We propose and analyse a mathematical model for infectious disease dynamics with a discontinuous control function, where the control is activated with some time lag after the density of the infected population reaches a threshold. The model is mathematically formulated as a delayed relay system, and the dynamics is determined by the switching between two vector fields (the so-called free and control systems) with a time delay with respect to a switching manifold. First we establish the usual threshold dynamics: when the basic reproduction number \(\,{\mathcal {R}}_0\le 1\), then the disease will be eradicated, while for \(\,{\mathcal {R}}_0>1\) the disease persists in the population. Then, for \(\,{\mathcal {R}}_0>1\), we divide the parameter domain into three regions, and prove results about the global dynamics of the switching system for each case: we find conditions for the global convergence to the endemic equilibrium of the free system, for the global convergence to the endemic equilibrium of the control system, and for the existence of periodic solutions that oscillate between the two sides of the switching manifold. The proof of the latter result is based on the construction of a suitable return map on a subset of the infinite dimensional phase space. Our results provide insight into disease management, by exploring the effect of the interplay of the control efficacy, the triggering threshold and the delay in implementation.

中文翻译：

---

控制不连续的SIR模型的周期轨道和全局稳定性

我们提出并分析了具有不连续控制功能的传染病动力学的数学模型，其中在被感染人口的密度达到阈值后，以一定的时间延迟激活控制。该模型在数学上被公式化为延迟继电器系统，并且动力学是通过两个矢量场（所谓的自由系统和控制系统）之间的切换（相对于切换歧管具有时间延迟）来确定的。首先，我们建立通常的阈值动力学：当基本繁殖数\（\，{\ mathcal {R}} _ 0 \ le 1 \）时，该病将被根除，而对于\（\，{\ mathcal {R} } _0> 1 \）这种疾病在人群中依然存在。然后，对于\（\，{\ mathcal {R}} _ 0> 1 \），我们将参数域划分为三个区域，并针对每种情况证明有关切换系统的全局动力学的结果：我们找到了自由系统的局部均衡的全局收敛性，自由系统的局部均衡的全局收敛性的条件。控制系统，以及存在于开关歧管两侧之间的周期性解的存在。后一个结果的证明是基于在无限维相空间的子集上构造合适的返回图的。我们的结果通过探索控制功效，触发阈值和实施延迟之间相互作用的影响，为疾病管理提供了见识。