In this study, we present an integro-differential model to simulate the local spread of infections. The model incorporates a standard susceptible-infected-recovered (\textit{SIR}-) model enhanced by an integral kernel, allowing for non-homogeneous mixing between susceptibles and infectives. We define requirements for the kernel function and derive analytical results for both the \textit{SIR}- and a reduced susceptible-infected-susceptible (\textit{SIS}-) model, especially the uniqueness of solutions. In order to optimize the balance between disease containment and the social and political costs associated with lockdown measures, we set up requirements for the implementation of control function, and show examples for three different formulations for the control: continuous and time-dependent, continuous and space- and time-dependent, and piecewise constant space- and time-dependent. Latter represent reality more closely as the control cannot be updated for every time and location. We found the optimal control values for all of those setups, which are by nature best for a continuous and space-and time dependent control, yet found reasonable results for the discrete setting as well. To validate the numerical results of the integro-differential model, we compare them to an established agent-based model that incorporates social and other microscopical factors more accurately and thus acts as a benchmark for the validity of the integro-differential approach. A close match between the results of both models validates the integro-differential model as an efficient macroscopic proxy. Since computing an optimal control strategy for agent-based models is computationally very expensive, yet comparatively cheap for the integro-differential model, using the proxy model might have interesting implications for future research.

翻译：本研究提出了一种模拟感染局部传播的积分-微分模型。该模型在标准易感-感染-恢复（SIR）模型基础上引入积分核函数，允许易感者与感染者之间的非均匀混合。我们定义了核函数需满足的条件，并推导出SIR模型及其简化版易感-感染-易感（SIS）模型的解析结果，特别是解的唯一性。为优化疾病控制与封锁措施带来的社会政治成本之间的平衡，我们设定了控制函数实施的要求，并展示了三种不同控制形式的实例：连续且随时间变化、连续且随空间与时间变化、以及分段常数且随空间与时间变化。后者更贴近现实，因为控制无法在每个时间点和空间位置都进行更新。我们求出了所有设置下的最优控制值——理论上连续且随空间与时间变化的控制效果最佳，但离散设置下也获得了合理结果。为验证积分-微分模型的数值结果，我们将其与一个成熟的智能体模型进行对比——该模型能更精确地纳入社会及其他微观因素，因此可作为积分-微分方法有效性的基准。两种模型结果的紧密吻合验证了积分-微分模型作为高效宏观替代模型的可行性。由于计算智能体模型的最优控制策略计算代价极高，而积分-微分模型相对较低，使用替代模型可能对未来研究具有重要启示。