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

In this paper, a fractional SIR model with birth and death rates on heterogeneous complex networks is proposed. Firstly, we obtain a threshold value R 0 based on the existence of endemic equilibrium point 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 E 0 and the endemic equilibrium point E ∗ of the model are investigated. That is, when R 0 1 , the disease-free equilibrium point E 0 is globally asymptotically stable and the disease always dies out; when R 0 > 1 , the disease-free equilibrium point E 0 becomes unstable and in the meantime there exists a unique endemic equilibrium point 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.