Dynamical analysis of a fractional SIR model with birth and death on heterogeneous complex networks

•A fractional order SIR model with birth and death on complex networks is regarded.•A threshold value for the transmission rate with or without immunization schemes is obtained.•The criteria of global stability of the equilibrium points are obtained.•Numerical simulations were performed to illustrate the analysis results.

In this paper, a fractional SIR model with birth and death rates on heterogeneous complex networks is proposed. Firstly, we obtain a threshold value R0R0 based on the existence of endemic equilibrium point E∗E∗, which completely determines the dynamics of the model. Secondly, by using Lyapunov function and Kirchhoff’s matrix tree theorem, the globally asymptotical stability of the disease-free equilibrium point E0E0 and the endemic equilibrium point E∗E∗ of the model are investigated. That is, when R0<1R0<1, the disease-free equilibrium point E0E0 is globally asymptotically stable and the disease always dies out; when R0>1R0>1, the disease-free equilibrium point E0E0 becomes unstable and in the meantime there exists a unique endemic equilibrium point E∗E∗, which is globally asymptotically stable and the disease is uniformly persistent. Finally, the effects of various immunization schemes are studied and compared. Numerical simulations are given to demonstrate the main results.