Optimal control of an avian influenza model with multiple time delays in state and control variables

In this paper, we consider an optimal control model governed by a class of delay differential equation, which describe the spread of avian influenza virus from the poultry to human. We take three control variables into the optimal control model, namely: slaughtering to the susceptible and infected poultry ( \begin{document}$ u_{1}(t) $\end{document} ), educational campaign to the susceptible human population ( \begin{document}$ u_{2}(t) $\end{document} ) and treatment to infected population ( \begin{document}$ u_{3}(t) $\end{document} ). The model involves two time delays that stand for the incubation periods of avian influenza virus in the infective poultry and human populations. We derive first order necessary conditions for existence of the optimal control and perform several numerical simulations. Numerical results show that different control strategies have different effects on controlling the outbreak of avian influenza. At the same time, we discuss the influence of time delays on objective function and conclude that the spread of avian influenza will slow down as the time delays increase.