Global stability of a time-delayed malaria model with standard incidence rate

The paper formulates a four-dimensional delayed malaria model with standard incidence, establishes well-posedness and boundedness, derives R0 and equilibria, and proves that E0 is GAS for R0<1 (GA for R0=1), while E* exists uniquely and is GAS for R0>1, with E0 unstable when R0>1. The proofs combine Lyapunov functionals on the limiting system and asymptotically autonomous systems theory, plus a weak-persistence argument; key steps are clearly exhibited and consistent throughout . The candidate solution reaches the same conclusions and correctly computes the characteristic equation and equilibria, proposing direct Lyapunov–Krasovskii functionals on the original system using p(t)=Iv/Nv and g(u)=u−1−ln u, which is a different but standard route. However, several steps (notably the R0=1 case and the LaSalle characterization of the largest invariant set for the direct functional, and an explicit statement of the domain D for E*) require more justification than provided in the outline. Overall, the paper’s argument is complete and correct, and the model’s proof strategy is sound but needs additional details for full rigor .