A fractal-fractional perspective on Lassa fever dynamics with saturated incidence and relapse

This paper presents a comprehensive investigation into the transmission dynamics of Lassa fever through a sophisticated mathematical model formulated with a fractal-fractional operator in the Atangana-Baleanu sense. The model's architecture is an SEIQR (susceptible-exposed-infected-quarantined-recovered) framework that incorporates crucial real-world epidemiological features, including a saturated incidence rate to account for behavioral changes at high infection levels and a relapse mechanism for recovered individuals. A rigorous qualitative analysis is conducted to establish the fundamental properties of the model, wherein we prove the existence, uniqueness, positivity, and boundedness of the solutions, ensuring the biological viability of the system. The stability of the model's equilibria is thoroughly examined. We derive the basic reproduction number ( $R_0$) using a next-generation matrix method, which serves as the critical threshold for disease persistence. We prove that the disease-free equilibrium is both locally and globally asymptotically stable when $R_0<1$. For the numerical investigation, we implement and validate a robust predictor-corrector scheme tailored for fractional-order systems, comparing its accuracy and convergence against the fractional Euler method and the classical Runge-Kutta scheme. Extensive numerical simulations are performed to visualize the system's dynamics. The results demonstrate that the fractional order provides enhanced flexibility in capturing memory effects, often leading to a more realistic, delayed epidemic peak. Furthermore, a series of novel three-dimensional dynamic surface plots reveals the geometric sensitivity of the solution manifold to key infection parameters, visually confirming that the transmission rate ( $\beta$) and quarantine rate ( $\eta$) are the most powerful levers in shaping the epidemic landscape. These findings offer a deeper understanding of Lassa fever's dynamics and highlight critical points for effective public health interventions.