  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.
