Temporal and Spacial Studies of Infectious Diseases: Mathematical Models and Numerical Solvers

The paper states and numerically verifies second-order accuracy in time for the staggered Crank–Nicolson (CN) scheme and first order for implicit Euler, and it derives a Peaceman–Rachford ADI splitting with an O(Δt^3) local factorization error, together with complexity claims (ADI O(n^2) per step; SOR O(n^3)), but it does not provide a full stability or global error proof for the coupled S–I system. The candidate solution outlines a plausible global ℓ∞-error analysis using staggered linearizations and Varah-type diagonal dominance bounds, but it contains a critical algebraic omission: in bounding A^{-1}B for the CN step it drops the Δt A^{-1}L_h term (coming from the diffusion operator), so the claimed bound ∥A^{-1}B∥∞ ≤ 1 + CΔt is not justified as written and may hide an h-dependent factor unless one proves a uniform bound on Δt∥A^{-1}L_h∥∞. Therefore, the paper’s argument is incomplete, and the model’s proof outline is also incomplete. Key paper facts used here: the PDE and discretizations (implicit Euler (6), staggered CN (8)), the ADI derivation and O(Δt^3) remainder (18)–(22), and the empirical second-order tables and complexity statements for SOR and ADI.