Stability analysis and optimal control in an epidemic model on directed complex networks with nonlinear incidence

In this paper, stability and optimal control for SIS epidemic systems with birth and death in directed complex networks are investigated. A class of epidemic systems are constructed based on SIS models combining with two kinds of topological characters as in-degree and out-degree in directed networks, in which the nonlinear incidence rate is introduced that relevant to network’s topology containing the psychological effect coefficient. The disease-free and the endemic equilibria are derived and the epidemic threshold for our systems is obtained. Then, the local stability about the disease-free and the endemic equilibria are discussed via the corresponding Jacobian matrices and the characteristic equation. In addition, global asymptotical stability about two kinds of equilibria are also considered based on the Lyapunov function and stable analysis method. From the view of Pontryagin Minimum Principle, the optimal control issue about the system is also considered, and the corresponding optimal control systems and optimal control tactics are obtained. Several numerical examples are provided to demonstrate the effectiveness of our main method. Meanwhile, by means of the Forward–Backward Sweep method, optimal control systems are solved in the aspect of numeric and the control aim under the derived strategies is achieved.