The dynamics and near‐optimal controls of a dengue model with threshold policy

In this paper, a piecewise smooth dengue model with threshold policy is developed, and the impacts of different control measures on the spread of dengue fever are also investigated. The number of dengue infections is used as a threshold level Ic$$ {I}_c $$ to determine whether to implement control, and control measures are triggered only if the number of infected individuals exceeds Ic$$ {I}_c $$. Using the Routh–Hurwitz criterion, the dynamic behaviors of the free system and control system are studied. Then, the existence of the sliding mode is verified, and the sliding dynamics are analyzed by using the Utkin equivalent control method. It is shown that model solutions eventually converge to one of two endemic equilibria or the sliding equilibrium depending on Ic$$ {I}_c $$. In addition, by using Ekeland's principle and maximum principle, the sufficient and necessary conditions for near‐optimal controls of this model are obtained. Finally, numerical simulations are carried out to explain and supplement the theoretical results.