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Mean field limits are an important tool in the context of large-scale dynamical systems, in particular, when studying multiagent and interacting particle systems. While the continuous-time theory is well-developed, few works have considered mean field limits for deterministic discrete-time systems, which are relevant for the analysis and control of large-scale discrete-time multiagent system. We prove existence results for the mean field limit of very general discrete-time dynamical systems, which in particular encompass typical multiagent systems. As a technical novelty, we utilize kernel mean embeddings, which are an established tool in machine learning and statistics, but have been rarely used in the context of multiagent systems and kinetic theory. Our results can serve as a rigorous foundation for many applications of mean field approaches for discrete-time dynamical systems, from analysis, simulation and control to learning.

In this talk, we present two novel variants of the contact process. In the first variant individuals carry a viral load. An individual with viral load zero is classified as healthy and otherwise infected. If an individual becomes infected it begins with a viral load of one, which then evolves according to a Birth-Death process. In this model, viral load indicates severity of the infection such that individuals with a higher load can be more infectious. Moreover, the recovery times of individual is not necessarily exponentially distributed and can even be chosen to follow a power-law distribution. In the second variant individuals are permanently infected albeit in two states: actively infected or dormant. The dynamics of these individual states are again governed by a Birth-Death process. Dormant infections do not interact with neighbouring individuals but may reactivate spontaneously. Active infections reactivate dormant neighbours at a constant rate and may become dormant themselves. We present a Poisson construction for both variants. For the first model, we study the phase transition of survival and discuss existence of a non-trivial upper invariant law. Additionally, we derive a duality relationship between the two variant, which we use to uncover a phase transition regarding invariant distributions in the second variant.

The initial-boundary value problem for a system of nonlinear parabolic equations modeling the growth and migration of glioma cells is studied. New a priori estimates of the solution are obtained, based on which the non-local in time unique solvability of the initial-boundary value problem is proved. The conducted computational experiments demonstrate the ability of the mathematical model to reflect the desired properties like development of hypoxia in the tumor microenvironment, the switching of proliferative tumor cells to invasive migratory ones due to hypoxia, the regression of the vasculature in the area occupied by the tumor and simultaneous angiogenesis at the periphery of the tumor.