Martin Kühn: Mathematical modeling of infectious diseases

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 are 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 metapopulation or hybrid graph-ODE approach can be used [M.J. Kühn et al, Math. Biosciences, 2021]. Further developments then allow for a rigorous testing of commuters coming from hot spots [M.J. Kühn et al, medrxiv, 2021] or the evaluation of vaccination effects [W. Koslow et al., medrxiv, 2021] while certain NPIs are relaxed. Other limitations can be removed by using integro-differential equation-based models which are also called age of infection models.  Another promising approach is the use of multi-scale models by combining metapopulation models with agent-based models. This talk will give a general introduction to the topic and lay out certain important aspects and approaches when modeling infectious diseases' spread.