Optimal vaccination and treatment of an epidemic network model

•Propose an optimally controlled SIRS epidemic model on heterogeneous networks.•Obtain criteria of global stability of the disease-free equilibrium and the endemic equilibrium.•Investigate existence of optimal control for the control problem.•The results be illustrated by some numerical simulations.

In this Letter, we firstly propose an epidemic network model incorporating two controls which are vaccination and treatment. For the constant controls, by using Lyapunov function, global stability of the disease-free equilibrium and the endemic equilibrium of the model is investigated. For the non-constant controls, by using the optimal control strategy, we discuss an optimal strategy to minimize the total number of the infected and the cost associated with vaccination and treatment. Table 1 and Figs. 1–5 are presented to show the global stability and the efficiency of this optimal control.