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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.

We develop and analyse a finite volume scheme for a nonlocal active matter system known to exhibit a rich array of complex behaviours. The model under investigation was derived from a stochastic system of interacting particles describing a foraging ant colony coupled to pheromone dynamics. In this work, we prove that the unique numerical solution converges to the unique weak solution as the mesh size and the time step go to zero. We also show discrete long-time estimates, which prove that certain norms are preserved for all times, uniformly in the mesh size and time step. In particular, we prove higher regularity estimates which provide an analogue of continuum parabolic higher regularity estimates. Finally, we numerically study the rate of convergence of the scheme, and we provide examples of the existence of multiple metastable steady states. Joint work with Maria Bruna and Markus Schmidtchen.