具有非线性传染率和易感者类具有Smith增长的SI传染病模型

An SI Epidemic Model with Smith Increasing and Nonlinear Incidence Rate

研究了一种具有非线性传染率且易感者类具有Sm ith增长的的传染病模型。以往的具有非线性传染率的传染病模型相比,这种模型引入了种群动力,也就是种群的总数不再为常数且种群的增长规律满足方程dxdt=rx(K-x)K+Dx。因此,该类模型更精确的描述了传染病传播的规律。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。进一步得出了在某些参数范围内会出现Hopf分支现象,并对上述模型进行了生物学讨论。

The paper deals with a kind of epidemic models with Smith increasing and Nonlinear incidence rate. Compared with the former epidemic models with nonlinear incidence rate, the model induces population dynamics. That means the total amount of populations is no longer a constant and increases by Simith function, so they can describe the communicable diseases＇ transmitting law more accurately. Discussed the positive invariant set, the existence 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. Further,for some range the parameters, they can undergo Hopf bifurcation.