具有周期传染率的SIR传染病模型的周期解

The Periodic Solution of a SIR Epidemic Model with Periodic Infection Rate

考虑了具有周期传染率的SIR流行病模型.定义了基本再生数■_0=■/(μ+γ),分析了该模型的动力学性态,证明了■_0<1时无病平衡点是全局稳定的;■_0>1时,无病平衡点是不稳定的,模型至少存在一个周期解.对小振幅的周期传染率模型,给出了模型周期解的近似表达式,证明了该周期解的稳定性,最后做了数值模拟,结果显示周期解可能是全局稳定的.

In this paper a SIR model with periodic infection rate β（t） is studied. The basic reproductive number ^-R0= β/（μ＋γ） is defined. The dynamical behavior of the model is analyzed. It is proved that the disease free equilibrium is globally stable if ^-R0〈 1. The disease free equilibrium is unstable when ^-R0〉 1. The existence of the periodic solution is investigated, and it is proved that the periodic model has at least one periodic solution if ^-R0〉 1. The unique- ness and stability of the periodic solution for sufficient small periodicity is obtained. We have the conjecture that the periodic endemic sohltion is globally stable when ^-R0〉 1. The numerical simulation suooorts our conjecture.