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Diabetes mellitus constitutes a major health problem and its clinical presentation and progression may vary considerably. A number of standardized diagnostic and monitoring tests are currently used for diabetes. They are based on measuring either plasma glucose, glycated haemoglobin or both. Their main goal is to assess the average blood glucose concentration. There are several sources of interference that can lead to discordances between measured plasma glucose and glycated haemoglobin levels. These include haemoglobinopathies, conditions associated with increased red blood cell turnover or the administration of some therapies, to name a few. Therefore, there is a need to provide new diagnostic tools for diabetes that employ clinically accessible biomarkers which, at the same time, can offer additional information allowing us to detect possible conflicting cases and to yield more reliable evaluations of the average blood glucose level concentration. We put forward a biomathematical model to describe the kinetics of two patient-specific glycaemic biomarkers to track the emergence and evolution of diabetes: glycated haemoglobin and its labile fraction. Our method incorporates erythrocyte age distribution and utilizes a large cohort of clinical data from blood tests to support its usefulness for diabetes monitoring.

In this article, we investigate two types of double dispersion equations in two different dimensions, which arise in several physical applications. Double dispersion equations are derived to describe long nonlinear wave evolution in a thin hyperelastic rod. Firstly, we obtain conservation laws for both these equations. To do this, we employ the multiplier method, which is an efficient method to derive conservation laws as it does not require the PDEs to admit a variational principle. Secondly, we obtain travelling waves and line travelling waves for these two equations. In this process, the conservation laws are used to obtain a triple reduction. Finally, a line soliton solution is found for the double dispersion equation in two dimensions.

Resistance to chemotherapy is a major cause of cancer treatment failure. The processes of resistance induction and selection of resistant cells (due to the over-expression of the membrane transporter P-glycoprotein, P-gp) are well documented in the literature, and a number of mathematical models have been developed. However, another process of transfer of resistant characteristics is less well known and has received little attention in the mathematical literature. In this paper, we discuss the potential of simple mathematical models to describe the process of resistance transfer, specifically P-gp transfer, in mixtures of resistant and sensitive tumor cell populations. Two different biological hypotheses for P-gp transfer are explored: (1) exchange through physical cell-cell connections and (2) through microvessicles released to the culture medium. Two models are developed which fit very well the observed population growth dynamics. The potential and limitations of these simple "global" models to describe the aforementioned biological processes involved are discussed on the basis of the results obtained.

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