One intervention procedure to control the spread of infectious diseases is to isolate some infectives, in order to reduce transmissions of the infection to susceptibles. Isolation may have been the first infection control method. Over the centuries quarantine has been used to reduce the transmission of human diseases such as leprosy, plague, cholera, typhus, yellow fever, smallpox, diphtheria, tuberculosis, measles, mumps, ebola, and lassa fever. Quarantine has also been used for animal diseases such as rinderpest, foot and mouth, psittacosis, Newcastle disease, and rabies. Studies of epidemic models with quarantine have become an important area in the mathematical theory of epidemiology, and they have largely been inspired by Refs. [1–4].
Hethcote et al. in  discuss an SIQS epidemic model:
where parameters A, μ, and β are positive constants, and α, γ, ε, and δ are non-negative constants. Here denotes the number of members who are susceptible to an infection at time t. denotes the number of members who are infective at time t. denotes the number of members who are removed and isolated either voluntarily or coercively from the infectious class. The parameters in the model are summarized in the following list:
A: the recruitment rate of susceptibles corresponding to births and immigration;
β: transmission coefficient between compartments S and I;
μ: the per capita natural mortality rate;
δ: the rate for individuals leaving the infective compartment I for the quarantine compartment Q;
γ: recovery rate of infectious individuals;
ε: the rate at which individuals return to susceptible compartment S from compartments Q;
α: disease-caused death rate of infectious individuals.
In a simple epidemic model, there is generally the basic reproduction number (or the threshold) . If , the disease-free equilibrium is a unique equilibrium in this type of epidemic model and it is globally asymptotically stable; if , this type of model has also a unique endemic equilibrium, which is globally asymptotically stable. The threshold of system (1.1) is
In , the system (1.1) always has the disease-free equilibrium . If , then is the unique equilibrium of (1.1) and it is globally stable in invariant set Γ, where
If , then is unstable and there is an endemic equilibrium
which is globally asymptotically stable under a sufficient condition in invariant set Γ.
In fact, epidemic models are inevitably affected by environmental white noise which is an important component in realism, because it can provide an additional degree of realism in comparison to their deterministic counterparts. Many stochastic models for epidemic populations have been developed in [6–17]. Dalal et al.  have previously used the technique of parameter perturbation to examine the effect of environmental stochasticity in a model of AIDS and condom use. They found that the introduction of stochastic noise changes the basic reproduction number of the disease and can stabilize an otherwise unstable system. Tornatore et al.  propose a stochastic disease model where vaccination is included. They prove existence, uniqueness, and positivity of the solution and the stability of the disease-free equilibrium. Zhao et al.  discuss the dynamics of a stochastic SIS epidemic model with vaccination. They obtain the condition of the disease extinction and persistence according to the threshold of the deterministic system and the noise. Ji et al.  discuss the SDE SIR model with no delay. They obtain if , then the disease-free equilibrium is stochastically asymptotically stable in the large and exponentially mean-square stable. If , then the solution of the system is fluctuating around , which is the endemic equilibrium of the corresponding deterministic system. The disease will prevail if the white noise is small.
However, compared to deterministic systems, it is difficult to give the threshold of stochastic systems. Recently, Gray et al. in  investigate the stochastic SIS epidemic model with fluctuations around the transmission coefficient β. They prove that this model has a unique global positive solution and establish conditions for extinction and persistence of according to the threshold of the stochastic model. In the case of persistence they show the existence of a stationary distribution and derive expressions for its mean and variance. Zhao and Jiang  continue to discuss the stochastic SIS epidemic model with vaccination. When the noise is small, they obtain a threshold of the stochastic system which determines the extinction and persistence of the epidemic. Besides, they find that large noise will suppress the epidemic from prevailing.
In this paper, taking into account the effect of randomly fluctuating environment, we assume that fluctuations in the environment will manifest themselves mainly as fluctuations in the parameter β,
where is standard Brownian motions with , and with intensity of white noise . The stochastic version corresponding to the deterministic model (1.1) takes the following form:
This paper is organized as follows. In Section 2, we show there is a unique positive solution of system (1.3). In Section 3, we investigate system (1.3) is exponential stability when the noise is large. In this case, the infective decays exponentially to zero. When the noise is small, we deduce the condition which will enable the disease to die out exponentially in Section 3 and the condition for the disease being persistent is given in Sections 4. Throughout the paper, outcomes of numerical simulations are reported to support the analytical results.
Next, we give some basic theory in stochastic differential equations (see ).
Throughout this paper, unless otherwise specified, let be a complete probability space with a filtration satisfying the usual conditions (i.e. it is right continuous and contains all P-null sets). Let
In general, consider the n-dimensional stochastic differential equation
with initial value . denotes n dimensional standard Brownian motion defined on the above probability space. Define the differential operator L associated with (1.4) by
If L acts on a function , then
where , , . By Itô’s formula, if , then
Consider (1.4), assume and for all . So is a solution of (1.4), called the trivial solution or equilibrium position.