Dynamical modeling and optimal control of brucellosis with mixed delays and environmental transmission

Brucellosis, a persistent zoonotic disease, presents significant challenges in understanding its transmission dynamics and in designing effective control strategies due to the complex interaction of biological and environmental factors. In this study, we develop a delayed differential equation model incorporating discrete time delays to represent both disease latency and the delayed effects of interventions, as well as environmental transmission mechanisms. These elements allow for a more accurate representation of the epidemiological processes that sustain brucellosis in both livestock and human populations. We derive threshold conditions for controlling disease prevalence and verify the existence of equilibrium points, analyzing their global asymptotic stability using Lyapunov functions. A key contribution of this work is the formulation and analysis of a mixed state-control delay optimal control problem, which explicitly considers the impact of delays on both state variables and control measures. This framework provides new insights into the timing of interventions and the optimal allocation of resources. Optimal strategies, including vaccination, culling, environmental disinfection and public awareness campaigns, are derived using the delay Pontryagin maximum principle. The study balances the direct costs of interventions with the indirect economic and societal impacts of the disease, offering actionable, cost-effective recommendations. Sensitivity analysis identifies key parameters influencing transmission and control effectiveness. The model is validated with epidemiological data from Inner Mongolia, demonstrating its applicability to endemic settings. Our findings highlight the critical role of mixed state-control delays in shaping optimal control strategies, showing that neglecting these factors can lead to ineffective or suboptimal results. This research advances both the theoretical and practical understanding of mixed delay optimal control problems, offering a valuable framework for managing brucellosis and other delay-sensitive infectious diseases.