Transmission of multiple pathogens across species

The paper derives the multitype branching-process PGFs (eq. (14)), the product-formula for extinction (eq. (15)), and the threshold alternative tied to R0 via the next-generation matrices Bv (eqs. (8)–(9), (41)). These match standard theory and are consistent with the candidate’s solution, including zℓ = (ε z i + d)/(ε + d) and the algebraic rearrangement of the i-fixed-point equation to z i = (γ + d)/(γ + d + ∑q β S0(1 − zℓ)). However, Theorem 3 in the paper claims that when R(1)0 > 1 there is a unique vector 0 < z < 1 such that F(z) = z, without stating the additional condition that every pathogen block be supercritical. In the reducible, block-decoupled setting documented by the paper, if some pathogen blocks have ρ(Bv) ≤ 1, the only fixed point for those subcritical blocks is 1, so a strictly interior fixed point in all 2PV coordinates need not exist. The candidate’s solution makes precisely this blockwise qualification (supercritical blocks have unique interior fixed points; subcritical blocks do not) and is therefore the more accurate statement of the alternative. All other components (PGFs, fixed-point characterization, reduction to latent-only with K v = (B v)T, and the threshold via ρ(B v)) agree with the paper’s derivations and proofs (Theorem 3; Appendix C). Citations: the PGFs and Theorem 3 statements are in 14–15 and the uniqueness/existence discussion in Appendix C; the R0 and B v structure appear in 8–11; the two-species/two-pathogen block forms in 41–44 corroborate the reducible, per-pathogen decomposition.