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In the context of multi-agent systems of binary interacting particles, a kinetic model for action potential dynamics on a neural network is proposed, accounting for heterogeneity in the neuron-to-neuron connections, as well as in the brain structure. Two levels of description are coupled: in a single area, pairwise neuron interactions for the exchange of membrane potential are statistically described; among different areas, a graph description of the brain network topology is included. Equilibria of the kinetic and macroscopic settings are determined and numerical simulations of the system dynamics are performed with the aim of studying the influence of the network heterogeneities on the membrane potential propagation and synchronization., Comment: 14 pages, 5 figures. arXiv admin note: text overlap with arXiv:2303.01829

In this work, we derive a system of Boltzmann-type equations to describe the spread of SARS-CoV-2 virus at the microscopic scale, that is by modeling the human-to-human mechanisms of transmission. To this end, we consider two populations, characterized by specific distribution functions, made up of individuals without symptoms (population $1$) and infected people with symptoms (population $2$). The Boltzmann operators model the interactions between individuals within the same population and among different populations with a probability of transition from one to the other due to contagion or, vice versa, to recovery. In addition, the influence of innate and adaptive immune systems is taken into account. Then, starting from the Boltzmann microscopic description we derive a set of evolution equations for the size and mean state of each population considered. Mathematical properties of such macroscopic equations, as equilibria and their stability, are investigated and some numerical simulations are performed in order to analyze the ability of our model to reproduce the characteristic features of Covid-19.

In this paper, we study a modification of the mathematical model describing inflammation and demyelination patterns in the brain caused by Multiple Sclerosis proposed in [Lombardo et al. (2017), Journal of Mathematical Biology, 75, 373--417]. In particular, we hypothesize a minimal amount of macrophages to be able to start and sustain the inflammatory response. Thus, the model function for macrophage activation includes an Allee effect. We investigate the emergence of Turing patterns by combining linearised and weakly nonlinear analysis, bifurcation diagrams and numerical simulations, focusing on the comparison with the previous model.

We propose and investigate general kinetic models %of Boltzmann type with transition probabilities that can describe the simultaneous change of multiple microscopic states of the interacting agents. These models can be applied to many problems in socio-economic sciences, where individuals may change both their compartment and their characteristic kinetic variable, as for instance kinetic models for epidemics or for international trade with possible transfers of agents. Mathematical properties of our kinetic model are proved, as existence and uniqueness of a solution for the Cauchy problem in suitable Wasserstein spaces. The quasi-invariant asymptotic regime, leading to simpler kinetic Fokker-Planck-type equations, is investigated and commented on in comparison with other existing models. Some numerical tests are performed in order to show time evolution of distribution functions and of meaningful macroscopic fields, even in case of non-constant interaction probabilities.

A general framework for the kinetic modelling of non-relativistic polyatomic gases is proposed,where each particle is characterized both by its velocity and by its internal state, and the Boltzmann collisionoperator involves suitably weighted integrals over the space of internal energies. The description of the internalstructure of a molecule is kept highly general, and this allows classical and semi-classical models, such asthe monoatomic gas description, the continuous internal energy structure, and the description with discreteinternal energy levels, to fit our framework. We prove the H-Theorem for the proposed kinetic equation ofBoltzmann type in this general setting, and characterize the equilibrium Maxwellian distribution and thethermodynamic number of degrees of freedom. Euler equations are derived, as zero-order approximation in asuitable asymptotic expansion. In addition, within this general framework it is possible to build up new models,highly desirable for physical applications, where rotation and vibration are precisely described. Examples ofmodels for the Hydrogen Fluoride gas are presented., Comment: 36 pages, 3 figures

In this article we deduce a mathematical model of Maxwell-Stefan type for a reactive mixture of polyatomic gases with a continuous structure of internal energy. The equations of the model are derived in the diffusive limit of a kinetic system of Boltzmann equations for the considered mixture, in the general non-isothermal setting. The asymptotic analysis of the kinetic system is performed under a reactive-diffusive scaling for which mechanical collisions are dominant with respect to chemical reactions. The resulting system couples the Maxwell-Stefan equations for the diffusive fluxes with the evolution equations for the number densities of the chemical species and the evolution equation for the temperature of the mixture. The production terms due to the chemical reaction and the Maxwell-Stefan diffusion coefficients are moreover obtained in terms of the collisional kernels and parameters of the kinetic model., Comment: 32 pages

We prove a linear inequality between the entropy and entropy dissipation functionals for the linear Boltzmann operator (with a Maxwellian equilibrium background). This provides a positive answer to the analogue of Cercignani's conjecture for this linear collision operator. Our result covers the physically relevant case of hard-spheres interactions as well as Maxwellian kernels, both with and without a cut-off assumption. For Maxwellian kernels, the proof of the inequality is surprisingly simple and relies on a general estimate of the entropy of the gain operator due to Matthes and Toscani (2012) and Villani (1998). For more general kernels, the proof relies on a comparison principle. Finally, we also show that in the grazing collision limit our results allow to recover known logarithmic Sobolev inequalities.

We consider the spatially homogeneous Boltzmann equation for inelastic hard-spheres (with constant restitution coefficient $\alpha \in (0,1)$) under the thermalization induced by a host medium with a fixed Maxwellian distribution. We prove uniqueness of the stationary solution (with given mass) in the weakly inelastic regime; i.e., for any inelasticity parameter $\alpha \in (\alpha_0,1)$, with some constructive $\alpha_0 \in [0, 1)$. Our analysis is based on a perturbative argument which uses the knowledge of the stationary solution in the elastic limit and quantitative estimates of the convergence of stationary solutions as the inelasticity parameter goes to 1. In order to achieve this we give an accurate spectral analysis of the associated linearized collision operator in the elastic limit. Several qualitative properties of this unique steady state $F_\alpha$ are also derived; in particular, we prove that $F_\alpha$ is bounded from above and from below by two explicit universal (i.e. independent of $\alpha$) Maxwellian distributions.

We show the existence of smooth stationary solutions for the inelastic Boltzmann equation under the thermalization induced by a host-medium with a fixed distribution. This is achieved by controlling the Lp-norms, the moments and the regularity of the solutions for the Cauchy problem together with arguments related to a dynamical proof for the existence of stationary states.