Necessary and sufficient conditions for near‐optimal controls of a stochastic West Nile virus system with spatial diffusion

Abstract

Near‐optimal controls are as meaningful as optimal controls given both theory and applications; however, they are usually much easier to be obtained than optimal ones, which are simply impossible to be obtained in many complicated systems. Therefore, this paper focuses on the near‐optimal control problems of a stochastic West Nile virus system with spatial diffusion describing the virus transmission among the bird, mosquito, and human populations. First, we introduce two control variables, namely, antimosquito and the treatment of the infected humans, into the system and prove the existence and uniqueness of the global positive solution of the system. Second, the necessary and sufficient conditions for two controls to be near‐optimal are acquired in terms of a Hamiltonian function and a small parameter ε$$ \varepsilon $$ based on the Ekeland principle, the adjoint equation, and some prior estimates. Finally, we use numerical simulations to illustrate the theoretical results and conclude that mosquito control is the most critical factor preventing West Nile virus.