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This paper is devoted to the multidisciplinary modelling of a pandemic initiated by an aggressive virus, speci cally the so-called SARS{CoV{2 Severe Acute Respiratory Syndrome; corona virus n.2. The study is developed within a multiscale framework accounting for the interaction of di erent spatial scales, from the small scale of the virus itself and cells, to the large scale of individuals and further up to the collective behaviour of populations. An interdisciplinary vision is developed thanks to the contributions of epidemiologists, immunologists and economists as well as those of mathematical mod- ellers. The rst part of the contents is devoted to understanding the complex features of the system and to the design of a modelling rationale. The modelling approach is treated in the second part of the paper by showing both how the virus propagates into infected individuals, successfully and not successfully recovered, and also the spatial patterns, which are subsequently studied by kinetic and lattice models. The third part reports the contribution of research in the elds of virology, epidemiology, immune competition, and economy focussed also on social behaviours. Finally, a critical analysis is proposed looking ahead to research perspectives. ; Granada University ; modelling Nature Group, Spain ; Italian Research Council, IMATI, CNR, Pavia, Italy ; Engineering & Physical Sciences Research Council (EPSRC) EP/R023204/1 ; Royal Society Wolfson Fellowship RSWF\R1\180009 ; European Union (EU) 822781 ; Consejo Nacional de Investigaciones Cientificas y Tecnicas (CONICET) PIP 11220150100500CO ; Secretaria de Ciencia y Tecnología (SECYT) 33620180100326CB ; National Science Foundation (NSF) DMS-1763272 ; Simons Foundation for a NSF-Simons Center for Multiscale Cell Fate Research 594598QN ; United States Department of Health & Human Services National Institutes of Health (NIH) - USA 1U54CA217378-01A1 P30CA062203 ; 110145 ; 110146 ; DMS-1714973 ; DMS-1936833

Juan Soler and Nicola Bellomo acknowledge the support by the MINECO-Feder (Spain) research Grant Number RTI2018-098850-B-I00, the Junta de Andalucia (Spain) Project PY18-RT-2422, A-FQM-311-UGR18 & B-FQM-580-UGR20. Youshan Tao is partially supported by the National Natural Science Foundation of China (No. 12171316), and he also acknowledges support of Shanghai Frontiers Science Center of Modern Analysis. MichaelWinkler acknowledges support of the Deutsche Forschungsgemeinschaft in the context of the Project Emergence of Structures and Advantages in CrossDiffusion Systems (No. 411007140, GZ: WI 3707/5-1). ; This paper proposes a review focused on exotic chemotaxis and cross-diffusion models in complex environments. The term exotic is used to denote the dynamics of models interacting with a time-evolving external system and, specifically, models derived with the aim of describing the dynamics of living systems. The presentation first, considers the derivation of phenomenological models of chemotaxis and cross-diffusion models with particular attention on nonlinear characteristics. Then, a variety of exotic models is presented with some hints toward the derivation of new models, by accounting for a critical analysis looking ahead to perspectives. The second part of the paper is devoted to a survey of analytical problems concerning the application of models to the study of real world dynamics. Finally, the focus shifts to research perspectives within the framework of a multiscale vision, where different paths are examined to move from the dynamics at the microscopic scale to collective behaviors at the macroscopic scale. ; Spanish Government RTI2018-098850-B-I00 ; Junta de Andalucia European Commission PY18-RT-2422 A-FQM-311-UGR18 B-FQM-580-UGR20 ; National Natural Science Foundation of China (NSFC) 12171316 ; Shanghai Frontiers Science Center of Modern Analysis ; German Research Foundation (DFG) 411007140 ; GZ: WI 3707/5-1

This paper deals with the modeling of social competition, possibly resulting in the onset of extreme conflicts. More precisely, we discuss models describing the interplay between individual competition for wealth distribution that, when coupled with political stances coming from support or opposition to a Government, may give rise to strongly self-enhanced effects. The latter may be thought of as the early stages of massive unpredictable events known as Black Swans, although no analysis of any fully-developed Black Swan is provided here. Our approach makes use of the framework of the kinetic theory for active particles, where nonlinear interactions among subjects are modeled according to game-theoretical principles.

This work aims to foster the interdisciplinary dialogue between mathematicians and socio-economic scientists. Interaction among scholars and practitioners traditionally coming from different research areas is necessary more than ever in order to better understand many real-world problems we face today. On the one hand, mathematicians need economists and social scientists to better address the methodologies they design in a more realistic way; on the other hand, economists and social scientists need to be aware of sound mathematical modelling tools in order to understand and, ultimately, solv