Highly-interconnected societies difficult to model the spread of infectious diseases such as COVID-19. Single-region SIR models fail to account for incoming forces of infection and expanding them to a large number of interacting regions involves many assumptions that do not hold in the real world. We propose using Universal Differential Equations (UDEs) to capture the influence of neighboring regions and improve the model's predictions in a combined SIR+UDE model. UDEs are differential equations totally or partially defined by a deep neural network (DNN). We include an additive term to the SIR equations composed by a DNN that learns the incoming force of infection from the other regions. The learning is performed using automatic differentiation and gradient descent to approach the change in the target system caused by the state of the neighboring regions. We compared the proposed model using a simulated COVID-19 outbreak against a single-region SIR and a fully data-driven model composed only of a DNN. The proposed UDE+SIR model generates predictions that capture the outbreak dynamic more accurately, but a decay in performance is observed at the last stages of the outbreak. The single-area SIR and the fully data-driven approach do not capture the proper dynamics accurately. Once the predictions were obtained, we employed the SINDy algorithm to substitute the DNN with a regression, removing the black box element of the model with no considerable increase in the error levels.

翻译：高度互联的社会使得COVID-19等传染病的传播建模变得困难。单区域SIR模型未能考虑外部感染压力，而将其扩展至大量交互区域需引入诸多在现实世界中难以成立的假设。我们提出在SIR+UDE联合模型中采用通用微分方程（Universal Differential Equations, UDEs）来捕捉邻域区域的影响并提升预测性能。UDEs是由深度神经网络（DNN）完全或部分定义的微分方程。我们在SIR方程中引入一个由DNN构成的加性项，用于学习其他区域的外部感染压力。该学习过程利用自动微分与梯度下降法，逼近由邻域区域状态引发的目标系统变化。我们通过模拟COVID-19暴发数据集，将所提模型与单区域SIR模型及纯数据驱动的DNN模型进行对比。结果表明，所提UDE+SIR模型能够更准确地捕捉暴发动态，但在暴发后期出现性能衰减。单区域SIR模型与纯数据驱动方法均无法准确反映实际动态。获取预测结果后，我们采用SINDy算法将DNN替换为回归模型，在未显著增加误差水平的前提下消除了模型的黑箱成分。