Stability analysis and optimal control of epidemic-like transmission model with nonlinear inhibition mechanism and time delay in both homogeneous and heterogeneous networks

Abstract

In this paper, considering the bilingual environment, we establish the 2SIR rumor models with a nonlinear inhibition mechanism and time delay in both homogeneous networks and heterogeneous networks. In the homogeneous network model, we prove the non-negativity of the solutions for the system and obtain the basic reproduction number R01 through the next-generation matrix. Then we discuss the local and global stability of the equilibrium point by means of analyzing the characteristic equation of linearized system and Lyapunov functional. In the heterogeneous model, based on the analysis of the existence of the rumor-spreading equilibrium points, we derive the basic reproduction numbers R02. Besides, by utilizing the linearized system and LaSalle's invariance principle, the stability of the rumor-free equilibrium point was proved. What's more, the system exhibits bifurcation behavior at R02=1 by choosing a system parameter as the bifurcation parameter. Additionally, optimal control problems in both homogeneous and heterogeneous networks with three control measures are discussed. Moreover, numerical simulations are carried out to verify the correctness of theoretical results.