Evolutionary Vaccination Games with premature vaccines to combat ongoing deadly pandemic

The paper’s Theorem 5 (Vaccinating-ESS-AS) in the non-deadly case de=0 states that an ESS-AS exists iff (i) the resident policy induces q(Υ*)=1 with strict truncation q̃(Υ*)≠1, and (ii) the endemic equilibrium is (θ*,ψ*)=(θE,ψE) with μρ>μ+1 and h(θE,ψE)<0; it also provides the policy-specific thresholds β*: FC β*>μρ; FR β*>(μρ)^2/(μρ−1); VFC1 β*>(μρ)^2/(μρ−1−μ) (using Tables 1–3) . The candidate solution derives exactly these conditions and thresholds via the same key steps: (a) linear utility in q implies only corner best responses (0 or 1); (b) for ρ>1, endemic steady states satisfy φ*=1/ρ via the ODE (11), and dψ/dt=0 yields ψ*=q*/(μρ), hence θ*=1−1/ρ−q*/(μρ) ; (c) hm:=h(1−1/ρ,0) from pI(θ)=λθ/(λθ+ν) controls whether q*=0 can be ESS, and hm<0 rules it out ; (d) strict truncation at (θE,ψE) is required for mutational stability, matching the paper’s use of q̃(Υ*)≠1 and Lemma 6 to control small static mixes . The final thresholds follow by enforcing q̃(θE,ψE)>1 as in the paper’s tables . Therefore, the model’s write-up aligns with the paper’s theorem and proof sketch; any differences are not substantive.