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Cervical intraepithelial neoplasia (CIN) is the development of abnormal cells on the surface of the cervix, caused by a human papillomavirus (HPV) infection. Although in most of the cases it is resolved by the immune system, a small percentage of people might develop a more serious CIN which, if left untreated, can develop into cervical cancer. Cervical cancer is the fourth most common cancer in women globally, for which the World Health Organization (WHO) recently adopted the Global Strategy for cervical cancer elimination by 2030. With this research topic being more imperative than ever, in this paper, we develop a nonlinear mathematical model describing the CIN progression. The model consists of partial differential equations describing the dynamics of epithelial, dysplastic and immune cells, as well as the dynamics of viral particles. We use our model to explore numerically three important factors of dysplasia progression, namely the geometry of the cervix, the strength of the immune response and the frequency of viral exposure.

During an epidemic, such as the COVID-19 pandemic, policy-makers are faced with the decision of implementing effective, yet socioeconomically costly intervention strategies, such as school and workplace closure, physical distancing, etc. In this study, we propose a rigorous definition of epidemiological strategies. In addition, we develop a scheme for comparing certain epidemiological strategies, with the goal of providing policy-makers with a tool for their systematic comparison. Then, we put the suggested scheme to the test by employing an age-based epidemiological compartment model introduced in Bitsouni et al. (2024), coupled with data from the literature, in order to compare the effectiveness of age-based and horizontal interventions. In general, our findings suggest that these two are comparable, mainly at a low or medium level of intensity.

We first introduce the generic versions of the fraction rules for monotonicity, i.e. the one that involves integrals known as the Gromov theorem and the other that involves derivatives known as L'H\^opital rule for monotonicity, which we then extend to high order antiderivatives and derivatives, respectively.

An inequality, which combines the concept of completely monotone functions with the theory of divided differences, is proposed. It is a straightforward generalization of a result, recently introduced by two of the present authors.

We introduce the multivariate analogue of the well known inequality $1+x\leq \mathrm{e}^x$, for an abstract non negative real number $x$. The result emerges from the study of the blow up time of certain solutions of the Cauchy problem for a particular ODE. It is also closely related to the notion of completely monotone functions and the theory of divided differences.

A model for the mathematical study of immune response to breast cancer is proposed and studied, both analytically and numerically. It is a simplification of a complex one, recently introduced by two of the present authors. It serves for a compact study of the dynamical role in cancer promotion of a relatively recently described subgroup of regulatory B cells, which are evoked by the tumour.

tBregs are a newly discovered subcategory of B regulatory cells, which are generated by breast cancer, resulting in the increase of Tregs and therefore in the death of NK cells. In this study, we use a mathematical and computational approach to investigate the complex interactions between the aforementioned cells as well as CD8$^+$ T cells, CD4$^+$ T cells and B cells. Furthermore, we use data fitting to prove that the functional response regarding the lysis of breast cancer cells by NK cells has a ratio-dependent form. Additionally, we include in our model the concentration of rituximab - a monoclonal antibody that has been suggested as a potential breast cancer therapy - and test its effect, when the standard, as well as experimental dosages, are administered.

We study, from a purely quantitative point of view, the quasi-steady-state assumption for the fundamental mathematical model of the general enzymatic reaction: we re-establish, on a rigorous basis, certain already known results and we propose a novel approach to the subject, that makes waivable an ambiguous, widely used, practice. In particular, we define the two parts of the assumption in a quantitative fashion, we employ a simple algorithm for the proper scaling of the corresponding problem which naturally provides us with the necessary and sufficient information, and we comment, among other issues, on a dispensable third part of the assumption.

We present a compartmental mathematical model with demography for the spread of the COVID-19 disease, considering also asymptomatic infectious individuals. We compute the basic reproductive ratio of the model and study the local and global stability for it. We solve the model numerically based on the case of Italy. We propose a vaccination model and we derive threshold conditions for preventing infection spread in the case of imperfect vaccines.

Cancer cells exhibit increased motility and proliferation, which are instrumental in the formation of tumours and metastases. These pathological changes can be traced back to malfunctions of cellular signalling pathways, and calcium signalling plays a prominent role in these. We formulate a new model for cancer cell movement which for the first time explicitly accounts for the dependence of cell proliferation and cell-cell adhesion on calcium. At the heart of our work is a non-linear, integro-differential (non-local) equation for cancer cell movement, accounting for cell diffusion, advection and proliferation. We also employ an established model of cellular calcium signalling with a rich dynamical repertoire that includes experimentally observed periodic wave trains and solitary pulses. The cancer cell density exhibits travelling fronts and complex spatial patterns arising from an adhesion-driven instability (ADI). We show how the different calcium signals and variations in the strengths of cell-cell attraction and repulsion shape the emergent cellular aggregation patterns, which are a key component of the metastatic process. Performing a linear stability analysis, we identify parameter regions corresponding to ADI. These regions are confirmed by numerical simulations, which also reveal different types of aggregation patterns and these patterns are significantly affected by \ca. Our study demonstrates that the maximal cell density decreases with calcium concentration, while the frequencies of the calcium oscillations and the cell density oscillations are approximately equal in many cases. Furthermore, as the calcium levels increase the speed of the travelling fronts increases, which is related to a higher cancer invasion potential. These novel insights provide a step forward in the design of new cancer treatments that may rely on controlling the dynamics of cellular calcium., Comment: 27 pages; 8 figures

This report summarises the work and results produced at the 146th European Study Group with Industry/co-creation event with society on the challenge \textit{Breaking barriers for women in Science}. The aim of this challenge, proposed by the Cyprus-based non-profit AIPFE Cyprus-Women of Europe, was to quantify the barriers that women face in science so that eventually policy changes can take place in Cyprus and elsewhere. Two distinct but related challenges were considered. The first challenge was to quantify the wage gap between men and women in 28 European countries. In this connection, we analysed Eurostat data and developed a mathematical model quantifying how probable it is for countries to decrease their wage gap. Secondly, we analysed data provided by the University of Cyprus and determined the percentage of women and men in STEM (Science, Technology, Engineering and Mathematics) departments as they move up the academic ladder, starting from the undergraduate level. Studying the latter challenge is a first step in studying the wage gap in all Cypriot universities and in other universities abroad. This work was supported financially by the EU project SciShops.eu, the EU Mathematics for Industry Network (MI-NET) and several other organisations.