一类带有时滞的SIR模型的稳定性及分支分析

Analysis of stability and bifurcation of a delayed SIR model

为了研究饱和发生率和时滞对传染病模型动力学性态的影响,建立了一类具有饱和发生率和指数出生且带有时滞的SIR模型,通过对模型特征方程的分析,判定了系统的地方病平衡点的稳定性,并找到了系统发生分支的临界值,通过数值模拟验证了理论分析结果的正确性。结果表明:当时滞小于临界值时,地方病平衡点是局部渐近稳定的;当时滞大于临界值时,地方病平衡点不稳定,并产生了Hopf分支。研究结果对解释传染病的周期性暴发、预防和控制传染病的传播具有借鉴作用。

In order to analyze the effects of saturation incidence and time delay on the dynamics of epidemic model, a delayed SIR model with a saturated incidence rate and exponential birth is constructed. By considering the characteristic equation of the system, the stability of the endemic equilibrium is analyzed, and the critical value of the bifurcation is found. The theoretical analysis results are verified by numerical simulations. The result shows that when the delay is less than the critical value, the endemic equilibrium is locally asymptotically stable; When the delay is larger than the critical value, the endemic equilibrium is unstable and there exists a Hopf bifurcation. The results of this study can be used to explain the periodic outbreaks of infectious diseases, and guide the prevention and control of the spread of the disease.