Theoretical Analysis of SIRVVD Model to Provide Insight on the Target Rate of COVID-19/SARS-CoV-2 Vaccination in Japan

The paper freezes S, V1, V2, and β to obtain a linear 3-compartment infected subsystem I'(t) = A I(t), with A = β u v^T − γ I, u = (S, (1−σ1)V1, (1−σ2)V2)^T, v = (1,1,1)^T. It diagonalizes A and finds eigenvalues −γ, −γ, and β(S + (1−σ1)V1 + (1−σ2)V2) − γ, leading to the growth/neutral/decay trichotomy and the targets V2^obj and P2^obj, along with the “limitation” and “no-vaccine-needed” conditions (their Eqs. (16)-(21), (19)) . The candidate solution proves the same results via the rank-one matrix-exponential formula e^{β u v^T t} = I + ((e^{β(v^T u)t} − 1)/(v^T u)) u v^T, yielding the identical spectrum and closed form for I(t). Minor wording in the paper (“does not change” at neutrality) is clarified by the model (converges to constants), but this is not a mathematical error. Hence both are correct; the proofs differ in technique (eigendecomposition vs. rank-one exponential).