Stability Analysis of SIR Model with Distributed Delay on Complex Networks.

In this paper, by taking full consideration of distributed delay, demographics and contact heterogeneity of the individuals, we present a detailed analytical study of the Susceptible-Infected-Removed (SIR) epidemic model on complex population networks. The basic reproduction number [Formula: see text] of the model is dominated by the topology of the underlying network, the properties of individuals which include birth rate, death rate, removed rate and infected rate, and continuously distributed time delay. By constructing suitable Lyapunov functional and employing Kirchhoff's matrix tree theorem, we investigate the globally asymptotical stability of the disease-free and endemic equilibrium points. Specifically, the system shows threshold behaviors: if [Formula: see text], then the disease-free equilibrium is globally asymptotically stable, otherwise the endemic equilibrium is globally asymptotically stable. Furthermore, the obtained results show that SIR models with different types of delays have different converge time in the process of contagion: if [Formula: see text], then the system with distributed time delay stabilizes fastest; while [Formula: see text], the system with distributed time delay converges most slowly. The validness and effectiveness of these results are demonstrated through numerical simulations.