Compartmental models provide simple and efficient tools to analyze the relevant transmission processes during an outbreak, to produce short-term forecasts or transmission scenarios, and to assess the impact of vaccination campaigns. However, their calibration is not straightforward, since many factors contribute to the rapid change of the transmission dynamics during an epidemic. For example, there might be changes in the individual awareness, the imposition of non-pharmacological interventions and the emergence of new variants. As a consequence, model parameters such as the transmission rate are doomed to change in time, making their assessment more challenging. Here, we propose to use Physics-Informed Neural Networks (PINNs) to track the temporal changes in the model parameters and provide an estimate of the model state variables. PINNs recently gained attention in many engineering applications thanks to their ability to consider both the information from data (typically uncertain) and the governing equations of the system. The ability of PINNs to identify unknown model parameters makes them particularly suitable to solve ill-posed inverse problems, such as those arising in the application of epidemiological models. Here, we develop a reduced-split approach for the implementation of PINNs to estimate the temporal changes in the state variables and transmission rate of an epidemic based on the SIR model equation and infectious data. The main idea is to split the training first on the epidemiological data, and then on the residual of the system equations. The proposed method is applied to five synthetic test cases and two real scenarios reproducing the first months of the COVID-19 Italian pandemic. Our results show that the split implementation of PINNs outperforms the standard approach in terms of accuracy (up to one order of magnitude) and computational times (speed up of 20%).

翻译：仓室模型为分析疫情暴发期间的相关传播过程、生成短期预测或传播情景、评估疫苗接种运动的影响提供了简单而有效的工具。然而，其校准并非易事，因为多种因素会导致流行病期间传播动态的快速变化。例如，个体意识的改变、非药物干预措施的实施以及新变异株的出现。因此，如传播率等模型参数必然随时间变化，使得评估更加困难。在此，我们提出使用物理信息神经网络（PINNs）来追踪模型参数的时间变化，并估算模型状态变量。PINNs因其能够同时考虑数据信息（通常具有不确定性）和系统控制方程，近年来在众多工程应用中备受关注。PINNs识别未知模型参数的能力使其特别适用于解决病态逆问题，例如流行病模型应用中出现的此类问题。我们开发了一种用于PINNs实现的简化拆分方法，基于SIR模型方程和感染数据估算疫情状态变量和传播率的时间变化。主要思想是首先对流行病数据进行训练，然后对系统方程残差进行训练。该方法应用于五个合成测试案例和两个真实场景，这些场景模拟了COVID-19疫情在意大利最初几个月的传播情况。我们的结果表明，与标准方法相比，PINNs的拆分实现在精度上（数量级提升高达一个数量级）和计算时间上（加速约20%）均表现出更优异的性能。