Heterogeneously structured compartmental models of epidemiological systems: from individual-level processes to population-scale dynamics

The paper carefully derives the Jacobian at the DFE, proposes the natural A/B split J*(DFE)=A−B, and—explicitly assuming B is invertible—uses the Next Generation Matrix to obtain R0=ρ(AB^{-1}) (see their equation (40) and ensuing discussion) . It also cautions that B need not be invertible in general (Gershgorin discs may touch the origin) and provides a simple sufficient condition for invertibility, namely a spontaneous outflow from each infectious compartment to some non-infectious class; only under this assumption do they state the NGM formula . By contrast, the candidate’s solution asserts that B is a nonsingular M-matrix via ρ(P)<1 from a “strict inequality in at least one column sum,” which is insufficient (the maximum column sum may still be 1), thereby overclaiming invertibility and B^{-1}≥0. The candidate also claims a componentwise differential inequality x'(t) ≤ [N1(t)A−B]x(t) ≤ (A−B)x(t) to prove global stability for R0<1; this comparison is directionally incorrect in general because interaction terms among infectious classes contribute with indefinite sign at the component level, so dropping positive parts does not yield an upper bound. The other elements—J*(DFE)=A−B and the conditional NGM expression—agree with the paper’s derivation. The paper’s remarks on forward/backward bifurcation and case studies (SIRS, SIRS with secondary infections, SIRWS) are consistent with the framework and examples provided .