### Near-optimal control and threshold behavior of an avian influenza model with spatial diffusion on complex networks

Near-optimization is as sensible and important as optimization for both theory and applications. This paper concerns the near-optimal control of an avian influenza model with saturation on heterogeneous complex networks. Firstly, the basic reproduction number $ \mathcal{R}_{0} $ is defined for the model, which can be used to govern the threshold dynamics of influenza disease. Secondly, the near-optimal control problem was formulated by slaughtering poultry and treating infected humans while keeping the loss and cost to a minimum. Thanks to the maximum condition of the Hamiltonian function and the Ekeland´s variational principle, we establish both necessary and sufficient conditions for the near-optimality by several delicate estimates for the state and adjoint processes. Finally, a number of examples presented to illustrate our theoretical results.