The CoSysM3 project aims to contribute to scientific advances in the mathematical modeling of behavioral epidemiology by building new hybrid models and generalizing results and methods from systems and control theory through their application to infectious and autoimmune diseases. The mathematical challenges we propose, applied to human diseases, are aligned with Goals 3 and 4 of the 2030 Agenda - Good Health and Well-being and Quality Education. Theoretical and numerical results will be proven, providing solutions to control the spread of epidemics and the effective treatment of some autoimmune diseases. We propose to develop a new approach to the construction of hybrid models that will allow the simulation of complex epidemic scenarios and their control. CoSysM3 is a research project in Biomathematics, with an emphasis on human diseases, and will contribute to the introduction of Applied Mathematics topics in the 1st, 2nd, and 3rd cycles of study, in line with Goal 4 of the 2030 Agenda - Quality Education. Mathematical tools will be used to study biological processes and provide optimal solutions for the prevention and treatment of human diseases. The main topics of mathematical research in this project are differential equation theory (ordinary, partial, and fractional), optimal control theory, dynamical systems theory, statistical methods, computational methods, and numerical simulations. The main objectives of the project are detailed in its four tasks: impact of human behavior on the control of infectious diseases; new generation of hybrid models - computational temporal automata and their validation; multiscale modeling of autoimmune diseases; optimal control and applications to human diseases. CoSysM3 aims to study the inclusion of human behavior in epidemiological models. New challenges arise in controlling disease transmission due to behavioral changes in the population. One example is vaccine refusal, which is closely linked to the perception of low risk when herd immunity is achieved. We intend to develop compartmental models that include human behavior and find interventions that take advantage of voluntary vaccination while maintaining an acceptable social cost, in the context of vaccines that provide only temporary protection. The study of these interventions will give rise to new optimal control problems. Changes in mobility behavior due to awareness of the risk of infection also have direct consequences for disease control. To describe this influence, we propose to use reaction-diffusion models with behavior-dependent coefficients and introduce superdiffusion models for disease transmission to better describe the spread of the disease globally. The study of these models involves defining appropriate numerical schemes and generalizing the concept of the basic reproduction number (R0) to the operators considered. Modeling the influence of social behaviors on the dynamics of disease propagation requires a coupling of macroscopic (collective behaviors) and microscopic (individual decisions) approaches, which is not always possible to capture with classical compartmental models. Thus, we propose to apply an innovative modeling approach that combines the classical formalisms retained at each macro-micro scale, leading to the development of a new generation of hybrid models. We will draw on recent studies to calibrate hybrid epidemic models by comparing them with statistical data. Motivated by the potential of mathematical models to provide valuable predictions about disease development and medical treatment, CoSysM3 members will continue to develop a line of research on mathematical modeling of autoimmune diseases. The approach used is based on kinetic theory to describe an autoimmune episode and extend optimal control policies for immunotherapeutic treatment. These models also present a multiscale description in terms of a kinetic system and its macroscopic analogue. They have the advantage of describing not only the collective behavior of cell populations, but also cellular activity and the individual behavior of cells. We will explore the micro-macro interaction of these models and study other challenging problems at the level of modeling descriptions, rigorous analyses, and biological predictions based on numerical simulations. One of the main objectives of CoSysM3 is to develop control strategies that minimize the negative impact of certain social behaviors on epidemic dynamics. By extending optimal control methods to the study of hybrid epidemic models and micro-macro models for autoimmune diseases, new challenges are expected that could enable the project to contribute to the generalization of systems and control theory to new contexts.

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