Ordinary differential equations (ODE) are a popular tool to model the spread of infectious diseases, yet they implicitly assume an exponential distribution to describe the flow from one infection state to another. However, scientific experience yields more plausible distributions where the likelihood of disease progression or recovery changes accordingly with the duration spent in a particular state of the disease. Furthermore, transmission dynamics depend heavily on the infectiousness of individuals. The corresponding nonlinear variation with the time individuals have already spent in an infectious state requires more realistic models. The previously mentioned items are particularly crucial when modeling dynamics at change points such as the implementation of nonpharmaceutical interventions. In order to capture these aspects and to enhance the accuracy of simulations, integro-differential equations (IDE) can be used. In this paper, we propose a generalized model based on integro-differential equations with eight infection states. The model allows for variable stay time distributions and generalizes the concept of ODE-based models as well as IDE-based age-of-infection models. In this, we include particular infection states for severe and critical cases to allow for surveillance of the clinical sector, avoiding bottlenecks and overloads in critical epidemic situations. We will extend a recently introduced nonstandard numerical scheme to solve a simpler IDE-based model. This scheme is adapted to our more advanced model and we prove important mathematical and biological properties for the numerical solution. Furthermore, we validate our approach numerically by demonstrating the convergence rate. Eventually, we also show that our novel model is intrinsically capable of better assessing disease dynamics upon the introduction of nonpharmaceutical interventions.

翻译：常微分方程（ODE）是模拟传染病传播的常用工具，但其隐含假设了描述从一种感染状态流向另一种状态的指数分布。然而，科学经验表明，更合理的分布是疾病进展或康复的可能性随个体在特定疾病状态中持续时间的改变而相应变化。此外，传播动力学在很大程度上取决于个体的传染性。个体在感染状态下已停留时间对应的非线性变化需要更现实的模型。前述因素在模拟诸如实施非药物干预措施等转折点处的动力学时尤为关键。为捕捉这些方面并提高模拟精度，可采用积分微分方程（IDE）。本文提出了一种基于积分微分方程的八状态广义感染模型。该模型允许可变的停留时间分布，并推广了基于ODE的模型以及基于IDE的感染年龄模型的概念。在此框架中，我们特别设置了重症和危重症感染状态，以便监测临床部门，避免在严重疫情情况下出现瓶颈和过载。我们将扩展最近提出的一种用于求解更简单IDE模型的非标准数值格式。该格式适用于我们更先进的模型，我们证明了数值解的重要数学和生物学性质。此外，我们通过展示收敛速率对方法进行了数值验证。最终，我们还证明我们的新模型本质上能更好地评估引入非药物干预措施后的疾病动态。