In this talk, we are concerned with modelling, analysis and the numerical simulation of coupled bulk and surface reaction-diffusion systems, where in general the surface can be time dependent, and its motion law might be coupled to the concentration fields. Such systems appear in many models in a variety of different fields. In this work, we consider several biological applications. In particular, we study symmetry breaking in a mathematical model for signalling networks. For the numerical treatment of the resulting system of partial differential equations and interface evolution laws, we follow a diffuse-interface ansatz, where the boundary of a domain (a surface) is implicitly described by a smeared out indicator function of the domain.

Throughout human history, infectious diseases have been the cause of hundreds of million of deaths and even nowadays tuberculosis, HIV/AIDS, and malaria cause more than one million deaths per year. The Sars-CoV-2 pandemic has still shown more drastically how humans and human society is impacted by infectious diseases. For almost two years, policy makers from all over the world have been searching for the right answers to the challenges posed by the virus spread.
Until full vaccination of the populations or the right medicine is available, nonpharmaceutical interventions (NPIs) have to be implemented. In order to find the right interventions, the future developments of the virus dynamics have to be estimated under different assumptions. A straightforward approach is to use numerical simulation of mathematical models in epidemiology. The work on mathematical models in infectious diseases already started in the 18th century with the works of Daniel Bernoulli and other major contribution were already made in 1927 [W.O. Kermack, A.G. McKendrick 1927]. During the Sars-CoV-2 pandemic, many models have seen renewed interest.
Mathematical models in epidemiology can be classified according to different categories, e.g., deterministic and stochastic or subpopulation-based and agent-based. While agent-based methods model individual behavior and natural transmission chains in a natural way, classical ODE models are subpopulation-based and hide important features such as mobility or superspreading events behind averaged effects. These models are, on the other hand, computationally much less demanding, and allow for an on-time simulation of many different model scenarios.
To overcome the limitations of simple models, different approaches are possible. In order to account for the most important features of virus dynamics, spatial and demographic resolution have to be considered. Age stratification of ODE models can be realized in a straightforward way. To avoid homogeneous mixing in all locations, a hybrid graph-ODE approach can be used [M.J. Kühn, D. Abele, T. Mitra, W. Koslow, M. Abedi, K. Rack, M. Siggel, S. Khailaie, M. Klitz, S. Binder, L. Spataro, J. Gilg, J. Kleinert, M. Häberle, L. Plötzke, C.D. Spinner, M. Stecher, X.X.Zhu, A. Basermann, M.Meyer-Hermann 2021]. Further developments then allow for a rigorous testing of commuters coming from hot spots [M.J. Kühn, D. Abele, S. Binder, K. Rack, M. Klitz, J. Kleinert, J. Gilg, L. Spataro, W. Koslow, M. Siggel, M. Meyer-Hermann, A. Basermann 2021] or the evaluation of vaccination effects [W. Koslow, M.J. Kühn, S. Binder, M. Klitz, D. Abele, A. Basermann, M. Meyer-Hermann 2021] while certain NPIs are relaxed. Future research will include age of infection effects, more realistic mobility, and comparisons to agent- or multi-scale models.

The Stochastic SEIR Model is a continuous-time stochastic model for the spread of an infectious disease in a (finite) population. At each time point, the population is divided into four so-called compartments, namely the subpopulations of susceptible, exposed (or latent), infective and recovered (or immune) individuals, respectively. Moreover, it is assumed that at time zero, there is exactly one infected individual and that all the other individuals are susceptible to infection. After being infected, an individual remains latent for some (random) time, then becomes and stays infectious for some other random time and finally recovers from the disease.
In this talk, we will focus on the case of a closed community with a homogeneous mixing behaviour. This means that there is no influx of susceptible individuals, that recovered individuals stay immune and that an infectious individual has contacts at the points of a homogeneous Poisson point process, where each contact is chosen uniformly at random from the other individuals. The famous General Stochastic Epidemic is the special case, in which the latent phase is missing and the duration of infectivity is also exponential.
The end of the epidemic is marked by the first (random) point in time, at which only susceptible and recovered individuals are left. In general, one is interested in the final size of the epidemic, i.e. in the total number of infections. By means of coupling with a suitable branching process, the final size and the qualitative properties of the epidemic are analyzed as the population size diverges to infinity. In this context, it will further be discussed how the basic reproduction number, i.e. the average number of infectious contacts of an infected individual, and the preventive measure of vaccination affect the nature of the epidemic. If time allows, then also modifications of the model as well as further mathematical properties of the process will be discussed.

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