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.

我们研究了带有疫苗接种的 SIR 模型的全球动态，我们假设个体对流行病动态的反应不同。它们的异构响应由 Preisach 滞后算子建模。我们提出了无感染平衡状态的全局稳定性的条件。如果此条件不成立，则该模型具有一组连接的地方性平衡状态，其特征是受感染和免疫个体的比例不同。在这种情况下，我们证明了每条轨迹要么收敛到一个地方性平衡，要么收敛到一个周期性轨道。在额外的自然假设下，周期性吸引子被排除在外，我们保证每个轨迹收敛到一个地方性平衡状态。