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 functions, and show examples for continuous and time-dependent, continuous and space- and time-dependent, and piecewise constant space- and time-dependent controls. 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：本研究提出一个积分-微分模型用于模拟传染病的局域传播。该模型在标准易感-感染-康复（SIR）模型基础上引入积分核函数，实现了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，推导了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。为平衡疫情控制与封锁措施带来的社会政治成本，我们设定了控制函数的实施要求，并分别展示了连续时间依赖型、连续时空依赖型以及分段常数时空依赖型控制的实现范例。其中后者因无法对每个时空点进行实时更新而更贴近现实场景。我们求得了所有情形下的最优控制值——自然以连续时空依赖型控制效果最佳，但离散设定下仍获得合理结果。为验证积分-微分模型的数值结果，我们将其与更精准融入社会及微观因素的成熟个体基模型进行对比，后者作为检验积分-微分方法有效性的基准。两种模型结果的紧密吻合证实了积分-微分模型作为高效宏观替代模型的可靠性。由于个体基模型的最优控制策略计算成本极高，而积分-微分模型计算相对廉价，该代理模型的应用可能为未来研究开辟重要方向。