Hybrid PDE-ODE Models for Efficient Simulation of Infection Spread in Epidemiology

The paper derives the Ω2 ODEs by integrating the full SEIR reaction–diffusion PDEs over Ω2, using Gauss–Green to convert the diffusion term to an interface flux, and invoking density/flux continuity across Γ; under a homogeneous-mixing closure it evaluates the Allee factor at an area-mean n̄2 and closes the ODEs with −|Ω2|−1∫Γ ν1^T D∇y1, then couples these with PDEs on Ω1 and Dirichlet matching y1=y2 on Γ. This is exactly the strategy and result presented in the candidate solution for Tasks (1)–(2) (see the paper’s Section 2.3 derivation and the final hybrid system) . For Task (3), the paper explicitly chooses n0=3A/2 to guarantee 1 − A/(n+n0) ≥ 1/3, matching the candidate’s inequality argument . Minor differences (e.g., the model’s regularity assumptions and D treated constant vs. spatially varying) are inessential to the derivation.