﻿ 一类离散SIR流行病模型的分岔和混沌分析 Bifurcation and Chaos Analysis of a Class of Discrete SIR Epidemic Models

Vol.05 No.03(2016), Article ID:18371,9 pages
10.12677/AAM.2016.53048

Bifurcation and Chaos Analysis of a Class of Discrete SIR Epidemic Models

Qin Pang, Jiangang Zhang, Tian Deng, Jun Yin, Jiarong Lu

School of Mathematics, Lanzhou Jiaotong University, Lanzhou Gansu

Received: Jul. 28th, 2016; accepted: Aug. 19th, 2016; published: Aug. 22nd, 2016

ABSTRACT

The paper discusses the dynamical behaviors of a discrete-time SI epidemic model. The local stability of the disease-free equilibrium and endemic equilibrium is obtained. It is shown that the model undergoes Flip bifurcation and Hopf bifurcation by using center manifold theorem and bifurcation theory. So it exhibits the complex dynamical behaviors. These results reveal far richer dynamical behaviors of the discrete epidemic model.

Keywords:Discrete-Time SIR System, Flip Bifurcation, Hopf Bifurcation, Chaos, Random Parameter

1. 前言

2. 离散的SIR系统

(1)

(2)

(3)

(4)

(5)

(6)

3. 分岔

3.1. Flip分岔

(7)

Jacobian矩阵的特征方程可以写成

， (8)

(9)

(10)

(11)

。我们变换不动点到原点，考虑的参数的变量，系统(3)变成

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

3.2. 霍普夫分岔

(23)

(24)

(25)

(26)

(27)

(28)

， (29)

(30)

， (31)

(32)

(33)

，(34)

(35)

4. 结论

“国家自然科学基金项目”(No 61364001)，和甘肃省科学与技术项目(No. 144GKCA018)。

Bifurcation and Chaos Analysis of a Class of Discrete SIR Epidemic Models[J]. 应用数学进展, 2016, 05(03): 390-398. http://dx.doi.org/10.12677/AAM.2016.53048

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