一类具有自然治愈率的传染病模型的全局稳定性分析.doc

文档介绍：
一类具有自然治愈率的传染病模型的全局
稳定性分析
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(空军工程大学理学院,西安 710051)
摘要:研究了一类具有自然治愈率、非线性发生率的流行病传染模型,得到了基本再生数,通过构造 Lyapunov
函数,证明了当基本再生数小于等于 1 时,无病平衡点是全局渐进稳定的;当基本再生数大于 1 时,地方
病平衡点是全局渐进稳定的。
关键词:生物数学;基本再生数;非线性发生率;全局渐进稳定;自然治愈率
中图分类号:0175
Global Stability of an Epidemic Model with Spontaneous
Cure Rate
SONG Xiuchao, REN Jinshen, SONG Hao, ZHENG Mingfa
(School of Science,Air Force Engineering University,Xi'an 710051)
Abstract: This paper deals with a kind of epidemic model with nonlinear incidence rate and
spontaneous cure rate. The basic reproduction number is obtained. By constructing the Lyapunov
function, we prove that the infection free equilibrium of the system is globally asymptotically stable
when the basic reproduction number is less than or equal to one, and that the unique infection
equilibrium of the system is globally asymptotically stable when the basic reproduction number is
greater than one.
Key words: Biomathematics; Basic reproduction number; Nonlinear incidence; Globally
asymptotically stable; Spontaneous cure rate
0 引言
数学模型是研究传染病动力学的重要工具,并且已经得到了许多重要结果,在临床医学
上也有一定的指导价值。传统的传染病模型通常将发生率设为双线性发生率或标准性发生率
[1]
种具有一般形式的非线性发生率f(S ,I ,N )满足如下性质:
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(1)
f(S ,0,N ) = f(0,I ,N ) = 0
(2)
¶f(S ,I ,N )
¶I

> 0,
¶f(S ,I ,N )
¶S

> 0
(3)
¶ 2f(S ,I ,N )
¶I 2

£ 0

S ,I > 0
文献[2]重点考虑了具有如上形式的发生率的 SIRS 和 SEIRS 模型的稳定性,并考虑了