Delay compartment models from a stochastic process

The paper derives the memory kernel via Laplace transforms so that Lt{K_i}(s)=μ_i e^{-s τ_i}, yielding K_i(t)=μ_i δ(t-τ_i) and the delayed term in the master equation dρ_i/dt = q_i^+(t) − ω_i(t)ρ_i(t) − μ_i [Φ_i(t,0)/Φ_i(t−τ_i,0)] ρ_i(t−τ_i) Θ(t−τ_i) (their Eqs. (24)–(26)) , exactly as in the candidate’s Step 1. With constant ω_i, the ratio Φ_i(t,0)/Φ_i(t−τ_i,0) collapses to e^{−ω_i τ_i}, giving the standard discrete-delay DDE (their Eqs. (27)–(28)) , matching Step 2. For the delay exponential Ψ(t)=dexp(−μ t; −μ τ), the paper states Lt{Ψ}(s)=1/(s+μ e^{−s τ}) (Eq. (21)) and that Ψ is a valid survival function if and only if 0 ≤ μτ ≤ e^{−1}, with positivity/monotonicity addressed in Appendix A and oscillation for μτ>e^{−1} noted via prior results (Agarwal et al.) . The candidate’s Step 3 follows the same transform and characteristic-root structure; it adds a spectral explanation (Lambert W) for the sign changes when μτ>1/e, consistent with the paper’s statements. Minor note: the paper’s appendix proves strict positivity for μτ<1/e and cites oscillation when μτ>1/e, while the main text asserts the sharp boundary μτ≤1/e; the candidate acknowledges the same boundary. Overall, both are correct and use substantially the same method (Laplace/shift/reduction), with the candidate offering slightly more spectral detail for Part 3 while remaining aligned with the paper.