Brucellosis remains a significant public health and economic burden in many regions, primarily transmitted from livestock to humans through direct contact and environmental contamination. In this paper, we develop a novel cross-species epidemic model that couples the transmission dynamics of brucellosis among sheep, humans, and the environmental reservoir of Brucella. The sheep population is divided into susceptible, exposed, infectious, and vaccinated compartments, while the human population is stratified into susceptible and infected classes. Environmental brucella load is explicitly modeled, and distributed time delays are incorporated to account for incubation periods and delayed exposure risks in humans. We prove that all solutions are non-negative and ultimately bounded, ensuring biological consistency. The basic reproduction number R0 is derived using the next-generation matrix method. Using Lyapunov functionals and LaSalle’s invariance principle, we establish that the disease-free equilibrium is globally asymptotically stable when R0≤1, whereas a unique endemic equilibrium exists and is globally asymptotically stable when R0>1. Sensitivity analysis identifies the environmental transmission rate, shedding rate, and disinfection as the most influential parameters. Treatment efficacy is shown to exhibit a critical threshold pcr=1−1/R0, above which eradication becomes feasible. Numerical simulations validate the theoretical findings and demonstrate that time delays affect outbreak timing but not asymptotic stability. These results provide quantitative guidance for brucellosis control strategies, emphasizing environmental sanitation, culling, and vaccination as key interventions.
Alharbi et al. (2026) studied this question.