Sufficient and necessary conditions of near-optimal controls for a stochastic listeriosis model with spatial diffusion

Random environment and human activities have important effects on the survival of listeria. In this paper, treating infected people and removing bacteria from the environment as control strategies, we developed a listeriosis model that considers random noise and spatial diffusion. By constructing a Lyapunov function, we demonstrated the existence and uniqueness of the global positive solution of the model. However, it was a challenging task to realize the optimal control of the model by solving the Pontryagin random maximum principle with the lowest control cost. Therefore, our study on near-optimal controls is of great significance for controlling the spread of listeriosis. Initially, we gave some adjoint equations and a priori estimates. Subsequently, the Pontryagin random maximum principle was utilized to establish the sufficient and necessary conditions for achieving near-optimal controls. Ultimately, the theoretical findings are corroborated through numerical analysis.