The impact of recovery rate heterogeneity in achieving herd immunity

The paper proves, under hypotheses (8)–(11), that the heterogeneous SIR model has herd immunity iff E[1/γ]<∞ (Proposition 3), and extends this to SEIR with 0<αmin≤α≤αmax and E(0)+I(0)>0 (Proposition 5), by a clean inequality-based argument built on equations (1)–(3) and (26)–(29) and bounds (9) (see Proposition 3 and its proof, and the SEIR extension ). The candidate solution’s SIR argument is essentially correct and even yields sharp time–integral identities. However, its SEIR part claims the identity α∫E=E0+S0−S∞ without the necessary −E(∞) term; this gap is nontrivial for the necessity direction and is not justified by an argument that E(∞)=0. Hence, as written, the model’s SEIR proof is flawed, while the paper’s proofs are correct.