Exact solutions and bounds for network SIR and SEIR models using a rooted-tree approximation

The paper proves that the rooted-tree system (26) yields upper bounds on Sk for general networks and is exact on rooted trees with a single source, via an inequality on ⟨IjSk⟩ that leads to (22) and a cooperative ODE comparison to (26), and shows exact reduction on trees to (9) . The candidate solution proves the same two claims using a different route: a generator-level monotone functional (edge-gap) to get a sharper inequality λ⟨IjSk⟩ ≥ [ (λ+γ)⟨Sk⟩ − λ⟨Sj⟩ − γ⟨Sk⟩(0) ]+ and structural zeros on rooted trees to show exactness. The logic aligns with the paper’s statements and conclusions, though the candidate omits an explicit cooperativity check (the paper supplies it), so both are correct but via different proofs.