一类具有非线性发生率的时滞传染病模型Hopf分支

Hopf Bifurcation of a Delayed Epidemic Model with Nonlinear Incidence Rate

以恢复个体临时免疫期时滞为分支参数,研究了一类具有阶段结构和非线性发生率的时滞SIRS传染病模型的局部Hopf分支.首先计算得到模型的有病毒平衡点,然后通过分析模型相应特征方程根的分布,得到模型有病毒平衡点局部渐近稳定和产生Hopf分支的时滞临界点τ_0.研究表明,当时滞的值低于临界点τ_0时,有病毒平衡点是局部渐近稳定的.而一旦时滞的值超越临界点,模型的有病毒平衡点将失去稳定性并在有病毒平衡点附近产生一簇分支周期解.最后,利用仿真示例对理论分析结果的正确性进行了验证.

Local Hopf bifurcation of a delayed epidemic model with nonlinear incidence rate and stagestructure is studied in this paper.The viral equilibrium of the model is obtained and then the critical value of the delay for local stability of the viral equilibrium and existence of local Hopf bifurcation is also obtained by analyzing distribution of roots of the corresponding characteristic equation.It is proved that the viral equilibrium is locally asymptotically stable when the value of the delay is below and it will lose its stability and generate a cluster of branching periodic solutions near a viral equilibrium point when the value of the delay is above.Finally,a numerical example is presented to testify the validity of theoretical results.