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
DOI: 10.12677/AAM.2016.54070, PDF,    国家自然科学基金支持
作者: 卢加荣, 张建刚, 罗宏伟, 殷俊, 庞琴：兰州交通大学数理学院，甘肃 兰州
关键词: 双传染病假设模型；时滞；稳定性；Hopf分岔；Epidemic Model with Double Epidemic Hypothesis； Time Delay； Stability； Hopf Bifurcation
摘要: 本文主要分析带有时滞影响下的双传染病假设模型的稳定性及Hopf分岔，研究了不同情况下的系统唯一的正平衡点的稳定性，并通过分析相应线性系统的特征根的分布，得到了使得系统平衡的条件。然而，当时滞的值通过一个临界值的时候，Hopf分岔就会发生。最后, 运用MATLAB数值模拟去验证得到的理论结果。
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 criti-cal value. Then we use the MATLAB numerical simulations for justifying the theoretical results.
文章引用：卢加荣, 张建刚, 罗宏伟, 殷俊, 庞琴. 带有时滞的双传染病假设模型稳定性及Hopf分岔分析[J]. 应用数学进展, 2016, 5(4): 605-613. http://dx.doi.org/10.12677/AAM.2016.54070

参考文献

[1] Kang, H.Y. and Fu, X.C. (2015) Epidemic Spreading and Global Stability of an SIS Model with an Infective Vector on Complex Networks. Communications in Nonlinear Science and Numerical Simulation, 27, 30-39. http://dx.doi.org/10.1016/j.cnsns.2015.02.018 [Google Scholar] [CrossRef] 
[2] Li, C.H. (2015) Dynamics of a Network-Based SIS Epidemic Model with Nonmonotone Incidence Rate. Physica A: Statistical Mechanics and its Applications, 427, 234-243. http://dx.doi.org/10.1016/j.physa.2015.02.023 [Google Scholar] [CrossRef] 
[3] Economoua, A., Gómez-Corral, A. and López-García, M. (2015) A Stochastic SIS Epidemic Model with Heterogeneous Contacts. Physica A: Statistical Mechanics and its Applications, 421, 78-97. http://dx.doi.org/10.1016/j.physa.2014.10.054 [Google Scholar] [CrossRef] 
[4] Zhang, J.C. and Sun, J.T. (2014) Stability Analysis of an SIS Epidemic Model with Feedback Mechanism on Networks. Physica A: Statistical Mechanics and its Applications, 394, 24-32. http://dx.doi.org/10.1016/j.physa.2013.09.058 [Google Scholar] [CrossRef] 
[5] Li, J., Zhao, Y.L. and Zhu, H.P. (2015) Bifurcation of an SIS Model with Nonlinear Contact Rate. Journal of Mathematical Analysis and Applications, 432, 1119-1138. http://dx.doi.org/10.1016/j.jmaa.2015.07.001 [Google Scholar] [CrossRef] 
[6] Cai, Y.L., Kang, Y., Banerjee, M. and Wang, W.M. (2015) A Stochastic SIRS Epidemic Model with Infectious Force under Intervention Strategies. Journal of Differential Equations, 259, 7463-7502. http://dx.doi.org/10.1016/j.jde.2015.08.024 [Google Scholar] [CrossRef] 
[7] Zhou, T.T., Zhang, W.P. and Lu, Q.Y. (2014) Bifurcation Analysis of an SIS Epidemic Model with Saturated Incidence Rate and Saturated Treatment Function. Applied Mathematics and Computation, 226, 288-305. http://dx.doi.org/10.1016/j.amc.2013.10.020 [Google Scholar] [CrossRef] 
[8] Luo, J.F., Wang, W.D., Chen, H.Y. and Fu, R. (2016) Bifurcations of a Ma-thematical Model for HIV Dynamics. Journal of Mathematical Analysis and Applications, 434, 837-857. http://dx.doi.org/10.1016/j.jmaa.2015.09.048 [Google Scholar] [CrossRef] 
[9] Hao, P.M., Fan, D.J., Wei, J.J. and Liu, Q.H. (2012) Dynamic Behaviors of a Delayed HIV Model with Stage-Structure. Communications in Nonlinear Science and Numerical Simulation, 17, 4753-4766. http://dx.doi.org/10.1016/j.cnsns.2012.04.004 [Google Scholar] [CrossRef] 
[10] Liu, B.W. (2015) Convergence of an SIS Epidemic Model with a Constant Delay. Applied Mathematics Letters, 49, 113-118. http://dx.doi.org/10.1016/j.aml.2015.04.012 [Google Scholar] [CrossRef] 
[11] Zhao, Y.N. and Jiang, D.Q. (2014) The Threshold of a Stochastic SIS Epidemic Model with Vaccination. Applied Mathematics and Computation, 243, 718-727. http://dx.doi.org/10.1016/j.amc.2014.05.124 [Google Scholar] [CrossRef] 
[12] Meng, X.Z., Zhao, S.N., Feng, T. and Zhao, T.H. (2016) Dynamics of a Novel Nonlinear Stochastic SIS Epidemic Model with Double Epidemic Hypothesis. Journal of Mathematical Analysis and Applications, 433, 227-242. http://dx.doi.org/10.1016/j.jmaa.2015.07.056 [Google Scholar] [CrossRef] 
[13] Huo, H.F. and Ma, Z.P. (2010) Dynamics of a Delayed Epidemic Model with Non-Monotonic Rate. Communications in Nonlinear Science and Numerical Simulation, 15, 459-468. http://dx.doi.org/10.1016/j.cnsns.2009.04.018 [Google Scholar] [CrossRef] 
[14] Çelik, C. (2008) The Stability and Hopf Bifurcation for a Predator-Prey System with Time Delay. Chaos, Solitons Fractals, 37, 87-99. http://dx.doi.org/10.1016/j.chaos.2007.10.045 [Google Scholar] [CrossRef] 
[15] Zhang, L.Y. (2015) Hopf Bifurcation Analysis in a Monod-Haldane Predator-Prey Model with Delays and Diffusion. Applied Mathematical Mod-elling, 39, 1369-1382. http://dx.doi.org/10.1016/j.apm.2014.09.007 [Google Scholar] [CrossRef] 
[16] Çelik, C. (2009) Hopf Bifurcation of a Ratio-Dependent Predator-Prey System with Time Delay. Chaos, Solitons Fractals, 42, 1474-1484. http://dx.doi.org/10.1016/j.chaos.2009.03.071 [Google Scholar] [CrossRef] 
[17] Hale, J.K. (1977) Theory of Functional Differential Equation. Springer-Verlag, New York. http://dx.doi.org/10.1007/978-1-4612-9892-2 [Google Scholar] [CrossRef] 
[18] Hassard, B.D., Kazarinoff, N.D. and Wan, Y.H. (1981) Theory and Applica-tions of Hopf Bifurcation. Cambridge University Press, Cambridge.

为你推荐