# American Institute of Mathematical Sciences

ISSN:
1556-1801

eISSN:
1556-181X

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## Networks & Heterogeneous Media

December 2021 , Volume 16 , Issue 4

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2021, 16(4): 513-533 doi: 10.3934/nhm.2021015 +[Abstract](626) +[HTML](193) +[PDF](9210.31KB)
Abstract:

In this paper, properties of a recently proposed mathematical model for data flow in large-scale asynchronous computer systems are analyzed. In particular, the existence of special weak solutions based on propagating fronts is established. Qualitative properties of these solutions are investigated, both theoretically and numerically.

2021, 16(4): 535-552 doi: 10.3934/nhm.2021016 +[Abstract](624) +[HTML](235) +[PDF](1039.55KB)
Abstract:

The spread of rumors is a phenomenon that has heavily impacted society for a long time. Recently, there has been a huge change in rumor dynamics, through the advent of the Internet. Today, online communication has become as common as using a phone. At present, getting information from the Internet does not require much effort or time. In this paper, the impact of the Internet on rumor spreading will be considered through a simple SIR type ordinary differential equation. Rumors spreading through the Internet are similar to the spread of infectious diseases through water and air. From these observations, we study a model with the additional principle that spreaders lose interest and stop spreading, based on the SIWR model. We derive the basic reproduction number for this model and demonstrate the existence and global stability of rumor-free and endemic equilibriums.

2021, 16(4): 553-567 doi: 10.3934/nhm.2021017 +[Abstract](650) +[HTML](210) +[PDF](464.8KB)
Abstract:

We study transport processes on infinite metric graphs with non-constant velocities and matrix boundary conditions in the \begin{document}${\mathrm{L}}^{\infty}$\end{document}-setting. We apply the theory of bi-continuous operator semigroups to obtain well-posedness of the problem under different assumptions on the velocities and for general stochastic matrices appearing in the boundary conditions.

2021, 16(4): 569-589 doi: 10.3934/nhm.2021018 +[Abstract](591) +[HTML](195) +[PDF](436.15KB)
Abstract:

In this paper, we study the concept of approximate controllability of retarded network systems of neutral type. On one hand, we reformulate such systems as free-delay boundary control systems on product spaces. On the other hand, we use the rich theory of infinite-dimensional linear systems to derive necessary and sufficient conditions for the approximate controllability. Moreover, we propose a rank condition for which we can easily verify the conditions of controllability. Our approach is mainly based on the feedback theory of regular linear systems in the Salamon-Weiss sense.

2021, 16(4): 591-607 doi: 10.3934/nhm.2021019 +[Abstract](629) +[HTML](191) +[PDF](351.11KB)
Abstract:

We study convex and quasiconvex functions on a metric graph. Given a set of points in the metric graph, we consider the largest convex function below the prescribed datum. We characterize this largest convex function as the unique largest viscosity subsolution to a simple differential equation, \begin{document}$u'' = 0$\end{document} on the edges, plus nonlinear transmission conditions at the vertices. We also study the analogous problem for quasiconvex functions and obtain a characterization of the largest quasiconvex function that is below a given datum.

2021, 16(4): 609-636 doi: 10.3934/nhm.2021020 +[Abstract](573) +[HTML](191) +[PDF](571.81KB)
Abstract:

This work deals with a mathematical analysis of sodium's transport in a tubular architecture of a kidney nephron. The nephron is modelled by two counter-current tubules. Ionic exchange occurs at the interface between the tubules and the epithelium and between the epithelium and the surrounding environment (interstitium). From a mathematical point of view, this model consists of a 5\begin{document}$\times$\end{document}5 semi-linear hyperbolic system. In literature similar models neglect the epithelial layers. In this paper, we show rigorously that such models may be obtained by assuming that the permeabilities between lumen and epithelium are large. We show that when these permeabilities grow, solutions of the 5\begin{document}$\times$\end{document}5 system converge to those of a reduced 3\begin{document}$\times$\end{document}3 system without epithelial layers. The problem is defined on a bounded spacial domain with initial and boundary data. In order to show convergence, we use \begin{document}${{{\rm{BV}}}}$\end{document} compactness, which leads to introduce initial layers and to handle carefully the presence of lateral boundaries. We then discretize both 5\begin{document}$\times$\end{document}5 and 3\begin{document}$\times$\end{document}3 systems, and show numerically the same asymptotic result, for a fixed meshsize.

2020 Impact Factor: 1.213
5 Year Impact Factor: 1.384
2020 CiteScore: 1.9