Compartmental models have long served as important tools in mathematical epidemiology, with their usefulness highlighted by the recent COVID-19 pandemic. However, most of the classical models fail to account for certain features of this disease and others like it, such as the ability of exposed individuals to recover without becoming infectious, or the possibility that asymptomatic individuals can indeed transmit the disease but at a lesser rate than the symptomatic. In the first part of this paper, we propose two new compartmental epidemiological models and study their equilibria, obtaining an endemic threshold theorem for the first model. In the second part of the paper, we treat the second model as an affine control system with two controls: vaccination and mitigation. We show that this system is static feedback linearizable, presents some simulations, and investigates an optimal control version of the problem. We conclude with some open problems and ideas for future research.

长期以来，区室模型一直是数学流行病学的重要工具，最近的 COVID-19 大流行凸显了其有用性。然而，大多数经典模型未能解释这种疾病和其他类似疾病的某些特征，例如暴露个体在不具有传染性的情况下康复的能力，或者无症状个体确实可以传播该疾病但传播率较低的可能性比有症状的。在本文的第一部分中，我们提出了两个新的区室流行病学模型并研究了它们的平衡，获得了第一个模型的地方性阈值定理。在本文的第二部分，我们将第二个模型视为具有两个控制的仿射控制系统：疫苗接种和缓解。我们证明该系统是静态反馈可线性化的，提出了一些模拟，并研究了问题的最优控制版本。最后我们提出了一些未解决的问题和未来研究的想法。