Transmissibility in Interactive Nanocomposite Diffusion: The Nonlinear Double-Diffusion Model

What the paper actually provides is a modeling proposal and numerical evidence: after non-dimensionalizing a Walgraef–Aifantis–type double-diffusion system (its form is given as Eqs. (2a–2b) in the manuscript) , the authors introduce a time-varying reproduction number R0(t) via a generation-time kernel (Eq. (8)) and motivate taking that kernel to be exponential from a random-walk/memoryless argument . They then compare “normalized autocorrelation” curves with R0(t) across spatial locations and report that only at the symmetric point x = 0.5 do the two profiles match (approximately) for both species, also asserting in the conclusion that at x = 0.5 the two will asymptotically match with time, enabling a closed-form mapped description . No theorem or rigorous proof of asymptotic equality is given, nor is a precise analytic definition of the plotted autocorrelation supplied in the text. The candidate solution supplies a plausible route to a limit statement by: (i) using the exponential kernel to reduce the convolution to a leaky-integrator ODE; (ii) appealing to cooperative parabolic PDE theory to claim convergence of bounded trajectories to equilibria; and (iii) using a non-centered normalized autocorrelation definition for which any function converging to a nonzero constant has autocorrelation identically 1. However, it does not establish key hypotheses (global boundedness; precise boundary conditions; preclusion of non-homogeneous equilibria), nor does it check that its autocorrelation notion matches the paper’s. It also suggests the asymptotic ratio limit would hold at any x once the solution homogenizes, which conflicts with the paper’s emphasis on x = 0.5. Hence the paper’s argument is incomplete (largely empirical/heuristic), and the model’s proof is incomplete (missing assumptions and a mismatch of definitions).