Mathematical Models and Methods in Applied Sciences

Very recently, a model for flocking was introduced by Cucker and Smale together with a proof of convergence. This proof established unconditional convergence to a common velocity provided the interaction between agents was strong enough and conditional convergence otherwise. The strength of the interaction is measured by a parameter β ≥ 0 and the critical value at which unconditional convergence stops holding is β = 1/2. This model was extended by Shen to allow for a hierarchical leadership structure among the agents and similar convergence results were proved. But, for discrete time, unconditional convergence was proved only for (k being the number of agents). In this note we improve on this result showing that unconditional convergence holds indeed for β < 1/2.

The present study is devoted to modelling the onset and the spread of epidemics. The mathematical approach is based on the generalized kinetic theory for active particles. The modelling includes virus mutations and the role of the immune system. Moreover, the heterogeneous distribution of patients is also taken into account. The structure allows the derivation of specific models and of numerical simulations related to real systems.

The aim of this work is to implement a statistical mechanics theory of social interaction, generalizing econometric discrete choice models. A class of simple mean-field discrete models is introduced and discussed both from the theoretical and phenomenological point of view. We propose a parameter evaluation procedure and test it by fitting the model against three families of data coming from different cases: the estimated interaction parameters are found to have similar positive values, giving a quantitative confirmation of the peer imitation behavior found in social psychology. Furthermore, all the values of the interaction parameters belong to the phase transition regime suggesting its possible role in the study of social systems.

We consider a model of socially interacting individuals that make a binary choice in a context of positive additive endogenous externalities. It encompasses as particular cases several models from the sociology and economics literature. We extend previous results to the case of a general distribution of idiosyncratic preferences, called here Idiosyncratic Willingnesses to Pay (IWP). When j, the ratio of the social influence strength to the standard deviation of the IWP distribution, is small enough, the inverse demand is a classical monotonic (decreasing) function of the adoption rate. However, even if the IWP distribution is mono-modal, there is a critical value of j above which the inverse demand is non-monotonic. Thus, depending on the price, there are either one or several equilibria.

Beyond this first result, we exhibit the generic properties of the boundaries limiting the regions where the system presents different types of equilibria (unique or multiple). These properties are shown to depend only on qualitative features of the IWP distribution: modality (number of maxima), smoothness and type of support (compact or infinite). The main results are summarized as phase diagrams in the space of the model parameters, on which the regions of multiple equilibria are precisely delimited. We also discuss the links between the model and the random field Ising model studied in the physics literature.

In this work we present a comprehensive account of our current knowledge on vascular morphogenesis, both from a biological and a mathematical point of view. To this end, we first describe the basic steps in the known mechanisms of blood vessel morphogenesis, whose structure, function and unfolding properties are examined. We then provide a wide, although by no means exhaustive, account of mathematical models which are used to describe and discuss particular aspects of the overall biological process considered. We finally summarize the approaches presented, and suggest possible directions for future research. Details about some of the major signalling molecules involved are included in a first Appendix at the end of the paper. A second Appendix provides a brief overview of design principles for vascular nets, a subject that has deserved considerable attention over the years.

The challenge to understand the dynamics of Complex Systems is attracting attention from a wide range of disciplines across the natural, biological and social sciences. Recent turmoil in the financial markets has brought this challenge into the public domain, with speculation rife as to the root cause of the observed fluctuations. At their heart, all Complex Systems share the common property of featuring many interacting objects from which the observed macroscopic dynamics emerge. Exactly how this happens cannot yet be specified in a generic way — however, an important milestone in this endeavor is to develop a quantitative understanding of any internal clustering dynamics within the population. Coalescence-fragmentation processes have been studied widely in conventional chemistry and physics — however, collective behavior in social systems is not limited by nearest-neighbor interactions, nor are the details of social coalescence or fragmentation processes necessarily the same as in physical and biological systems. Here we discuss the general phenomenon of coalescence and fragmentation problems with a focus on social systems in which a typical fragmentation process corresponds to an entire group breaking up, as opposed to the typical binary splitting studied in physical and biological systems. Having discussed situations under which power-laws for the group distribution size emerge from such internal clustering dynamics, we move on to look at the specific application to financial markets. We propose a new model for financial market dynamics based on the combination of internal clustering (i.e. herding) dynamics with human decision-making. The resulting fluctuation in price movements is closer to what is observed empirically, leading us to speculate that the combination of dynamical clustering and decision-making are key for developing quantitative models of social dynamical phenomena.