Mathematical Insight Research Articles

Modeling tuberculosis transmission dynamics in Kazakhstan using SARIMA and SIR models

Tuberculosis (TB) is a highly contagious disease that remains a global concern affecting numerous countries. Kazakhstan has been facing considerable challenges in TB prevention and treatment for decades. This study aims to understand TB transmission dynamics by developing and comparing statistical and deterministic models: Seasonal Autoregressive Integrated Moving Average (SARIMA) and the basic Susceptible-Infected-Recovered (SIR). TB data from 2014 to 2019 were collected from the Unified National Electronic Health System (UNEHS) using retrospective quantitative analysis. SARIMA models were able to capture seasonal variations, with Model 2 exhibiting superior predictive accuracy. Both models showed declining TB incidence and revealed a notable predictive performance evaluated by statistical metrics. In addition, the SIR model calculated the basic reproduction number (\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${R_0}$$\\end{document}) below 1, indicating a receding epidemic. Models proved the capability of each to accurately capture trends (SARIMA) and provide mathematical insights (SIR) into TB transmission dynamics. This study contributes to the general knowledge of TB transmission dynamics in Kazakhstan thus laying the foundation for more comprehensive studies on TB and control strategies.

Mathematical insights into the influence of delay and external recruitment on coral-macroalgae system

As anthropogenic pressures on coral reef ecosystems continue to increase due to global warming and mass coral bleaching events, there is growing interest in developing conservation strategies to restore degraded coral reefs. One such approach is the transplantation of coral larvae onto degraded reefs. In this paper, we use a delayed coral-macroalgae model to explore the effects of external recruitment of coral. By analyzing the local and global stability of macroalgae-free equilibrium and coexistence equilibrium in the system, we find that sustained external recruitment of coral will favor coral competition with macroalgae. In addition, we choose delay as a bifurcation parameter and demonstrate that Hopf bifurcation may occur at a critical delay near the coexistence equilibrium. Interestingly, the delay may cause the globally asymptotically stable equilibrium in the ODE system to become unstable, resulting in the appearance of periodic solutions. Furthermore, we analyze population dynamics using optimal control theory and determine the effect of minimum external recruitment on the population dynamics. In the numerical simulation section, parameters of coral-macroalgae dynamics are estimated by using a 12-year (2005–2017) benthic cover dataset of coral reefs in the Gulf of Mannar, southeastern India. The theoretical results are validated and supported by numerical simulations.

Mathematical analysis of busy tone in full-duplex optical MAC for hidden node mitigation

This article provides the mathematical analysis of carrier sense multiple access with collision avoidance (CSMA/CA) media access control (MAC) protocol of IEEE 802.15.7 optical wireless communication (OWC). While the prior works have performed the OWC CSMA/CA mathematical analysis using the Markov models, deviation from the simulation results has been observed. We address this by improving the Markov model calculations, which display a mere 0.2% throughput deviation, nearly matching the simulation results. Furthermore, we work on the hidden node problem of the OWC networks. This problem is solved in literature by using various full-duplex communication methods, such as bi-directional data transmission and the busy tone signal; the latter is employed in our previous work on full-duplex optical MAC (FD-OMAC). These techniques increase the coverage area of the nodes by utilizing an access point (AP) as a relay node. However, the AP response is delayed by the processing time, causing an unexpected network behavior. The quantitative effect of this delay remains unexplored, which is critical for optimizing the OWC network. We bridge this gap by extending the proposed Markov analysis to model CSMA/CA and the aforementioned full-duplex techniques. This work equips readers with mathematical insights for future OWC MAC layer enhancements.

Mathematical modeling insights into improving CAR T cell therapy for solid tumors with bystander effects

As an adoptive cellular therapy, Chimeric Antigen Receptor T cell (CAR T cell) therapy has shown remarkable success in hematological malignancies but only limited efficacy against solid tumors. Compared with blood cancers, solid tumors present a series of challenges that ultimately combine to neutralize the function of CAR T cells. These challenges include, but are not limited to, antigen heterogeneity - variability in the expression of the antigen on tumor cells, as well as trafficking and infiltration into the solid tumor tissue. A critical question for solving the heterogeneity problem is whether CAR T therapy induces bystander effects, such as antigen spreading. Antigen spreading occurs when CAR T cells activate other endogenous antitumor CD8 T cells against antigens that were not originally targeted. In this work, we develop a mathematical model of CAR T cell therapy for solid tumors that considers both antigen heterogeneity and bystander effects. Our model is based on in vivo treatment data that includes a mixture of target antigen-positive and target antigen-negative tumor cells. We use our model to simulate large cohorts of virtual patients to better understand the relationship involving bystander killing. We also investigate several strategies for enhancing bystander effects, thus increasing CAR T cell therapy’s overall efficacy for solid tumors.

A mathematical insight to control the disease psoriasis using mesenchymal stem cell transplantation with a biologic inhibitor

Psoriasis is a chronic, non-contagious, immune-mediated skin disorder. Inflammation of the skin’s surface is characterised by scaly white, red, or silvery spots that occur due to the hyper-proliferation of keratinocytes in the epidermal layer. Primarily, pharmaceutical drugs or immune therapy are used to treat psoriasis. We are all aware that, certain therapeutic strategies can have some adverse effects, and over time, that hidden inflammation may manifest. This article introduces a mathematical model for psoriasis, formulated by employing a set of nonlinear ordinary differential equations (ODEs) that describe the densities of T-cells, dendritic cells (DCs), keratinocytes, and mesenchymal stromal cells (MSCs) as basic cell populations. A tumor necrosis factor-α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} (TNF-α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$TNF-\\alpha$$\\end{document}) inhibitor has been imposed from the initial stage of the treatment regime, using the optimal control theoretic approach, and the numerical results have been observed. After 80 days of monitoring using only biologic TNF-α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$TNF-\\alpha$$\\end{document} inhibitors, if this approach did not provide the intended outcomes (when severity arises), stem cells are administered a few times in a pulsed manner as a cell replacement technique in addition to this anti TNF-α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$TNF-\\alpha$$\\end{document} medicine. We have observed the combined therapeutic benefit of stem cell replacement with a TNF-α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$TNF-\\alpha$$\\end{document} inhibitor from a mathematical point of view. The theoretical analysis and the numerical results revealed that stem cell transplantation, along with a TNF-α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$TNF-\\alpha$$\\end{document} inhibitor, is a promising psoriasis treatment option moving forward.

Additional applications of the Lambert W function to solid state physics

Analytical solutions are always desirable in physics for the sake of mathematical exactness and physical insight. In this study, I obtain closed-form analytical expressions for various physical quantities taken from solid state physics. More precisely, I derive analytical expressions in terms of the Lambert W function for the work function and the field-enhancement factor of a field emitting material from the Fowler–Nordheim equation. Additionally, I derive similar analytical expressions for the localization length and the density of states in amorphous semiconductors from the Mott hopping conductivity. Similarly, I also derive analytical formulae based on the Lambert W function to compute the extrinsic-intrinsic transition temperature in a partially compensated semiconductor and the Kondo exchange coupling constant. All the obtained results are exact and explicit. Moreover, some of them allow a direct determination of some physical quantities of interest compared to their indirect determination from semi-logarithmic experimental plots. The findings of this paper are accessible and suitable for students enrolled in graduate solid state physics courses.

Overcompensation of transient and permanent death rate increases in age-structured models with cannibalistic interactions

There has been renewed interest in understanding the mathematical structure of ecological population models that lead to overcompensation, the process by which a population recovers to a higher level after suffering a permanent increase in predation or harvesting. Here, we apply a recently formulated kinetic population theory to formally construct an age-structured single-species population model that includes a cannibalistic interaction in which older individuals prey on younger ones. Depending on the age-dependent structure of this interaction, our model can exhibit transient or steady-state overcompensation of an increased death rate as well as oscillations of the total population, both phenomena that have been observed in ecological systems. Analytic and numerical analysis of our model reveals sufficient conditions for overcompensation and oscillations. We also show how our structured population partial integrodifferential equation (PIDE) model can be reduced to coupled ODE models representing piecewise constant parameter domains, providing additional mathematical insight into the emergence of overcompensation.

The forgotten p-versine and p-coversine family of functions revisited.

This study delves into special class of generalized p-trigonometric functions, examining their connection to the established counterparts like p-cosine and p-sine functions. We here explore the new class of functions called the p-versine, p-coversine, p-haversine, and p-hacovercosine, by providing comprehensive definitions and properties. Grounded in the characteristics of p-cosine and p-sine functions, the newly proposed functions offer unique mathematical insights. Our work contributes towards a thorough understanding of these new special functions, showcasing their potential applications in diverse scientific domains, from mathematical analysis to physics and engineering. This paper contributes as a valuable resource for future applied mathematics researchers, engaging with these new mathematical functions, enhancing the ability to model complex patterns from diverse real-world applications.

On the role of geometric phase in the dynamics of elastic waveguides.

The geometric phase provides important mathematical insights to understand the fundamental nature and evolution of the dynamic response in a wide spectrum of systems ranging from quantum to classical mechanics. While the concept of geometric phase, which is an additional phase factor occurring in dynamical systems, holds the same meaning across different fields of application, its use and interpretation can acquire important nuances specific to the system of interest. In recent years, the development of quantum topological materials and its extension to classical mechanical systems have renewed the interest in the concept of geometric phase. This review revisits the concept of geometric phase and discusses, by means of either established or original results, its critical role in the design and dynamic behaviour of elastic waveguides. Concepts of differential geometry and topology are put forward to provide a theoretical understanding of the geometric phase and its connection to the physical properties of the system. Then, the concept of geometric phase is applied to different types of elastic waveguides to explain how either topologically trivial or non-trivial behaviour can emerge based on the geometric features of the waveguide. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 2)'.

Variational Methods for Evolution

Variational principles for evolutionary systems arise in many settings, both in those describing the physical world and in man-made algorithms for data science and optimization tasks. Variational principles are available for Hamiltonian systems in classical mechanics, gradient flows for dissipative systems, as well as in time-incremental minimization techniques for more general evolutionary problems. Additional challenges arise via the interplay of two or more functionals (e.g. a free energy and a dissipation potential), thus encompassing a large variety of applications in the modeling of materials and fluids, in biology, and in multi-agent systems. Variational principles and associated evolutions are also at the core of the modern approaches to machine learning tasks, since many of them are posed as minimizing an objective functional that models the problem. The discrete and random nature of these problems and the need for accurate computation in high dimension present a set of challenges that require new mathematical insights. Variational methods for evolution allow for the usage of the rich toolbox provided by the calculus of variations, metric-space geometry, partial differential equations, and other branches of applied analysis. The variational methods for evolution have seen a rapid growth over the last two decades. This workshop continued the successful line of meetings (2011, 2014, 2017, and 2020), while evolving and branching into new directions. We have brought together a wide scope of mathematical researchers from calculus of variations, partial differential equations, numerical analysis, and stochastics, as well as researchers from data science and machine learning, to exchange ideas, foster interaction, develop new avenues, and generally bring these communities closer together.

Identifying mathematics teachers’ competency to look at elementary mathematics from an advanced standpoint: a pilot study

This study evaluates the impact of “the ability to look at elementary mathematics from an advanced standpoint (EMFAS),” which was originally proposed by Felix Klein, in mathematics education. By strengthening the connection between elementary and tertiary mathematics, mathematics teachers are expected to have advanced knowledge to offer students higher mathematical insights. Thus, EMFAS serves as a great indicator for measuring teaching effectiveness. By investigating newly qualified and pre-service teachers’ ability to apply EMFAS techniques, we analyze the logic involved in solving some specially constructed problems and how teachers could deliver mathematical knowledge to students at the elementary level with the advanced knowledge they possess. We explain the importance of the EMFAS course, in terms of pedagogy and content knowledge. This course prepares pre-service mathematics teachers by equipping them with advanced knowledge for their future profession. This study evaluates mathematics teachers’ proficiency in applying EMFAS techniques via a problem set and interviews. We summarize the hierarchical nature of how teachers perceive EMFAS, which could play an important role in guiding pedagogy in mathematics education.

Innovative Mathematical Insights through Artificial Intelligence (AI): Analysing Ramanujan Series and the Relationship between e and π\pi

This paper presents a comprehensive case study on the application of Artificial Intelligence (AI) in mathematics, focusing on the Ramanujan series and the intricate relationship between the mathematical constants e and π. The study explores how AI, particularly machine learning and pattern recognition techniques, can be harnessed to discover new mathematical series and patterns, thereby extending the pioneering work of the legendary mathematician Srinivasa Ramanujan. The paper begins with an overview of the Ramanujan series, illustrating their significance and applications in mathematical computations. It then delves into the specifics of AI methodologies employed to unearth new series for e and π, highlighting the algorithms and models used. Through detailed analysis and experimentation, the study demonstrates how AI can generate new series expansions for e and π, offering enhanced convergence rates and computational efficiencies. Furthermore, the paper examines the relationship between these two constants, providing insights into their interconnected nature through AI-discovered series and patterns. Practical applications of these new series in fields such as numerical methods, cryptography, and theoretical physics are also discussed. By showcasing the successful integration of AI in the realm of mathematical research, this case study underscores the potential of AI to revolutionize traditional mathematical approaches, fostering the discovery of new knowledge and the refinement of existing theories. The findings contribute to a deeper understanding of the interplay between e and π, reinforcing the profound impact of Ramanujan's work in modern mathematics and the transformative power of AI in advancing this legacy.

Darboux and generalized Darboux transformations for the fractional integrable derivative nonlinear Schrödinger equation

Analytical methods provide crucial mathematical insights into the stable solitary waves hidden in nonlinear phenomena. The nonlinear Schrödinger (NLS) equation is one of the most important typical integrable soliton models. From a mathematical perspective, the essence of the celebrated derivative NLS (DNLS) equation’s difference from the classical NLS equation lies in its cubic potential being differentiated once by the spatial variable and multiplied by the imaginary unit, which leads to the former having some characteristics that the latter cannot have. This paper extends the DNLS equation to the fractional integrable case with conformable derivative operators, and uses Darboux transformations (DTs) and generalized DT (GDT) to solve it exactly. Specifically, Lax pairs generating the fractional DNLS equation are first given. Based on the given Lax pairs, then the n-fold DTs and GDT for the fractional DNLS equation are derived. Some special exact solutions of the fractional DNLS equation are further obtained by employing the derived n-fold DTs and GDT. Finally, several novel space–time structures and dynamical evolutions of the obtained exact solutions are analyzed. This paper reveals through the DT and GDT methods that the double power-law fractional orders in the exact solutions of the fractional DNLS equation can be used to dominate the variable velocity propagation and anomalous diffusion in fractional dimensional media at different geometric scales.

Mathematical and experimental insights into acid-resistant enhancements: Graphene plates' influence on rubber-incorporated cementitious materials

To mitigate the strength loss of rubberised cementitious materials, either concrete or mortar, graphene plates were incorporated. Limited studies have highlighted significant strength enhancement in rubberised cementitious materials incorporating graphene plates under non-acidic conditions. However, a notable gap persists in their performance when exposed to harsh conditions, particularly those induced by acid attack. Therefore, this study aimed to assess the inclusion of graphene plates, alongside Arabic gum as a surfactant for graphene dispersion, on the acid resistance of rubberised mortar exposed to sulphuric acid (H2SO4). One purpose of this study was to experimentally examine their behaviour through visual inspection, compressive strength measurement, and weight loss analysis. The second purpose was to utilise response surface methodology for prediction, optimisation, and developing two quadratic equations for further exploration. The results exhibit a strong alignment between equations and experimental data, with a coefficient of variation below 2.8 % and an impressive R-squared value of 0.99, confirming their reliability and accuracy. Additionally, the proposed mortar incorporating graphene plates and rubber demonstrates superior resistance to acid attacks. Even with up to an 18 % increase in rubber content, it shows reduced strength and weight losses of 47.1 % and 25.2 % after 40 days, respectively, compared to the control mixture's losses of 62.1 % and 33.6 %. It can be concluded that the remarkable results obtained for graphene-based rubberised cementitious material hold promise for real-world applications, particularly where resistance to energy absorption and penetration by aggressive agents is required such concrete pavement.

Mathematical Insights into Large Language Models

Purpose: The paper presents an exhaustive examination of the mathematical frameworks that support the creation and operation of large language models. The document commences with an introduction to the core mathematical concepts that are foundational to large language models. It delves into the mathematical algorithms employed in training these models and scrutinizes how various mathematical notions influence their efficacy. Methodology: Furthermore, it dissects the structure of large language models, analyzing the mathematical tenets that dictate their design and functionality. It also considers the mathematical logic underpinning these models' performance and the intricacies involved in their expansion. Additionally, it probes into the mathematical underpinnings of attention mechanisms within large language models, assessing how these mechanisms bolster the models' effectiveness and comprehensibility. Findings: Subsequently, it examines the mathematical bases of attention mechanisms in large language models, considering how these mechanisms augment the models' efficiency and clarity. It also debates the mathematical methods for refining large language models and the hurdles faced in enhancing their interpretability. By understanding the mathematical foundations of LLMs, we can leverage insights from the algorithms and principles driving these models, thus enhancing their inventive output and broadening the horizons of design and artistic expression. Unique contribution to theory, policy and practice: Lastly, it ventures into the ethical considerations surrounding large language models, scrutinizing the mathematical aspects related to these concerns.

Mathematical insights into the [formula omitted] model with Allee and fear dynamics in the context of COVID-19

In this study, we introduce and investigate the SEIQRD COVID-19 model incorporating Allee and fear effects. Employing the next-generation matrix method, we derive the basic reproduction number, a crucial epidemiological metric. Our model exhibits two significant equilibrium points: one representing the disease-free state and the other representing the endemic state. Utilizing the Routh–Hurwitz criteria, we demonstrate that the disease-free equilibrium is locally asymptotically stable when the basic reproduction number (R0) is less than 1, while the endemic equilibrium attains local stability when R0 exceeds 1. Furthermore, we establish global stability for the endemic equilibrium through the application of Poincare–Bendixson properties and Dulac’s criteria. To complement our theoretical results, we conduct comprehensive simulation studies, validating the practical implications of our findings.

Deep-water and shallow-water limits of the intermediate long wave equation

In this paper, we study the low regularity convergence problem for the intermediate long wave equation (ILW), with respect to the depth parameter δ > 0, on the real line and the circle. As a natural bridge between the Korteweg–de Vries (KdV) and the Benjamin–Ono (BO) equations, the ILW equation is of physical interest. We prove that the solutions of ILW converge in the Hs -Sobolev space for s>12 , to those of BO in the deep-water limit (as δ→∞ ), and to those of KdV in the shallow-water limit (as δ → 0). This improves previous convergence results by Abdelouhab et al (1989 Physica D 40 360–92), which required s>32 in the deep-water limit and s⩾2 in the shallow-water limit. Moreover, the convergence results also apply to the generalised ILW equation, i.e. with nonlinearity ∂x(uk) for k⩾2 . Furthermore, this work gives the first convergence results of generalised ILW solutions on the circle with regularity s⩾34 . Overall, this study provides mathematical insights for the behaviour of the ILW equation and its solutions in different water depths, and has implications for predicting and modelling wave behaviour in various environments.

The PDI model system for parameterizing soil hydraulic properties

AbstractThe Peters–Durner–Iden (PDI) model system for describing soil hydraulic properties (SHP) has been developed over a decade. Inspired by Rien van Genuchten's seminal work, the PDI system focuses on an efficient and simple parameterization of water retention curves and hydraulic conductivity curves (HCC) across the entire soil moisture spectrum. By combining capillary and non‐capillary components for water retention and conductivity, it aims to reconcile mathematical simplicity and insights on water adsorption and film flow in soils. Recent developments have reduced the number of free parameters of the conductivity model to zero, enhancing the model's applicability in cases of limited data availability. The first reduction was achieved by a prediction of absolute non‐capillary conductivity based on the consideration of film and corner flow on the pore scale, and the second by a prediction of absolute capillary conductivity by a capillary bundle model. This allows a complete characterization of SHP over the entire moisture range with only four retention curve parameters. The inclusion of a maximum pore size in the capillary conductivity model prevents an unrealistic drop of the HCC near saturation. This paper provides a comprehensive overview of the PDI model system, emphasizing its conceptual features and mathematical details. An Excel sheet and a Python code stored in a repository are provided for accessibility.

Mammalian Adaptation Risk in HPAI H5N8: A Comprehensive Model Bridging Experimental Data with Mathematical Insights

ABSTRACT Understanding the mammalian pathogenesis and interspecies transmission of HPAI H5N8 virus hinges on mapping its adaptive markers. We used deep sequencing to track these markers over five passages in murine lung tissue. Subsequently, we evaluated the growth, selection, and RNA load of eight recombinant viruses with mammalian adaptive markers. By leveraging an integrated non-linear regression model, we quantitatively determined the influence of these markers on growth, adaptation, and RNA expression in mammalian hosts. Furthermore, our findings revealed that the interplay of these markers can lead to synergistic, additive, or antagonistic effects when combined. The elucidation distance method then transformed these results into distinct values, facilitating the derivation of a risk score for each marker. In vivo tests affirmed the accuracy of scores. As more mutations were incorporated, the overall risk score of virus heightened, and the optimal interplay between markers became essential for risk augmentation. Our study provides a robust model to assess risk from adaptive markers of HPAI H5N8, guiding strategies against future influenza threats.

Kinematic Batch Estimator for Angular Velocity and Associated Uncertainty

Estimating the relative pose and angular velocity of an uncooperative space object is a critical component for many applications such as on-orbit assembly, inspection, and servicing. This paper presents new mathematical insights that permit a kinematic batch filter formulation for angular velocity estimation using quaternion measurements for relative orientation, thereby eliminating the need to know the moment of inertia of the target. In particular, a novel batch approach to computing the associated uncertainties for the angular velocity magnitude, spin-axis direction, and overall angular velocity vector estimates is presented and compared against previous recursive methodologies. Finally, a realistic use-case scenario is demonstrated in real-time on flightlike hardware.