Hopf bifurcation and optimal control of a delayed reaction–diffusion brucellosis disease model

Abstract

This paper presents a brucellosis disease model with reaction–diffusion and time delay. The model takes into account both the direct and indirect transmission of infected animals and pathogens in the environment. By analyzing the associated characteristic equation, the local stability of the unique positive equilibrium point is established. The existence of Hopf bifurcations at the positive equilibrium point is also examined by considering the discrete time delay as a bifurcation parameter. Additionally, an optimal control analysis is conducted to minimize disease outbreaks and control costs. This includes reducing the exposure of susceptible animals to infected animals, removing infected animals from herds, and reducing emissions of brucella into the environment. By constructing Hamiltonian function and applying Pontryagin’s maximum principle, the necessary conditions for the existence of optimal control are given. Finally, the existence of bifurcation periodic solutions and the effectiveness of control strategies are illustrated through numerical simulations.