LAUSR

Considering that environmental factors, diet, subconscious mind and other uncertainties play an important role in the process of delaying and treating diseases, we propose, in this paper, an amended Hepatitis B virus (HBV) model with stochastic perturbation, and investigate the longtime dynamics of this stochastic model. First, if the basic reproductive number of the corresponding deterministic model is less than 1, some sufficient conditions for almost surely exponentially stable in the sense of the infected cells and free virus are established, and the stationary probability density function of the uninfected sell is also obtained. Further, some sufficient conditions for the existence of the stationary distribution are obtained for the basic reproductive number more than 1. In addition, oscillatory behaviors of this model about the equilibrium of the corresponding deterministic model are discussed. Finally, numerical simulations demonstrate the main theoretical results and show stochastic virus model has more dynamic behaviors relative to its corresponding deterministic model. Theoretical results and numerical simulations imply that the intensity and “type (divided into positive and negative)” of white noise play very important roles in the treatment of infectious disease, which can make the disease more and more repetitive and unpredictable. Of course, comfortable environment, reasonable diet, optimistic mood and other positive uncertainty factors have active effects on the treatment and delaying of diseases, but not the converse.