A stochastic reaction–diffusion avian influenza model with fast and slow time scales: Near-optimality of control strategies

The growth rates of avian and human populations are remarkably different on time scales pertaining to different biological characteristics, and the spread of avian influenza virus is often affected by external environmental perturbations and spatial heterogeneity. In this paper, we formulate a stochastic partial differential equation for avian influenza model with fast and slow time scales. The existence and uniqueness of the positive solutions are proved by using the Yamada–Watanabe approximation principle. The existence and uniqueness of the invariant measure within avian sub-system are demonstrated by applying the averaging principle and ergodic property. By using relaxed control expression along with martingale technique, the human subsystem’s weak convergence is proved, which means that the avian subsystem weakly converges to the solutions of a corresponding averaged equation. Furthermore, by implementing poultry culling and medical treatment of infected cases as control strategies, the near-optimality of control measures of the multiscale stochastic system is obtained by establishing the near-optimality of control measures of the average equations for human subsystem. Finally, numerical simulations are given to prove and understand our theoretical results.