Hybrid multi-step fractional numerical schemes for human-wildlife zoonotic disease dynamics

In this study, the transmission dynamics of zoonotic diseases between baboons and humans were explored by examining increased interactions between humans and wild animals. We established the model's well-posedness through proofs of existence, uniqueness, non-negativity, and boundedness of solutions. Stability and sensitivity analyses identified key parameters affecting disease dynamics, particularly the baboon-to-human transmission rate $ (\beta_h) $, the human recovery rate $ (\gamma_h) $, and the human-side contact control parameter $ (H_i) $. The basic reproduction number $ (R_0) $ governed disease outcomes: If $ R_0 < 1 $, the disease died out and the infection-free equilibrium was globally asymptotically stable; if $ R_0 > 1 $, a unique endemic equilibrium emerged and was locally asymptotically stable, indicating the potential for disease persistence. Numerical simulations were conducted using the Multi-Step Generalized Differential Transform Method and the Adams-Bashforth-Moulton scheme, confirming the model's biological relevance. Our results indicated that sterilization reduced infected baboons by up to 40%, while food access restrictions lowered human infections by approximately 25%. By leveraging fractional calculus and advanced numerical methods, this study provides a robust framework for modeling zoonotic diseases and offers actionable insights for public health and wildlife management.