Epidemic spreading with time delay on complex networks

In this paper, we propose a susceptible-infected-susceptible (SIS) model on complex networks, small-world (WS) networks and scale-free (SF) networks, to study the epidemic spreading behavior with time delay which is added into the infected phase. Considering the uniform delay, the basic reproduction number R ₀ on WS networks and (bar R_0) on SF networks are obtained respectively. On WS networks, if R ₀ ≤ 1, there is a disease-free equilibrium and it is locally asymptotically stable; if R ₀ > 1, there is an epidemic equilibrium and it is locally asymptotically stable. On SF networks, if (bar R_0 leqslant 1), there is a disease-free equilibrium; if (bar R_0 > 1), there is an epidemic equilibrium. Finally, we carry out simulations to verify the conclusions and analyze the effect of the time delay τ, the effective rate λ, average connectivity 〈k〉 and the minimum connectivity m on the epidemic spreading.