Existence of stationary solutions in kinetic models with Gaussian thermostats

Math Methods in App Sciences

Menale

2019

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A differential equation-based framework is suitable for the modeling of nonequilibrium complex systems if its solution is able to reach, as time goes to infinity, the related nonequilibrium steady states. The thermostatted kinetic theory framework has been recently proposed for the modeling of complex systems subjected to an external force field. The present paper is devoted to the mathematical proof of the convergence of the solutions of the thermostatted kinetic framework towards the related nonequilibrium stationary states. The proof of the main result is gained by employing the Fourier transform and distribution theory arguments.

“…The existence of a positive solution g(u) ∈ C(R∖{ }), with = 1 E 1 [ ] , of the stationary problem (3) has been proved in Bianca. 15 In particular, one has…”

Section: The Convergence Theoremmentioning

confidence: 99%

On the convergence toward nonequilibrium stationary states in thermostatted kinetic models

Math Methods in App Sciences

Menale

2019

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“…The analytical existence and uniqueness of solutions for (2.18) has been investigated in [6] and the convergence, when t → +∞, of the solution of the evolution equation (2.18) to the solution of the stationary equation (2.19) has been obtained in [8]. The results have been proved in the absence of proliferative/destructive and mutative events.…”

Section: Remark 22mentioning

confidence: 99%

Modeling the antigen recognition by B-cell and T-cell receptors through thermostatted kinetic theory methods

Brézin

2017

Int. J. Biomath.

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The activation and the resulting response of the immune system to antigens comprise different complex processes and cells. This paper aims at modeling the processes of recognition and learning of the immune system by means of the thermostatted kinetic theory methods. Specifically, the thermostatted kinetic framework is firstly generalized for taking into account that in some processes of proliferation of the cells, the rate is also function of the degree of information exchanged amongst cells. In particular, within the new framework, a mathematical model is proposed for miming the recognition process of the immune system through the definition of interactions between the cytotoxic and humoral components of the adaptive immune system via T-and B-cells. The model validation is obtained by performing a sensitivity analysis on the parameters which depicts the main emerging phenomena and the different phases of the recognition and learning of the immune system.

“…Indeed PDE solvers are even more computationally intensive than ODE solvers. In contrast to the general framework with internal structure, the thermostatted kinetic theory for active particles (that can be considered as a generalized model with internal structure) has the advantage that can take into account the modeling of nonequilibrium steady states that are typical of the biological systems [168,170].…”

Section: A Critical Analysis On the Different Modeling Approachesmentioning

confidence: 99%

Towards a unified approach in the modeling of fibrosis: A review with research perspectives

Amar

Physics of Life Reviews

2016

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Pathological fibrosis is the result of a failure in the wound healing process. The comprehension and the related modeling of the different mechanisms that trigger fibrosis is a challenge of many researchers that work in the field of medicine and biology. The modern scientific analysis of a phenomenon generally consists of three major approaches: theoretical, experimental, and computational. Different theoretical tools coming from mathematics and physics have been proposed for the modeling of the physiological and pathological fibrosis. However a complete framework is missing and the development of a general theory is required. This review aims at finding a unified approach in the modeling of fibrosis diseases that takes into account the different phenomena occurring at each level: molecular, cellular and tissue. Specifically by means of a critical analysis of the different models that have been proposed in the mathematical, computational and physical biology, from molecular to tissue scales, a multiscale approach is proposed, an approach that has been strongly recommended by top level biologists in the past decades.