Mathematical Modelling in Biosciences and Discrete Optimization

This thesis explores innovative methodologies in two distinct areas of Applied Mathematics: Mathematical Modelling in Biosciences and Discrete Optimization. The work is structured into two parts, each addressing critical challenges in its respective field while offering practical applications and advancements. The first part focuses on Mathematical Modelling in Biosciences, presented in two chapters. The first chapter introduces a statistical monitoring approach using funnel plots for the early detection of COVID-19 Variants of Concern (VoCs). This methodology demonstrates remarkable utility in epidemiological surveillance by providing a simple, cost-effective, and real-time tool for identifying anomalous patterns in regional reproduction numbers. Its practical impact lies in its ability to complement genomic sequencing efforts by enabling more targeted and efficient investigations, ultimately supporting timely public health interventions. The second chapter reconstructs the temporal profile of new COVID-19 cases in Italy during the first wave of 2020, addressing the significant underreporting that hindered accurate epidemiological assessments. By leveraging dynamic system identification and regularized inverse problem-solving techniques, this work not only offers a quantitative correction for underreported data but also provides a robust framework for evaluating the impact of nonpharmaceutical interventions. The second part focuses on Discrete Optimization and consists of two chapters. The first chapter addresses the design and optimization of fewbit neural networks tailored for classification problems under few-shot learning scenarios. A novel voting structure is proposed to extend the framework to multi-class classification, offering practical applications in scenarios where computational efficiency and adaptability are paramount. The second chapter investigates the unassigned Distance Geometry Problem in the Manhattan norm, applied to the Mobile Positioning Problem in grid-like geometries. This formulation is particularly suited to scenarios such as mobile device positioning in urban environments, where the assignment of distances between devices is unknown.