Natural killer (NK) cells belong to the first line of host defense against infection and cancer. Cytokines, including interleukin-15 (IL-15), critically regulate NK cell activity, resulting in recognition and direct killing of transformed and infected target cells. NK cells have to adapt and respond in inflamed and often hypoxic areas. Cellular stabilization and accumulation of the transcription factor hypoxia-inducible factor-1α (HIF-1α) is a key mechanism of the cellular hypoxia response. At the same time, HIF-1α plays a critical role in both innate and adaptive immunity. While the HIF-1α hydroxylation and degradation pathway has been recently described with the help of mathematical methods, less is known concerning the mechanistic mathematical description of processes regulating the levels of HIF-1α mRNA and protein. In this work we combine mathematical modeling with experimental laboratory analysis and examine the dynamic relationship between HIF-1α mRNA, HIF-1α protein, and IL-15-mediated upstream signaling events in NK cells from human blood. We propose a system of non-linear ordinary differential equations with positive and negative feedback loops for describing the complex interplay of HIF-1α regulators. The experimental design is optimized with the help of mathematical methods, and numerical optimization techniques yield reliable parameter estimates. The mathematical model allows for the investigation and prediction of HIF-1α stabilization under different inflammatory conditions and provides a better understanding of mechanisms mediating cellular enrichment of HIF-1α. Thanks to the combination of in vitro experimental data and in silico predictions we identified the mammalian target of rapamycin (mTOR), the nuclear factor-κB (NF-κB), and the signal transducer and activator of transcription 3 (STAT3) as central regulators of HIF-1α accumulation. We hypothesize that the regulatory pathway proposed here for NK cells can be extended to other types of immune cells. Understanding the molecular mechanisms involved in the dynamic regulation of the HIF-1α pathway in immune cells is of central importance to the immune cell function and could be a promising strategy in the design of treatments for human inflammatory diseases and cancer.

Quorum sensing is a bacterial cell-to-cell communication mechanism and is based on gene regulatory networks, which control and regulate the production of signaling molecules in the environment. In the past years, mathematical modeling of quorum sensing has provided an understanding of key components of such networks, including several feedback loops involved. This paper presents a simple system of delay differential equations (DDEs) for quorum sensing of Pseudomonas putida with one positive feedback plus one (delayed) negative feedback mechanism. Results are shown concerning fundamental properties of solutions, such as existence, uniqueness, and non-negativity; the last feature is crucial for mathematical models in biology and is often violated when working with DDEs. The qualitative behavior of solutions is investigated, especially the stationary states and their stability. It is shown that for a certain choice of parameter values, the system presents stability switches with respect to the delay. On the other hand, when the delay is set to zero, a Hopf bifurcation might occur with respect to one of the negative feedback parameters. Model parameters are fitted to experimental data, indicating that the delay system is sufficient to explain and predict the biological observations.

We investigate the temporal evolution of the distribution of immunities in a population, which is determined by various epidemiological, immunological and demographical phenomena: after a disease outbreak, recovered individuals constitute a large immune population, however their immunity is waning in the long term and they may become susceptible again. Meanwhile, their immunity can be boosted by repeated exposure to the pathogen, which is linked to the density of infected individuals present in the population. This prolongs the length of their immunity. We consider a mathematical model formulated as a coupled system of ordinary and partial differential equations, that connects all these processes, and systematically compare a number of boosting assumptions proposed in the literature, showing that different boosting mechanisms lead to very different stationary distributions of the immunity at the endemic steady state. In the situation of periodic disease outbreaks, the waveforms of immunity distributions are studied and visualized. Our results show that there is a possibility to infer the boosting mechanism from the population level immune-dynamics. AMS Classification: 92D30, 34K60, 34K34, 37M05

Commonly known as head lice, Pediculus humanus capitis are human ectoparasites which cause infestations in children worldwide. Understanding the life cycle of head lice is an important step in knowing how to treat lice infestations, as the parasite behavior depends considerably on its age and gender. In this work we propose a mathematical model for head lice population dynamics in hosts who could be or not quarantined and treated. Considering a lice population structured by age and gender we formulate the model as a system of hyperbolic PDEs, which can be reduced to compartmental systems of delay or ordinary differential equations. Besides studying fundamental properties of the model, such as existence, uniqueness and nonnegativity of solutions, we show the existence of (in certain cases multiple) equilibria at which the infestation persists on the host's head. Aiming to assess the performance of treatments against head lice infestations, by mean of computer experiments and numerical simulations we investigate four possible treatment strategies. Our main results can be summarized as follows: (i) early detection is crucial for quick and efficient eradication of lice infestations; (ii) dimeticone-based products applied every 4 days effectively remove lice in at most three applications even in case of severe infestations and (iii) minimization of the reinfection risk, e.g. by mean of synchronized treatments in families/classrooms is recommended.