A degenerate reaction-diffusion SIR model in interconnected regions

Abstract

This paper presents a novel time-space SIR (Susceptible-Infected-Recovered) model for simulating infectious disease dynamics in two interconnected regions. The model is formulated as a coupled reaction-diffusion system with boundary conditions that dynamically switch from Robin to Neumann types, effectively modelling policy-driven interventions such as lockdowns. A key innovation lies in the incorporation of degenerate diffusion, arising from vanishing population density, which significantly influences transmission behaviour near regional borders. The wellposedness of the model is rigorously established using the Faedo-Galerkin method, ensuring the existence, uniqueness, and positivity of weak solutions. Numerical simulations, performed using the Finite Volume Method, validate the theoretical findings and demonstrate the impact of migration and mobility restrictions on epidemic progression. This framework offers valuable insights for understanding and controlling disease spread in spatially heterogeneous and interconnected settings.