A Koopman Operator Framework for Nonlinear Epidemic Dynamics: Application to an SIRSD Model

The paper states the SIRSD model’s well‑posedness (existence, uniqueness, positivity, boundedness) and claims NL(t)=S+I+R stays strictly positive for all finite t, thereby avoiding the division by zero in the frequency-dependent incidence term, but the key step is only argued heuristically (“deaths occur continuously rather than instantaneously”) and not quantified. It then invokes Picard–Lindelöf on the positive orthant “excluding division by zero since NL(t)>0” to conclude global solutions, without a rigorous proof that the trajectory cannot approach NL=0 in finite time. By contrast, the candidate solution gives a complete quantitative bound: (e^{μt} NL(t))' = μ e^{μt}(S+R) ≥ 0, hence NL(t) ≥ NL(0) e^{-μ t} > 0 for all finite t, which closes the gap and yields global continuation on compact subsets of the open domain Ω={S+I+R>0}. Therefore, the result is correct, but the paper’s proof is incomplete; the model’s solution is correct and supplies the missing argument (see the model and well-posedness passages, including equations (1)–(2), the positivity and Corollary 1 claims, and the normalization/simplex discussion in the paper: ).