There is an urgent need to study the dynamics of disease spreading as recently infectious diseases such as SARS, H1N1, and foot and mouth disease greatly affect people's daily life. Thus, we formulate a stochastic white Gaussian noise (SWGN) model by introducing the noise item and variant rates into a compartment model with a susceptibleinfective-hospitalized-recovered framework. Analytical results of the basic reproduction number, the maximum infectious population and the disease invasive influence are derived to investigate the dynamics of disease spreading. Furthermore, we apply a proposed random Runge-Kutta method to the model and link this to a case study of SARS outbreaks in Great Toronto Area (GTA) to study the impact of related parameters on the number of new reported emerging cases. Numerical results show that the introduction of white Gaussian noise and the variant rates makes the model fit with the real SARS data better than previous deterministic models. Both the theoretical results and numerical simulations provide instructive suggestions and feasible countermeasures for responses to disease propagation.