﻿ 带有时滞的双传染病假设模型稳定性及Hopf分岔分析 Analysis on Stability and Hopf Bifurcation in a Delayed Epidemic Model with Double Epidemic Hypothesis

Vol.05 No.04(2016), Article ID:18972,9 pages
10.12677/AAM.2016.54070

Analysis on Stability and Hopf Bifurcation in a Delayed Epidemic Model with Double Epidemic Hypothesis

Jiarong Lu, Jiangang Zhang, Hongwei Luo, Jun Yin, Qin Pang

Department of Mathematics, Lanzhou Jiaotong University, Lanzhou Gansu

Received: Oct. 26th, 2016; accepted: Nov. 10th, 2016; published: Nov. 18th, 2016

ABSTRACT

This paper mainly investigates the stability and Hopf bifurcation in a delayed epidemic model system with double epidemic hypothesis. We study the stability of the unique positive equilibrium for the system under different conditions. By analyzing the distribution of characteristic roots of corresponding linearized system, we obtain the conditions for keeping the system to be stable. Moreover, it is illustrated that the Hopf bifurcation will occur when the delay passes through a critical value. Then we use the MATLAB numerical simulations for justifying the theoretical results.

Keywords:Epidemic Model with Double Epidemic Hypothesis, Time Delay, Stability, Hopf Bifurcation

1. 引言

(1)

2. 平衡点及局部稳定性

(2)

(3)

，(3)式变为

(4)

3. Hopf分岔的存在性

1) 当时。

(5)

(6)

(7)

(8)

，有

(9)

，通过以上的分析，可以得到以下引理。

2) 当时。

(10)

(11)

，当时，方程(3)有一对纯虚根，因此，当时，可以总结出以下引理。

i) 当时，系统(1)在正平衡点处是局部渐进稳定的，当时趋于不稳定。

ii) 当时，系统(1)在正平衡点处发生Hopf分岔。

4. 数值模拟

(12)

(a)(b)

Figure 1. The positive equilibrium of system (1) is locally asymptotically stable when (Figure 1(a)), the hopf bifurcation will occur in the equilibrium of system (1) when (Figure 1(b)), where initial value is “10, 10, 0. 1”

(a)(b)

Figure 2. The positive equilibrium of system (1) is locally asymptotically stable when (Figure 2(a)) , the hopf bifurcation will occur in the equilibrium of system (1) when (Figure 2(b)) , where initial value is “11, 0. 1, 1”

5. 总结

Analysis on Stability and Hopf Bifurcation in a Delayed Epidemic Model with Double Epidemic Hypothesis[J]. 应用数学进展, 2016, 05(04): 605-613. http://dx.doi.org/10.12677/AAM.2016.54070

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