Sliding dynamics and optimal control of avian influenza model with saturated incidence rate

Abstract

In this paper, we develop a susceptible‐infective‐susceptible‐infected‐recovered avian influenza model with saturated incidence rate and three control measures, which is proposed to evaluate the comprehensive effects of different control measures on the spread of avian influenza. Based on the threshold policy, the threshold level is used as the reference index of whether to take control measures. No control measures will be taken when the total number of infected birds and humans is less than the threshold level. Otherwise, the three control measures will be implemented simultaneously to mitigate the spread of the disease. Furthermore, by using the Utkin equivalent control method, the existence of sliding mode equation and its dynamics are investigated. Additionally, the method of dynamic programming is introduced to study the optimal control for the model. Through constructing and proving the existence of viscosity solution of the corresponding Hamilton–Jacobi–Bellman equation, the optimal control pair is further obtained. Finally, numerical simulations are performed to illustrate the corresponding theoretical analysis results.