Résumé: This work proposes a modeling for providing a Multi Agents System (MAS) based decision tool in case of pandemics with the objective to preserve vital functions of society while containing the pandemics. Some works integrate people behaviors in their daily life to get a more realistic model for simulation. They mainly use a mathematical modeling based on random graphs for handling social network concept. However, that modeling suers of a major inconvenience: social relationships are modeled only by one undirected graph which does not sounds realistic. Our approach consists in generalizing random graphs by stochastic pretopology. Stochastic pretopology concepts are then inserted in a MAS coupled with a Geographical Information System (GIS).We are thus able to deal with spatial information and directly run simulations on georeferenced data. Such simulations provide usefull results for the decision making concerning healthcare system management.

Ref HAL: hal-00605389_v1
Résumé: BELMANDT is the collective pseudonym under which a group of mathematicians and computer scientists have decide to publish their common works on Pretopology and to give a nod to BOURBAKI. These works would never have been realized without the dynamic leadership of professor Marcel Brissaud, now retired, and who is of course associated to the publication of this book. Marcel Brissaud, during the seventies, has been the first to observe the interest of developing mathematical concepts adapted to humanities. Pretopology theory relies on the statement that the axiomatic of mathematical topology is too strong to model the phenomena of social sciences. This axiomatic has to be reconsidered to give birth to a theory that is adapted to data issued from these fields of expertise, better than trying to force data to fit to an existing and inadequate theory. This book is the first English revised version of a first book, published in French, which is planned to be the first of a series dedicated to the whole set of results obtained in Pretopology: continuity, connectedness, associated metric... Many applications of Pretopology were conducted in various fields: economy with the structural analysis of a economy, game theory with the study of the dynamics of coalitions in cooperative games, sociology with analysis of social networks, data analysis with the construction of clustering algorithms, pattern recognition with digital image analysis as in mathematical morphology, etc. Purely mathematical work in Pretopology has enabled the building of processing algorithms that have been put in a software library, PRETOPOLIB, allowing researchers to implement a pretopological approach in their work. We wish the reader of this book the pleasure of exploring the basics of Pretopology and we hope that he will find the materials for his own work and we give him a rendezvous for the next books on the subject.

Résumé: Social sciences widely use the concept of network for modeling links between individuals. (Social networks) A lot of works have been proposed on different problems related to this concept. For example, what percentage of nodes of the network is required to observe a si- gnificant change in the connectivity of the network ? Or, what are nodes the most linked to others ? The common point of these Works is that they all model a social network by a graph, in the mathematical sense. However, we think a social network is composed from different types of links between individuals, which implies using more than one graph. So we propose a new formalism for modeling social networks, based on families of graphs. This formalism enables to study topological properties of a network by means of a weakened version of mathematical topology (pretopology). As a second point, we assume that links between individuals cannot be considered as certain and that many events can influence existence of these links. In most cases, these events are not predictable, so we introduce stochastic aspects in modeling a network. This leads us to the concept of stochastic network. In this paper, we first give basic definitions of a network, of a stochastic network and some elements about underlying mathematical theories : pretopologie and random sets. Then, we give first results about modeling the topology of a stochastic network, with some particu- lar points concerning transmission and connectivity. We conclude with perspectives of further works.

Résumé: Face à une situation de crise sanitaire liée à l’apparition d’une pandémie de grande envergure, des mesures adéquates doivent rapidement être prises pour la contenir tout en préservant la capacité de production de l’économie. C’est pourquoi nous nous sommes focalisés sur le développement d’un modèle d’aide à la décision à base d’agents hybrides simulant la diffusion d’une pandémie, fondé sur les caractéristiques médicales de la pandémie ainsi que sur la structure socio-économique de la zone géographique concernée. Cette structure socio-économique étant au cœur du modèle, une modélisation prétopologique du concept de réseau social est proposée et intégrée dans l’approche agent.

Ref HAL: halshs-00610696_v1
Résumé: In this paper, we present the various types of connectivities in pre- topology then we show that only the strong connectivity and the con- nectivity allow to de ne a partition of a space in components and we develop more particularly the case of the connectivity.