Modelling the Spatial Spread of COVID-19 in a German District using a Diffusion Model

TL;DR

This paper introduces an integro-differential SIR model with a kernel for simulating COVID-19 spread, deriving analytical solutions, optimizing control strategies, and validating results against an agent-based benchmark to enable efficient large-scale analysis.

Contribution

The study develops a novel integro-differential model for COVID-19 spread with analytical solutions and optimal control strategies, validated against an agent-based model for efficiency.

Findings

The integro-differential model closely matches agent-based simulations.

Optimal control strategies vary with control implementation complexity.

The model provides an efficient alternative for large-scale epidemic analysis.

Abstract

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.