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This paper deals with the modeling and simulation of the in-host dynamics of a virus. The modeling approach was developed according to the idea that mathematical models should go beyond deterministic single-scale population dynamics by taking into account the multiscale, heterogeneous features of the complex system under consideration. Here, we considered modeling the competition between the virus, the epithelial cells it infects, and the heterogeneous immune system with evolving activation states that induce a range of different effects on virus particles and infected cells. The subsequent numerical simulations showed different types of model outcomes: from virus elimination, to virus persistence and periodic relapse, to virus uncontrolled growth that triggers a blow-up in the fully activated immune response. The simulations also showed the existence of a threshold in the immune response that separates the regimes of higher re-infections from lower re-infections (compared to the magnitude of the first viral infection). [ABSTRACT FROM AUTHOR]

In this study, a two-dimensional "double-horizon peridynamics" formulation was presented for membranes. According to double-horizon peridynamics, each material point has two horizons: inner and outer horizons. This new formulation can reduce the computational time by using larger horizons and smaller inner horizons. To demonstrate the capability of the proposed formulation, various different analytical and numerical solutions were presented for a rectangular plate under different boundary conditions for static and dynamic problems. A comparison of peridynamic and classical solutions was given for different inner and outer horizon size values. [ABSTRACT FROM AUTHOR]

The Lie symmetry analysis and generalized Riccati equation expansion methods were performed on the inviscid and viscous incompressible magnetohydrodynamic equations. Using the Lie symmetry analysis method, symmetries and similarity reductions of (2 + 1)- and (3 + 1)-dimensional magnetohydrodynamic equations were derived. Different forms of trigonometric function solutions and rational solutions were obtained, which yielded periodic solutions, single soliton solutions, and lump solutions. Furthermore, using the generalized Riccati equation expansion method, we obtained abundant new solutions of magnetohydrodynamic equations, including kink, kink-like, breather, and interaction solutions. Moreover, the stability of magnetohydrodynamic equations was investigated from both qualitative and quantitative perspectives. The exact solutions and stability analysis could provide accurate mathematical descriptions and theoretical basis for numerical analysis and regulation of magnetohydrodynamic systems. [ABSTRACT FROM AUTHOR]

The entropy-based moment method is a well-known discretization for the velocity variable in kinetic equations which has many desirable theoretical properties but is difficult to implement with high-order numerical methods. The regularized entropy-based moment method was recently introduced to remove one of the main challenges in the implementation of the entropy-based moment method, namely the requirement of the realizability of the numerical solution. In this work we use the method of relative entropy to prove the convergence of the regularized method to the original method as the regularization parameter goes to zero and give convergence rates. Our main assumptions are the boundedness of the velocity domain and that the original moment solution is Lipschitz continuous in space and bounded away from the boundary of realizability. We provide results from numerical simulations showing that the convergence rates we prove are optimal. [ABSTRACT FROM AUTHOR]

Georg's trajectory in mathematics started at Aachen University of Technology (RWTH Aachen) in Germany, where he received a master's degree (Staatsexamen) in mathematics and physics education in 1970 and a Ph.D. in mathematics in 1973. John Neuberger knew Georg very well; I don't know where and how John and Georg met, but John and his wife Barbara visited us at Aachen. While visiting Auburn, Georg got along very well with everybody, especially the T's, as he was fluent in their language; he had even taught a course in algebraic topology at Aachen, in the early eighties. HT