具有种群动力和非线性传染率的传染病模型研究

【作者】 石磊；

【导师】 俞军；

【作者基本信息】 南京理工大学 ， 生物医学工程， 2005， 硕士

【摘要】 本文研究了三种形式的具有种群动力和非线性传染率的传染病模型。我们建立了具有常数迁入率和非线性传染率βI~PS/(1+αI~P)的SI模型、具有常数迁入率和非线性传染率βI~pS~q的SI模型以及具有非线性传染率和易感者类具有Smith增长的SI模型。与以往的具有非线性传染率的传染病模型相比,这三种模型都引入了种群动力,也就是种群的总数不再为常数,因此,该类模型更精确的描述了传染病传播的规律。本文讨论了模型的正不变集,运用微分方程稳定性理论分析了模型平衡点的存在性及稳定性,得出了疾病消除平衡点和地方病平衡点的全局渐进稳定的充分条件。进一步的,得出了在某些参数范围内会出现Hopf分支现象。最后对上述三种模型进行了比较。更多还原

【Abstract】 The paper deals with three kinds of epidemic models with population dynamics and Nonlinear incidence rate.We have established an SI epidemic model with constant inputand nonlinear incidence rate (βI~pS)/(1+αI~p), an SI epidemic model with constant input andnonlinear incidence rate βI~pS~q and an SI epidemic model with a nonlinear incidencerate and Smith growth in susceptible class.Compared with the former epidemic models with nonlinear incidence rate ,the three models all introduce population dynamics .That means the total amount of populations is no longer a constant ,so they can describe the communicable diseases’ transmitting law more accurately.We discussed the positive invariant set,the exsistence and the stability of the equilibrium by the stability theory of ordinary differential equation and the conditions for the global asymptotic stability of the disease-free equilibrium and the endemic equilibrium are obtained.Farther,for some range the parameters,they can undergo Hopf bifurcation.At last, a comparison is also made between the three models.更多还原