Kamonchat Trachoo

Introduction: Human papillomavirus (HPV) is the most common sexually transmitted infection and the cause of virtually all cervical cancer, yet elimination is complicated by two features of real vaccination programmes: available vaccines are imperfect, leaving vaccinated individuals partially susceptible, and naturally acquired immunity wanes. We ask whether reducing the basic reproduction number below unity is sufficient for elimination when both mechanisms are present. Methods: We formulate and analyse a six-compartment ( SEIRVC ) deterministic model combining an imperfect vaccine and waning immunity with explicit progression to an HPV-related cancer class. We establish positivity, boundedness, and an invariant region; derive R ₀ by the next-generation matrix method; characterize the equilibria and their stability using Routh–Hurwitz criteria, Lyapunov functions, and the geometric approach of Li and Muldowney; and analyse the bifurcation structure, supported by sensitivity indices and simulation. Results: The reproduction number decomposes additively into a susceptible and a vaccine-breakthrough route, and the disease-free equilibrium is locally asymptotically stable precisely when R ₀ < 1, and globally so under an explicit condition. Reducing the endemic equilibria to the roots of a quadratic shows that a leaky vaccine together with waning immunity produces a backward bifurcation, so that R ₀ < 1 is necessary but not sufficient for elimination and control must clear a turning point R 0 c < 1 . Above threshold the unique endemic equilibrium can lose stability through a Hopf bifurcation, giving sustained oscillations whose amplitude is magnified in the cancer class, which varies by a factor of order 10 ² within a cycle under constant parameters. Both phenomena require both mechanisms and vanish when the vaccine becomes perfectly protective or immunity permanent, the limit in which global stability of the endemic equilibrium is proved; two-parameter continuation shows the two regimes to be disjoint alternatives. Discussion: Sensitivity analysis gives sgnΥ _β = sgn(θ−λ _H ) so scaling up an insufficiently protective vaccine raises R ₀ , while the waning rate has zero elasticity yet determines both non-classical regimes. Threshold quantities alone are therefore an unreliable guide to HPV control, and surveillance must track prevalence over time.

This paper proposes a novel fractional-order asset flow model based on the Atangana–Baleanu–Caputo (ABC) derivative to analyze asset price dynamics in financial markets. Compared to classical models, the proposed model incorporates a nonlocal and non-singular fractional operator, allowing for a more accurate representation of investor behavior and market adjustment processes. The model captures both short-term trend-driven responses and long-term valuation-based decisions. We establish key theoretical properties of the system, including the existence and uniqueness of solutions, positivity, boundedness, and both local and global stability using Lyapunov functions. Numerical simulations under varying fractional orders demonstrate how the ABC derivative governs the convergence speed and equilibrium behavior of the system. Compared to classical integer-order models, the ABC-based approach provides smoother dynamics, greater flexibility in modeling behavioral heterogeneity, and better alignment with observed long-term financial phenomena.

Bone remodeling is a dynamic biological process that preserves bone strength and structure through the coordinated actions of osteoblasts, osteoclasts, osteocytes, and bone mass density. Traditional models based on ordinary differential equations often fail to capture the memory-dependent nature of these interactions. In this study, we propose a novel mathematical model of bone remodeling using the Atangana–Baleanu–Caputo fractional derivative, which accounts for the non-local and hereditary characteristics of biological systems. The model introduces fractional-order dynamics into a previously established ODE framework while maintaining the intrinsic symmetry between bone-forming and bone-resorbing mechanisms, as well as the balance mediated by porosity-related feedback. We establish the existence, uniqueness, and positivity of solutions, and analyze the equilibrium points and their global stability using a Lyapunov function. Numerical simulations under various fractional orders demonstrate symmetric convergence toward equilibrium across all biological variables. The results confirm that fractional-order modeling provides a more accurate and balanced representation of bone remodeling and reveal the underlying symmetry in the regulation of bone tissue. This work contributes to the growing use of fractional calculus in modeling physiological processes and highlights the importance of symmetry in both mathematical structure and biological behavior.

In this paper, we propose an improved mathematical model that can describe bone remodeling. This process involves the activities of osteoblasts, osteoclasts, osteocytes, and bone mass density. Bone remodeling is an important process that maintains bones in the human body by balancing formation and resorption. The proposed model is formulated as a system of differential equations that incorporate recruitment rates and porosity interaction. The local stability analysis was carried out using the Routh–Hurwitz criteria. The global stability analysis is conducted by using Lyapunov functions. The stability analysis provides the conditions for stable remodeling. Numerical simulations are illustrated to validate stability conditions and demonstrate the effects of parameter variations on cell populations and bone density. The results provide a framework for understanding bone-remodeling mechanisms and have potential applications for predicting bone health.