具有垂直传染和接触传染的传染病模型的稳定性研究
ON THE STABILITY PROPERTY OF A EPIDEMIC MODEL WITH VERTICAL TRANSMISSION AND CONTACT TRANSMISSION

本文研究了一类具有垂直传染和接触传染的传染病模型.利用常微分方程定性与稳定性方法,分析了该模型非负平衡点的存在性及其局部稳定性.同时,利用LaSalle不变性原理和通过构造适当的Lyapunov函数,获得了平凡平衡点、无病平衡点和地方病平衡点全局渐近稳定的充分条件.结果表明当基本再生数小于等于1时,所有种群趋于灭绝;当基本再生数大于1和病毒主导再生数小于1时,病毒很快被清除;当基本再生数大于1和病毒主导再生数大于1以及满足一定条件时,病毒持续流行并将成为一种地方病.
In this paper, a class of epidemic model with vertical transmission and contact transmission is established. By means of qualitative method and stability method of ordinary diffierential equations, the model and the existence of nonnegative equilibrium point are analyzed. And by constructing proper Lyapunov function and LaSalle invariance principle, sufficient conditions of the global asymptotic stability of the trivial equilibrium point, disease-free equilibrium point and endemic equilibrium point are obtained. The results show that when the basic reproduction number is less than or equal to 1, all populations tend to be extinct; when the basic reproduction number is greater than 1 and virus dominant reproduction number is less than 1, the viruses was quickly cleared; when the basic reproduction number is greater than 1 and virus dominant reproduction number is greater than 1 and satisfy certain conditis, the viruses continue to prevail and will become a local disease