arXiv subjects

Romina Travaglini

Publications and source records attributed to Romina Travaglini.

11 recordsLinked to original sources

arXiv2026

Weakly nonlinear analysis of a reaction-diffusion model for demyelinating lesions in Multiple Sclerosis

Multiple Sclerosis is a chronic autoimmune disorder characterized by the degradation of the myelin sheath in the central nervous system, leading to neurological impairments. In this work, we analyze a reaction-diffusion model derived from kinetic theory to study the formation of demyelinating lesions. We perform a Turing instability analysis and a weakly nonlinear analysis to investigate different spatial patterns that may emerge. Our study examines how key parameters, including the squeezing probability of immune cells and the chemotactic response, impact pattern formation. Numerical simulations confirm the analytical results, revealing the emergence of distinct spatial structures.

q-bio.TO↗

arXiv2026

A kinetic derivation of spatial distributed models for tumor-immune system interactions

We propose a mathematical kinetic framework to investigate interactions between tumor cells and the immune system, focusing on the spatial dynamics of tumor progression and immune responses. We develop two kinetic models: one describes a conservative scenario where immune cells switch between active and passive states without proliferation, while the other incorporates immune cell proliferation and apoptosis. By considering specific assumptions about the microscopic processes, we derive macroscopic systems featuring linear diffusion, nonlinear cross-diffusion, and nonlinear self-diffusion. Our analysis provides insights into equilibrium configurations and stability, revealing clear correspondences among the macroscopic models derived from the same kinetic framework. Using dynamical systems theory, we examine the stability of equilibrium states and conduct numerical simulations to validate our findings. These results highlight the significance of spatial interactions in tumor-immune dynamics, paving the way for a structured exploration of therapeutic strategies and further investigations into immune responses in various pathological contexts.

q-bio.CB↗

arXiv2026

A nonconservative kinetic framework for a closed-market society subject to shock events

Recently, several events have shockingly impacted society, carrying tough consequences. However, not all individuals are similarly affected by shock events. Among other factors, the consequences can vary depending on the income class. In our presented work, the approach typical of kinetic theory is used to analyze the dynamics of a closed-market society exposed to various types of shock events. To achieve this, we introduce non-conservative equations, incorporating proliferative and destructive binary interactions as well as external actions. Specifically, the latter term reproduces the shock events, and to accomplish this, we introduce an appropriate external force field into the kinetic framework, modeled using Gaussian functions. Several numerical simulations are presented to illustrate the behavior of the solution predicted by the model and an application in comparison to real data relative to the Hurricane Katrina catastrophe is carried out.

physics.soc-ph↗

arXiv2026

A nonconservative kinetic model under the action of an external force field for modeling the medical treatment of autoimmune response

In this paper, we develop a nonconservative kinetic framework to be applied to the study of immune system dysregulation. From the modeling viewpoint, the model regards a system composed of stochastically interacting agents, under the action of an eternal force field. According to the application perspectives of this paper, the external force field has a specific analytical shape. In this case, some analytical results are proved, i.e. existence, uniqueness, positivity, and boundedness of solution of the related Cauchy problem, at least locally in time. Then, the model is refined to be implemented for the study of treatment strategies in case of autoimmune response. Specifically, we distinguish the autonomous case from the nonautonomous one, representing the absence or delivery of drugs, respectively. The former allows us to gain some stability results. Whereas, the latter is qualitatively studied. Numerical simulations are provided for both schemes.

physics.med-ph↗

arXiv2026

Investigating dynamics and asymptotic trend to equilibrium in a reactive BGK model

We investigate numerically a recent BGK-type model for a multi-component mixture of monatomic gases, undergoing a reversible bimolecular chemical reaction. The model replaces each collisional term of the Boltzmann equation with a relaxation term, thereby describing separately the effects of the mechanical processes and the chemical reaction. Additionally, the model exhibits consistency properties. The correct entropy production is ensured when auxiliary temperatures in the chemical contributions share a common value. We assume isotropic distributions and perform numerical simulations for the macroscopic fields to appraise how the dynamics push the mixture toward thermalization and chemical equilibrium. We show that the hypothesis on the equalization of fictitious species temperatures is justifiable to ensure the monotonicity of the classical $H$-Boltzmann functional. Simulations show that, when initial temperatures are far from equilibrium, the relaxation towards equilibrium occurs at a later stage and the classical $H$-Boltzmann functional is not monotone during the initial transient.

cond-mat.stat-mech↗

arXiv2026

Derivation from kinetic theory and 2-D pattern analysis of chemotaxis models for Multiple Sclerosis

In this paper, a class of reaction-diffusion equations for Multiple Sclerosis is presented. These models are derived by means of a diffusive limit starting from a proper kinetic description, taking account of the underlying microscopic interactions among cells. At the macroscopic level, we discuss the necessary conditions for Turing instability phenomena and the formation of two-dimensional patterns, whose shape and stability are investigated by means of a weakly nonlinear analysis. Some numerical simulations, confirming and extending theoretical results, are proposed for a specific scenario.

q-bio.QM↗

arXiv2026

Reaction-diffusion systems from kinetic models for bacterial communities on a leaf surface

Many mathematical models for biological phenomena, such as the spread of diseases, are based on reaction-diffusion equations for densities of interacting cell populations. We present a consistent derivation of reaction-diffusion equations from systems of suitably rescaled kinetic Boltzmann equations for distribution functions of cell populations interacting in a host medium. We show at first that the classical diffusive limit of kinetic equations leads to linear diffusion terms only. Then, we show possible strategies in order to obtain, from the kinetic level, macroscopic systems with nonlinear diffusion and also with cross-diffusion effects. The derivation from a kinetic description has the advantage of relating reaction and diffusion coefficients to the microscopic parameters of the interactions. We present an application of our approach to the study of the evolution of different bacterial populations on a leaf surface. Turing instability properties of the relevant macroscopic systems are investigated by analytical methods and numerical tools, with particular emphasis on pattern formation for varying parameters in two-dimensional space domains.

math.AP↗

arXiv2026

A BGK-type model for multi-component gas mixtures undergoing a bimolecular chemical reaction

We propose a new kinetic BGK-type model for a mixture of four monatomic gases, undergoing a bimolecular and reversible chemical reaction. The elastic and reactive interactions are described separately by distinct relaxation terms and the mechanical operator is the sum of binary BGK contributions, one for each pair of interacting species. In this way, our model separately incorporates the effects of mechanical processes and chemical reactions. Additionally, it retains the effects of inter-species interactions which are proper of the mixture. The dependence of Maxwellian attractors on the main macroscopic fields is explicitly expressed by assuming that the exchange rates for momentum and energy of mechanical and chemical operators coincide with the ones of the corresponding Boltzmann terms. Under suitable hypotheses, the relaxation of the distribution functions to equilibrium is shown through entropy dissipation. Some numerical simulations are included to investigate the trend to equilibrium.

cond-mat.stat-mech↗

arXiv2025

Reaction-diffusion systems derived from kinetic theory for Multiple Sclerosis

We present a mathematical study for the development of Multiple Sclerosis in which a spatio-temporal kinetic { theory} model describes, at the mesoscopic level, the dynamics of a high number of interacting agents. We consider both interactions among different populations of human cells and the motion of immune cells, stimulated by cytokines. Moreover, we reproduce the consumption of myelin sheath due to anomalously activated lymphocytes and its restoration by oligodendrocytes. Successively, we fix a small time parameter and assume that the considered processes occur at different scales. This allows us to perform a formal limit, obtaining macroscopic reaction-diffusion equations for the number densities with a chemotaxis term. A natural step is then to study the system, inquiring about the formation of spatial patterns through a Turing instability analysis of the problem and basing the discussion on the microscopic parameters of the model. In particular, we get spatial patterns oscillating in time that may reproduce brain lesions characteristic of different phases of the pathology.

math.AP↗

arXiv2025

Turing instability and 2-D pattern formation in reaction-diffusion systems derived from kinetic theory

We investigate Turing instability and pattern formation in two-dimensional domains for two reaction-diffusion models, obtained as diffusive limits of kinetic equations for mixtures of monatomic and polyatomic gases. The first model is of Brusselator type, which, compared with the classical formulation, presents an additional parameter whose role in stability and pattern formation is discussed. In the second framework, the system exhibits standard nonlinear diffusion terms typical of predator-prey models, but differs in reactive terms. In both cases, the kinetic-based approach proves effective in relating macroscopic parameters, often set empirically, to microscopic interaction mechanisms, thereby rigorously identifying admissible parameter ranges for the physical description. Furthermore, weakly nonlinear analysis and numerical simulations extend previously known one-dimensional results and reveal a wider scenario of spatial structures, including spots, stripes, and hexagonal arrays, that better reflect the richness observed in real-world systems.

math-ph↗

arXiv2025

A reaction-diffusion model for relapsing-remitting multiple sclerosis with a treatment term

We present a mathematical study for the development of multiple sclerosis based on a reaction-diffusion system. The model describes interactions among different populations of human cells, motion of immune cells stimulated by cytokines, consumption of myelin sheath due to anomalously activated lymphocytes and its restoration by oligodendrocytes. Successively, we introduce a therapy term representing injection of low-dose IL-2 interleukine. A natural step is then to study the system, investigating the formation of spatial patterns by means of a Turing instability analysis of the problem. In particular, we get spatial patterns oscillating in time that may reproduce brain lesions characteristic of the early stage of the pathology, in both non-treatment and treatment scenarios.

q-bio.TO↗