Epidemic Population Games And Evolutionary Dynamics

The paper proves that E* is globally asymptotically stable for the coupled epidemic–population-game dynamics by constructing a Lyapunov function L and deriving the key decay inequality d/dt L ≤ −P(x,p) − (I−Î)^2 − (ω/γ)(R−Ř)^2, see (13a)–(13e), (14), and (17) in the paper . It then avoids standard LaSalle due to non-compactness of the state space Y, proving instead that q(t) is bounded (Appendix A, Step 2), constructing accumulation points in E*, and concluding lim L=0 and GAS via Steps 3–4 (including (43)) . The candidate solution mirrors the main outline (Lyapunov, characterization of the zero set, and E*) and correctly identifies G, H, and the structure of Q* in Cases I–II . However, it crucially invokes LaSalle and argues with an ω-limit set without establishing boundedness of q or precompactness of trajectories; it also sketches domain invariance with a flawed boundary argument (using I+R=B), which is not a valid state constraint in the original EPG model (cf. EPG and Y definitions) . These gaps invalidate the purported proof even though the conclusion matches the paper’s theorem.