Epidemic modeling has long been an important field of research, and the recent Covid-19 pandemic has shown its importance to everyday life. Often, in order to increase accuracy and applicability of the models, a PDE approach is used, thus allowing for modeling spread of infections over countries or accounting for age-depend model parameters such as mortality, infectivity, etc. In this talk, we analyze the structure of epidemic models with age and space structure where, instead a diffusion process, the spatial spread is modeled by a wave-like process. This remedies the infinite propagation speed of diffusion equations, but has its own set of interesting mathematical problems, such as the equation neither being purely parabolic nor purely hyperbolic. We discuss the existence of solutions and the convergence of the model towards a diffusion model when the relaxation parameter becomes arbitrary small.

We discuss transmission problems between a (thermo-)viscoelastic system with Kelvin-Voigt damping, and a purely elastic system. It is shown that neither the elastic damping by Kelvin-Voigt mechanisms nor the dissipative effect of the temperature in one material can assure the exponential stability of the total system when it is coupled through transmission to a purely elastic system. The approach shows the lack of exponential stability using Weyl's theorem on perturbations of the essential spectrum. Instead, strong stability can be shown using the principle of unique continuation. To prove polynomial stability we provide an extended version of the characterizations of Borichev and Tomilov. Observations on the lack of compacity of the inverse of the arising semigroup generators are included too. The results apply to thermo-viscoelastic systems, to purely elastic systems as well as to the scalar case consisting of wave equations.