In this paper, a delayed virus model with two different transmission methods and treatments is investigated. This model is a time-delayed version of the model in (Zhang et al. in Comput. Math. Methods Med. 2015:758362, 2015). We show that the virus-free equilibrium is locally asymptotically stable if the basic reproduction number is smaller than one, and by regarding the time delay as a bifurcation parameter, the existence of local Hopf bifurcation is investigated. The results show that time delay can change the stability of the endemic equilibrium. Finally, we give some numerical simulations to illustrate the theoretical findings.

In this paper, a stochastic SIS epidemic model with nonlinear incidence rate and double epidemic hypothesis is proposed and analysed. We explain the effects of stochastic disturbance on disease transmission. To this end, firstly, we investigated the dynamic properties of the system neglecting stochastic disturbance and obtained the threshold and the conditions for the extinction and the permanence of two kinds of epidemic diseases by considering the stability of the equilibria of the deterministic system. Secondly, we paid prime attention on the threshold dynamics of the stochastic system and established the conditions for the extinction and the permanence of two kinds of epidemic diseases. We found that there exists a significant difference between the threshold of the deterministic system and that of the stochastic system. Moreover, it has been established that the persistent of infectious disease analysed by use of deterministic system becomes extinct under the same conditions due to the stochastic disturbance. This implies that a stochastic disturbance has significant impact on the spread of infectious diseases and the larger stochastic disturbance leads to control the epidemic diseases. In order to illustrate the dynamic difference between the deterministic system and the stochastic system, there have been given a series of numerical simulations by using different noise disturbance coefficients.