Mathematical Institute of the Silesian University, Na Rybníčku 1, 746 01, Opava, Czech Republic.
University of Texas at Dallas, 800 W Campbell, Richardson, TX, 75080, United States.
J Math Biol. 2021 Jul 4;83(2):11. doi: 10.1007/s00285-021-01629-8.
We study global dynamics of an SIR model with vaccination, where we assume that individuals respond differently to dynamics of the epidemic. Their heterogeneous response is modeled by the Preisach hysteresis operator. We present a condition for the global stability of the infection-free equilibrium state. If this condition does not hold true, the model has a connected set of endemic equilibrium states characterized by different proportion of infected and immune individuals. In this case, we show that every trajectory converges either to an endemic equilibrium or to a periodic orbit. Under additional natural assumptions, the periodic attractor is excluded, and we guarantee the convergence of each trajectory to an endemic equilibrium state. The global stability analysis uses a family of Lyapunov functions corresponding to the family of branches of the hysteresis operator.
我们研究了具有疫苗接种的 SIR 模型的全局动态，其中我们假设个体对传染病动态的反应不同。他们的异质反应由 Preisach 滞后算子建模。我们提出了一个条件，用于确定无感染平衡点的全局稳定性。如果这个条件不成立，那么模型就有一个由不同比例的感染和免疫个体组成的传染病平衡点的连通集。在这种情况下，我们表明每条轨迹要么收敛到传染病平衡点，要么收敛到周期轨道。在附加的自然假设下，排除了周期吸引子，我们保证每条轨迹收敛到传染病平衡点。全局稳定性分析使用与滞后算子的分支族相对应的一族 Lyapunov 函数。
