Kinetic theory methods are applied in this paper to model the dynamics of vehicular traffic. The basic idea is to consider each vehicular-driver system as a single part, or micro-system, of a large complex system, in order to capture the heterogeneous behavior of all the micro-systems that compose the overall system. The evolution of the system is ruled by nonlinearly additive interactions described by stochastic games. A qualitative analysis for the proposed model with discrete states is developed, showing well-posedness of the related Cauchy problem for the spatially homogeneous case and for the spatially nonhomogeneous case, the latter with periodic boundary conditions. Numerical simulations are also performed, with the aim to show how the model proposed is able to reproduce empirical data and to show emerging behavior as the formation of clusters.
This paper provides a survey of mathematical models and methods dealing with the analysis and simulation of tumor dynamics in competition with the immune system. The characteristic scales of the phenomena are identified and the mathematical literature on models and problems developed on each scale is reviewed and critically analyzed. Moreover, this paper deals with the modeling and optimization of therapeutical actions. The aim of the critical analysis and review consists in providing the background framework towards the development of research perspectives in this promising new field of applied mathematics.
This paper deals with the development of suitable general mathematical structures including a large variety of Boltzmann type models. The contents are organized in three parts. The first part is devoted to modeling the above general framework. The second part to the development of specific models of interest in applied sciences. The third part develops a critical analysis towards research perspectives both on modeling and analytic problems.
This paper deals with the modeling of the early stage of cancer phenomena, namely mutations, onset, progression of cancer cells, and their competition with the immune system. The mathematical approach is based on the kinetic theory of active particles developed to describe the dynamics of large systems of interacting cells, called active particles. Their microscopic state is modeled by a scalar variable which expresses the main biological function. The modeling focuses on an interpretation of the immune-hallmarks of cancer.
This paper proposes a multicell model to describe the evolution of tumour growth from the avascular stage to the vascular one through the angiogenic process. The model is able to predict the formation of necrotic regions, the control of mitosis by the presence of an inhibitory factor, the angiogenesis process with proliferation of capillaries just outside the tumour surface with penetration of capillary sprouts inside the tumour, the regression of the capillary network induced by the tumour when angiogenesis is controlled or inhibited, say as an effect of angiostatins, and finally the regression of the tumour size. The three-dimensional model is deduced both in a continuum mechanics framework and by a lattice scheme in order to put in evidence the relation between microscopic phenomena and macroscopic parameters. The evolution problem can be written as a free-boundary problem of mixed hyperbolic-parabolic type coupled with an initial-boundary value problem in a fixed domain. *
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.