Virus models in complex frameworks: Towards modeling space patterns of SARS-CoV-2 epidemics

Abstract

This paper deals with the micro–macro-derivation of virus models coupled with a reaction–diffusion system that generates the dynamics in space of the virus particles. The first part of the presentation focuses, starting from [N. Bellomo, K. Painter, Y. Tao and M. Winkler, Occurrence versus absence of taxis-driven instabilities in a May–Nowak model for virus infection, SIAM J. Appl. Math. 79 (2019) 1990–2010; N. Bellomo and Y. Tao, Stabilization in a chemotaxis model for virus infection, Discrete Contin. Dyn. Syst. S 13 (2020) 105–117], on a survey and a critical analysis of some phenomenological models known in the literature. The second part shows how a Hilbert type can be developed to derive models at the macro-scale from the underlying description delivered by the kinetic theory of active particles. The third part deals with the derivation of macroscopic models of various virus models coupled with the reaction–diffusion systems. Then, a forward look to research perspectives is proposed.

Cross diffusion models in complex frameworks From microscopic to macroscopic

“…the derivation of models of diffusion of epidemics in a territory, where diffusion and reaction can be referred to networks and transportation dynamics, see [1,4,15,16]. Technical developments by studying models with anomalous diffusions this would be in the line of [19,20] referring to models in complex environments [9] with the challenging objective of understanding how a physically consistent interpretation of interactions at the microscopic scale can generate new models of nonlinear diffusion. These models have generated interesting analytic problems [12].…”

“…The survey of analytical studies generally correspond to initial-boundary value problems. Some hints towards a further refinement and improvement are therein reported, but the derivation of models at the macroscopic scale from the underlying description at the microscopic scale is still open with the exception of the preliminary results on the transport of virus models by a Keller-Segel system [20], briefly KS system. The contents of previous papers by the same authors [18,19] will contribute to the development of the present article.…”

“…Our paper refers to the micro-macro derivation of exotic models from the underlying description provided by methods of the kinetic theory of active particles [5] by further developments of the method developed in [20] somehow inspired by the celebrated sixth Hilbert problem [29,46], generally related to the kinetic theory of classical particles [27,46], but subsequently developed for biological systems [18]. We focus mainly on the micro-macro derivation of virus models with space dynamics.…”

“…Section 3 deals with the first step towards the derivation of macroscale models from the underlying microscale description. In detail, a general methodological approach is derived based on developments, with respect to those proposed in [20], of the Hilbert approach. We refer to the general structure proposed in Section 2.…”