The past COVID—pandemic has shown the need for mathematical models of disease dynamics. Worldwide, researchers have developed models and tried to evaluate the effect of potential countermeasures like lock-down, testing or vaccinations. However, disease dynamics is not just driven by individual factors but also social aspects. Living conditions, household structures or even beliefs and media consumption play a crucial role in determining the progression of the epidemic and the impact of countermeasures. Individual opinions about the disease influence the transmission dynamics and are influenced themselves by the prevalence of the disease in the population. In this talk we will discuss the combination of epidemiological models with social aspects in some exemplary situations.

Understanding microbial communities is crucial for advancing fields like ecology, biotechnology, and human health. Recently, constraint-based metabolic models of individual organisms have been combined in various ways to study microbial consortia. In this work, we present a comprehensive geometric approach to characterize all feasible microbial interactions. First, we project community models onto the relevant variables for interaction, namely exchange fluxes and community compositions. Next, we compute 'elementary' compositions/exchange fluxes, thereby extending the concept of minimal metabolic pathways from single species to entire communities. Every feasible community is a combination of these elementary compositions/exchange fluxes, and, surprisingly, every elementary vector represents a fundamental ecological interaction (such as specialization, commensalism, or mutualism). Hence, our geometric approach allows us to decode the metabolic interactions underlying microbial cooperation. Moreover, it provides a foundation for rational community design. Since it treats exchange fluxes and community compositions equally, we can directly apply existing constraint-based methods and algorithms.

Travelling waves ocurring in MIN-D protein concentration of E. coli has been known to play a vital role in cell center localization during mitosis. Building on the work of S. Meindlhumer et al. we investigate a simplified reaction-diffusion-advection model for reaction kinetics of the MIN proteins and identify different instabilities as well as travelling waves arising from them. Analytic approaches are chosen when possible, but the complexity of the equations makes numerical analysis necessary to some degree. Of particular interest are two conservation laws that lead to an infinite family of equilibium states as well as some numerical difficulties.

In this talk, a mathematical model of dengue fever transmission, incorporating delay terms to reflect realistic time delays in the disease dynamics is presented. Using the next-generation matrix method, the basic reproduction number is derived as a key threshold parameter. The dynamics of the model are analyzed with and without delays, showing that recovery delays can destabilize the system or induce oscillations. Numerical simulations illustrate how delay parameters affect disease spread, providing insights into the role of timing in transmission and control strategies. The framework is applicable to other infectious diseases.

The detailed balance is a property of macroscopic systems that are obtained from an underlying time-reversible microscopic model. It states that each elementary process (for instance each chemical reaction) is in equilibrium with its reverse process. Even if, at the fundamental level, we expect chemical reactions to satisfy the so-called detailed balance condition, biochemical systems are often modeled by systems of equations for which detailed balance fails. This can be justified if the system is in contact with "reservoirs" that are out of equilibrium, as is usually the case in biological systems. In this talk I will discuss the relation between the detailed balance property and two important properties of certain chemical systems. The first property that I will discuss is the capability of discrimination between different ligands of the classical Hopfield-Ninio kinetic proof-reading network. The second property that we will study is the adaptation property (i.e. the fact that some chemical networks respond to gradients instead of absolute values of signals). In both cases we will see that the failure of detailed balance plays an important role in order to attain the desired property.

I plan to present some old and new results on Lotka-Volterra systems $$\dot x_i = x_i (r_i + \sum_{j=1}^n a_{ij}x_j), \quad i = 1, \dots, n$$ and mention some open problems.

We give an overview of the recent results on the systematic studies of bifurcations in small mass-action reaction networks, i.e., ones with a few species and a few reactions. Further, we provide a brief introduction to the inheritance theory of mass-action systems, which allows us to infer dynamical properties of larger, more realistic reaction networks from their subnetworks.

Put simply, reaction networks pose two main challenges for dynamical analysis: they often involve large systems and they depend on many positive parameters. In this talk, I will present algebraic conditions that allow to infer the existence of periodic solutions based solely on the structure of the network. In particular, when the network is equipped with kinetics sufficiently rich in parameters — such as rational functions of Michaelis-Menten type — these conditions can be interpreted directly in terms of reaction stoichiometry. Through perturbation arguments and reduction methods, such structural conditions reveal network motifs for oscillations, as they guarantee the existence of parameter values that yield periodic behavior in any network — of any size — that contains the motif. In the more delicate case of mass-action kinetics, the link between the algebraic conditions and the raw stoichiometry is more subtle. At present, the analysis provides only necessary stoichiometric criteria, along with more elaborate inheritance results, as explored by other participants in this minisymposium. The main mathematical ingredients are the linear algebra concept of D-stability and the theory of global Hopf bifurcation.