Mathematical modeling of the impact of HPV vaccine uptake in reducing cervical cancer using a graph-theoretic approach via Caputo fractional-order derivatives

Human papillomavirus (HPV) is a highly prevalent sexually transmitted infection and the primary cause of cervical cancer, which remains a leading cause of cancer-related mortality among women globally. Despite ongoing vaccination efforts, challenges such as latency, persistent infections, and imperfect vaccine coverage complicate disease control. In this study, we develop a novel fractional-order compartmental model using Caputo derivatives to capture the memory and non-local transmission effects inherent in HPV dynamics. We analyze the model’s epidemiological properties by proving positivity, boundedness, and deriving the effective reproduction number ($R_e$) via a Graph Theoretic approach. Stability of disease-free and endemic equilibria is established through Lyapunov theory, complemented by Hyers–Ulam stability to ensure robustness. Parameter estimation is performed using Markov Chain Monte Carlo (MCMC), and sensitivity analysis utilizes Partial Rank Correlation Coefficients (PRCC) to identify key drivers of transmission. Our results indicate that achieving 56% vaccination coverage with 45.5% efficacy can reduce $R_e$ below one, supporting herd immunity. Numerical simulations demonstrate that vaccination coverage, timely treatment, and vaccine efficacy critically reduce infection prevalence and disease burden. Furthermore, higher fractional orders accelerate convergence to equilibrium without changing equilibrium values. This work lies in integrating fractional calculus with time-dependent vaccination and treatment controls to realistically model HPV progression and intervention impact. This approach provides a more accurate representation of HPV transmission dynamics, especially the long-term memory effects, thereby offering valuable insights for optimizing public health strategies.