一类具有非线性发生率的SIR模型的稳定性和Hopf分支

Stability and Hopf Bifurcation of an SIR Model with Nonlinear Incidence Rate

【目的】研究了一类具有非线性发生率的SIR传染病模型,分析该系统在非平凡平衡点处的稳定性和Hopf分支。【方法】运用正规形理论和中心流形投影定理,讨论了该系统在平衡点处的稳定性。【结果】得到第一Laypunov系数,当l1(0)>0时,该系统是不稳定的亚临界分支;当l1(0)<0时,该系统是稳定的超临界分支。【结论】得到了系统在非平凡平衡点附近会产生唯一、稳定的极限环,此时传染病会发生但不会大规模流行。

[Purposes]It presents a class of model, which hosts the SIR contagion with the nonlinear incidence rate, and the stability and Hopf branch at nontrivial equilibrium point have been analyzed. [Methods]The normal form theory and center manifold projec- tion theorem have been employed to discuss the stability at the equilibrium point in this system. [Findings]This system is not the stable subcritical branch when the first Laypunov coefficient l1（0）〉0, while the system hosts the stable supercritical branch if l1 （0）〈0.[Conelusions]The investigation indicates that the unique and stable limit cycle of the system is generate, and the contagion would be occur but not be popular.