The Stability of SI Epidemic Model in Complex Networks with Stochastic Perturbation

Abstract

We investigate a stochastic SI epidemic model in the complex networks. We show that this model has a unique global positive solution. Then we consider the asymptotic behavior of the model around the disease-free equilibrium and show that the solution will oscillate around the disease-free equilibrium of deterministic system when $R_0\le 1$. Furthermore, we derive that the disease will be persistent when $R_0>1$. Finally, a series of numerical simulations are presented to illustrate our mathematical findings. A new result is given such that, when $R_0\le 1$, with the increase of noise intensity the solution of stochastic system converging to the disease-free equilibrium is faster than that of the deterministic system.