Model analogies and exchange of ideas between physics or chemistry with biology or epidemiology have often involved inter-sectoral mapping of techniques. Material mechanics has benefitted hugely from such interpolations from mathematical physics where dislocation patterning of platstically deformed metals and mass transport in nanocomposite materials with high diffusivity paths such as dislocation and grain boundaries, have been traditionally analyzed using the paradigmatic Walgraef-Aifantis (W-A) double-diffusivity (D-D) model. A long standing challenge in these studies has been the inherent nonlinear correlation between the diffusivity paths, making it extremely difficult to analyze their interdependence. Here, we present a novel method of approximating a closed form solution of the ensemble averaged density profiles and correlation statistics of coupled dynamical systems, drawing from a technique used in mathematical biology to calculate a quantity called the basic reproduction number R0, which is the average number of secondary infections generated from every infected. We show that the R0 formulation can be used to calculate the correlation between diffusivity paths, agreeing closely with the exact numerical solution of the D-D model. The method can be generically implemented to analyze other reaction-diffusion models.
|Journal||Frontiers in Applied Mathematics and Statistics|
|Publication status||Published - 3 Jun 2022|
Bibliographical note© 2022 Chattopadhyay, Kundu, Nath and Aifantis. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
Funding: he authors gratefully acknowledge partial financial support from the H2020-MSCA-RISE-2016 program, Grant No. 734485, entitled Fracture Across Scales and Materials, Processes and Disciplines (FRAMED). BK acknowledges funding by UKRI Grant No. (MR/T046619/1), part of the NSF/CIHR/DFG/FRQ/UKRI-MRC Next Generation Networks for Neuroscience Program. SKN acknowledges the Leverhulme Trust (RPG-2018-137) for supporting his time and resources for research in the UK.
- Applied Mathematics and Statistics
- double diffusion
- reproduction number
- spatiotemporal correlation
- Fick's diffusion