Deterministic and stochastic modeling and simulation of complex phenomena in Biology, Social Sciences and Economy

Leader/s: Antonio Cuevas González

Members:
José Ramón Berrendero Díaz
Rodolfo Cuerno Rejado
Antonio Cuevas González
Esteban Moro Egido
Julio Daniel Rossi
Ángel Sánchez Sánchez

Description:
In recent years, much effort has continued to be made in Applied Mathematics in order to provide important contributions in the fields of Life Sciences and Social Sciences. An increasingly common factor in both these fields is the need to question the classical models, which are frequently too simplistic. Indeed, the proposal of new models may be regarded as one of the most outstanding contributions in the whole process of mathematical treatment. This is a further area to which our team wishes to contribute decisively by focussing its efforts on three different fronts: modeling, analysis and simulation.

In diverse scientific fields, and in particular in biomedical sciences, one may currently observe the phenomenon of "overabundance of data", due largely to the availability of increasingly sophisticated and precise instruments of measurement. This situation brings into question the validity of some classical statistical models based on finite-dimensional observations, and leads to the consideration of studying alternative models based on functional observations. Recently in this regard there have been applications in the field of experimental cardiology, with the proposal of new methodologies for the analysis of variance based on functional data. Further areas of interest addressed in the present project will therefore be statistical methods in the detection of patterns and statistical techniques in image analysis, with a strong presence of numerical simulation that will be dealt with in the following section.

Much remains to be done in the field of modeling, with the purpose of incorporating “long-range” interaction between particles and/or distant individuals. Modern ideas based on the renormalization group modeling the criticality of complex systems could play an important role in this regard. A further classical branch of mathematics that is gaining importance in this field is game theory, a framework traditonally employed for understanding social behavior and biological evolution, combined with graph theory (complex networks). Understanding the relation between the different time scales involved is an open problem with important consequences for the situations modeled, and in general for the comprehension of emerging phenomena in complex systems. Furthermore, and unlike what occurs in the usual Gaussian diffusion models, which are modeled by heat equation variants, the diffusion operator has no regulating effect, which poses new problems regarding the possible formation of solution singularities, as well as giving rise to many questions related with the analysis and numerical simulation of these models. As regards applications, we intend to conduct research into the problem of "optimal mass transport". This concerns a mathematical tool with multiple applications, originally conceived by Monge and subsequently developed by Kantorovich, but which has recently been profusely used in diverse spheres, such as Fluid Mechanics, Kinetics of Gases, Economics , etc.. The possibility of addressing optimal transport plans by asymptotic limits of nonlinear diffusion processes will also be explored, thereby opening up a broad range of future applications for the development of effective numerical approximation methods.

Furthermore, nonlinearity is a fundamental factor in the modeling of functionally relevant molecules in Biology, especially those of DNA. On the basis of these models, our intention is to develop a mesoscopic description to enable the form-function relationship to be analyzed, as well as to provide results with relevant applications (molecules with a specific function, biological computation, genomic analysis, etc).

Finally, the modeling and simulation of economic processes (particularly in finance) is one of the most active fields in financial mathematics. Our aim is to use stochastic techniques for the understanding, modeling and subsequent simulation of important problems in industry, such as the dissemination of information throughout social networks and its application to techniques of viral marketing (a project in co-operation with IBM Europe), risk auditing of financial products (of interest for the Safei company), and the use and efficient simulation of stochastic models for the evaluation of assets and derivatives.