Carlo Bianca

Complexity analysis and mathematical tools towards the modelling of living systems

This paper is a review and critical analysis of the mathematical kinetic theory of active particles applied to the modelling of large living systems made up of interacting entities. The first part of the paper is focused on a general presentation of the mathematical tools of the kinetic theory of active particles. The second part provides a review of a variety of mathematical models in life sciences, namely complex social systems, opinion formation, evolution of epidemics with virus mutations, and vehicular traffic, crowds and swarms. All the applications are technically related to the mathematical structures reviewed in the first part of the paper. The overall contents are based on the concept that living systems, unlike the inert matter, have the ability to develop behaviour geared towards their survival, or simply to improve the quality of their life. In some cases, the behaviour evolves in time and generates destructive and/or proliferative events.

Thermostatted kinetic equations as models for complex systems in physics and life sciences

Statistical mechanics is a powerful method for understanding equilibrium thermodynamics. An equivalent theoretical framework for nonequilibrium systems has remained elusive. The thermodynamic forces driving the system away from equilibrium introduce energy that must be dissipated if nonequilibrium steady states are to be obtained. Historically, further terms were introduced, collectively called a thermostat, whose original application was to generate constant-temperature equilibrium ensembles. This review surveys kinetic models coupled with time-reversible deterministic thermostats for the modeling of large systems composed both by inert matter particles and living entities. The introduction of deterministic thermostats allows to model the onset of nonequilibrium stationary states that are typical of most real-world complex systems. The first part of the paper is focused on a general presentation of the main physical and mathematical definitions and tools: nonequilibrium phenomena, Gauss least constraint principle and Gaussian thermostats. The second part provides a review of a variety of thermostatted mathematical models in physics and life sciences, including Kac, Boltzmann, Jager-Segel and the thermostatted (continuous and discrete) kinetic for active particles models. Applications refer to semiconductor devices, nanosciences, biological phenomena, vehicular traffic, social and economics systems, crowds and swarms dynamics.

Modelling aggregation-fragmentation phenomena from kinetic to macroscopic scales

This paper deals with the modelling of aggregation and/or fragmentation physical phenomena for large systems of interacting living entities in the framework of the mathematical kinetic theory for active particles. After introducing various mathematical structures in terms of systems of nonlinear integro-differential equations with quadratic type nonlinearities and variable number of equations, the relative qualitative analysis of the initial value problem is presented. Finally, the paper deals with the derivation of macroscopic equations based on the underlying description at the microscopic scale delivered by the kinetic theory models.

MATHEMATICAL MODELING FOR KELOID FORMATION TRIGGERED BY VIRUS: MALIGNANT EFFECTS AND IMMUNE SYSTEM COMPETITION

This paper deals with the modeling of a wound healing disease, the keloid, which may provoke onset of malignant cells with higher progression feature, thus generating cells with heterogeneous phenotype. According to medical hypothesis, it is assumed that viruses and the genetic susceptibility of patients are the main causes that trigger the formation. The mathematical model is developed by means of the tools of the kinetic theory for active particles. The competition of the immune system cells with viruses, keloid fibroblast cells, and malignant cells is taken into account. Numerical simulations, obtained by considering the sensitivity analysis of the parameters in the model, show the emerging phenomena that are typical of this disease.

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

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 are 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.

Immune System Network and Cancer Vaccine

This paper deals with the mathematical modelling of the immune system response to cancer disease, and specifically with the treatment of the mammary carcinoma in presence of an immunoprevenction vaccine. The innate action of the immune system network, the external stimulus represented by repeated vaccine administrations and the competition with cancer are described by an ordinary differential equations‐based model.The mathematical model is able to depict preclinical experiments on transgenic mice. The results are of great interest both in the applied and theoretical sciences.

Mathematical modeling of the immune system recognition to mammary carcinoma antigen

The definition of artificial immunity, realized through vaccinations, is nowadays a practice widely developed in order to eliminate cancer disease. The present paper deals with an improved version of a mathematical model recently analyzed and related to the competition between immune system cells and mammary carcinoma cells under the action of a vaccine (Triplex). The model describes in detail both the humoral and cellular response of the immune system to the tumor associate antigen and the recognition process between B cells, T cells and antigen presenting cells. The control of the tumor cells growth occurs through the definition of different vaccine protocols. The performed numerical simulations of the model are in agreement with in vivo experiments on transgenic mice.

On the modelling of space dynamics in the kinetic theory for active particles

This paper proposes an analysis of the modelling of space dynamics focused on a general class of models of the kinetic theory for active particles in space homogeneity. Various deterministic and stochastic developments are treated and referred to specific applications. These new classes of equations present different aspects of hybrid characteristic coupling deterministic and stochastic structures, as well as continuous and discrete variables, and constitute a background paradigm for the derivation of models whose qualitative properties are analyzed referring to modelling of complex systems in life and applied sciences.

On the modeling of nonlinear interactions in large complex systems

Abstract This work deals with the modeling of large systems of interacting entities in the framework of the mathematical kinetic theory for active particles. The contents are specifically focused on the modeling of nonlinear interactions which is one of the most important issues in the mathematical approach to modeling and simulating complex systems, and which includes a learning–hiding dynamics. Applications are focused on the modeling of complex biological systems and on immune competition.

The nonequilibrium Ehrenfest gas: a chaotic model with flat obstacles?

It is known that the nonequilibrium version of the Lorentz gas (a billiard with dispersing obstacles [Ya. G. Sinai, Russ. Math. Surv. 25, 137 (1970)], electric field, and Gaussian thermostat) is hyperbolic if the field is small [N. I. Chernov, Ann. Henri Poincare 2, 197 (2001)]. Differently the hyperbolicity of the nonequilibrium Ehrenfest gas constitutes an open problem since its obstacles are rhombi and the techniques so far developed rely on the dispersing nature of the obstacles [M. P. Wojtkowski, J. Math. Pures Appl. 79, 953 (2000)]. We have developed analytical and numerical investigations that support the idea that this model of transport of matter has both chaotic (positive Lyapunov exponent) and nonchaotic steady states with a quite peculiar sensitive dependence on the field and on the geometry, not observed before. The associated transport behavior is correspondingly highly irregular, with features whose understanding is of both theoretical and technological interests.