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This paper considers a system consisting of a number of interacting living entities whose state at the microscopic scale is heterogeneously distributed among the said entities. This state includes, in addition, the classical mechanical variables, such as position and velocity, also a behavioral variable which is modi ed by interactions. It is shown how the pioneering ideas proposed in Bellomo et al. [Towards a mathematical theory of behavioral swarms, ESAIM Control Optim. Calc. Var. 26 (2020) 125] can be developed towards modeling behavioral swarms within a quest towards a mathematical theory of living systems. The rst part of the paper presents a qualitative analysis of the emerging behaviors predicted by the model in aforementioned work. Some simulations follow to depict the said emerging behaviors. The last part of the paper is devoted to derive a new, more general theory in view of applications to model living systems., University of Granada, National Research Foundation of Korea NRF-2020R1A2C3A01003881

This article presents, within a multiscale framework, a search for a unified approach towards modelling the COVID-19 pandemic, from contagion to within-host dynamics. The modelling is focused on vaccination and therapeutical actions in general. The first part of our article is devoted to understanding the complex features of the system and to the design of a modelling rationale. Then, the modelling approach follows mainly focused on the competition between the proliferating virus and the immune system. Modelling considers also the action of vaccination plans related to the onset of new variants. This article is part of the theme issue ‘Kinetic exchange models of societies and economies’.

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

The complex interactions between human behaviors and social economic sciences is critically analyzed in this paper in view of possible applications of mathematical modeling as an attainable interdisciplinary approach to understand and simulate the aforementioned dynamics. The quest is developed along three steps: Firstly an overall analysis of social and economic sciences indicates the main requirements that a contribution of mathematical modeling should bring to these sciences; subsequently the focus moves to an overview of mathematical tools and to the selection of those which appear, according to the authors bias, appropriate to the modeling; finally, a survey of applications is presented looking ahead to research perspectives.