixsonDulac定理排除了三维系统的周期解,从而证明了地方病平衡点的全局稳定性.更多还原

【Abstract】 In this paper,we study four kinds of epidemic models with nonlinear incidence rates.First, we consider an SEIR epidemic model with nonlinear incidence rate and constant immigration, which includes susceptibles、exposeds and infectives. It has been shown that this model has only unique endemic equilibrium. The local asymptotical stable results of the epidemic equilibrium was proved by using the Hurwitz criterion and the global asymptotical stable results of its by means of the geometric approach developed by Li and Muldowney .Secondly, we study two kinds of SIQR epidemic models. The first one is the SIQR model with general form nonlinear saturated infectivity and constant inflows, which includes susceptibles、infectives and recovered, and the second one is the SIQR model with nonlinear incidence ratesβI~pS~q. For the first model, we reduce the four-dimensional model to a two-dimensional asymptotical autonomous system by means of a transformation of variables. Furthermore, we prove the global asymptotical stability of the epidemic equilibrium by means of Dulac’s function and the theory of limit systems. For the second model, we analyse the stability of the equilibria by using the Hurwitz criterion, and obtained the existence of periodic solutions by Hopf bifurcation for some parameter values. Furthermore,using Dulac’s function and the theory of limit systems,we prove the global asymptotical stability of the epidemic equilibrium when 0 < p≤1.Finally, we formulate a kind of SIRI model with the vertical transmission,general form nonlinear saturated infectivity and constant immigration which includes new susceptibles and infectives. It is also found that the system exists only one equilibrium. Using the Routh-Hurwitz criterion, we prove the local asymptotical stability of the epidemic equilibrium. For the important cases of mass action incidence and standard incidence,applying Bendixson-Dulac theorem, the existence of the periodic solutions of the three-dimensional system is excluded, thereby the global stability of the endemic equilibrium is proved provided the endemic equilibrium exists.更多还原