For a population dynamics system, studying its equilibrium points is the precondition for predicting the development trend of populations within the system.

Theorem 3.1 System (1) has two equilibrium points, E 0 = ( A μ , 0 , 0 , 0 ) for all parameter values and E * = ( S * , I * , Q * , R * ) for R 0 > 1 , here S * ∈ ( 0, A μ ) , I * , Q * and R * > 0 .

Proof. Summing up all the equations of model (1), we find the following differential equation: d N d t = A − μ N − μ 1 I − μ 2 Q . By comparison theorem, we obtain that the solutions of model (1) exist in the region defined by Γ = { ( S , I , Q , R ) ∈ ℝ + 4 : S + I + Q + R ≤ A μ , S ≥ 0 , I ≥ 0 , Q ≥ 0 , R ≥ 0 } . To get the equilibrium points, we set the right-side of equations to be 0,

{ A − μ S − β S I f ( S , I ) = 0 , β S I f ( S , I ) − ( δ + γ + μ + μ 1 ) I = 0 , δ I − ( ρ + μ + μ 2 ) Q = 0 , γ I + ρ Q − μ R = 0 , (4)

which yields

I = A − μ S δ + γ + μ + μ 1 ,   Q = δ ρ + μ + μ 2 I ,   R = ( γ μ + ρ δ μ ( ρ + μ + μ 2 ) ) I ,

β S f ( S , A − μ S δ + γ + μ + μ 1 ) = δ + γ + μ + μ 1 .

If I = 0 , the model (1) has a disease-free equilibrium E 0 = ( A μ , 0 , 0 , 0 ) for all parameter values. And we can get the basic reproduction number R 0 = β A ( μ + α 1 A ) ( δ + γ + μ + μ 1 ) by using next generation method. The value R 0 represents the average number of secondary infections when an infected person enters fully susceptible population. If I ≠ 0 , I = A − μ S δ + γ + μ + μ 1 > 0 implies S < A μ . Hence, there is no positive equilibrium point if S > A μ . Now, we consider the function g ( S ) defined on the interval [ 0, A μ ] , where

g ( S ) = β S f ( S , A − μ S δ + γ + μ + μ 1 ) − ( δ + γ + μ + μ 1 ) ≜ h ( S , I ) − ( δ + γ + μ + μ 1 ) .

Obviously, g ( 0 ) = − ( δ + γ + μ + μ 1 ) < 0 and g ( A μ ) = β A μ + α 1 A − ( δ + γ + μ + μ 1 ) = ( δ + γ + μ + μ 1 ) ( R 0 − 1 ) > 0 when R 0 > 1 . Simultaneously, differentiating the function g, we gain g ′ ( S ) = ∂ h ∂ S − μ δ + γ + μ + μ 1 ∂ h ∂ I > 0 . Because g ( S ) is monotonically increasing in the interval [ 0, A μ ] , g ( 0 ) < 0 and g ( A μ ) > 0 , the equation g ( S ) = 0 has only one positive root by the zero theorem. That is, there exists a unique endemic equilibrium E * = ( S * , I * , Q * , R * ) with S * ∈ ( 0, A μ ) .

In the biological sense, we analyze the stability of the disease-free equilibrium point and the endemic equilibrium point.

Theorem 3.2. The disease-free equilibrium E 0 of system (1) is globally asymptotically stable if R 0 < 1 and unstable if R 0 > 1 .

Proof. Consider the Jacobian matrix of system (1) at E 0

J ( E 0 ) = ( − μ − β A μ + α 1 A 0 0 0 β A μ + α 1 A − ( δ + γ + μ + μ 1 ) 0 0 0 δ − ( ρ + μ + μ 2 ) 0 0 γ ρ − μ ) .

The characteristic equation of system (1) at E 0 is

( λ + μ ) 2 [ λ + ( ρ + μ + μ 2 ) ] [ λ − β A μ + α 1 A + ( δ + γ + μ + μ 1 ) ] = 0.

Clearly, λ 1 , 2 = − μ < 0 , λ 3 = − ( ρ + μ + μ 2 ) < 0 and the positive and negative of the fourth eigenvalue depends on R 0 . That is, λ 4 = β A μ + α 1 A − ( δ + γ + μ + μ 1 ) = ( δ + γ + μ + μ 1 ) ( R 0 − 1 ) < 0 when R 0 < 1 , λ 4 > 0 when R 0 > 1 . Hence the disease-free equilibrium E 0 is locally asymptotically stable if R 0 < 1 and unstable if R 0 > 1 .

Then we prove the global stability of the system (1) at the equilibrium E 0 when R 0 < 1 . Taking the Lyapunov function W 1 ( t ) = I ( t ) into consideration, we get

W ˙ 1 = ( β S f ( S , I ) − ( δ + γ + μ + μ 1 ) ) I ≤ ( β A μ + α 1 A − ( δ + γ + μ + μ 1 ) ) I = ( δ + γ + μ + μ 1 ) ( R 0 − 1 ) I ≤ 0.

Thus if R 0 < 1 , W ˙ 1 ≤ 0 . And W ˙ 1 = 0 if and only if I = 0 . In this case, d S d t = A − μ S indicates S → A μ as t → ∞ . Similarly, Q → 0 and R → 0 as t → ∞ . So the largest positive invariant set in { ( S , I , Q , R ) ∈ Γ : W ˙ 1 = 0 } is the singleton E 0 . By Liapunov-Lasalle theorem, E 0 = ( A μ , 0 , 0 , 0 ) is globally asymptotically stable in Γ .

Theorem 3.3. If R 0 > 1 , the endemic equilibrium point E * of the system (1) is globally asymptotically stable in the region Ω = Γ − { ( S , I , Q , R ) ∈ Γ : I = 0 } .

Proof. Consider the Jacobian matrix of system (1) at E *

J ( E * ) = ( − μ − β I * + α 2 β I * 2 f 2 ( S * , I * ) − β S * + α 1 β S * 2 f 2 ( S * , I * ) 0 0 β I * + α 2 β I * 2 f 2 ( S * , I * ) β S * + α 1 β S * 2 f 2 ( S * , I * ) − ( δ + γ + μ + μ 1 ) 0 0 0 δ − ( ρ + μ + μ 2 ) 0 0 γ ρ − μ ) .

Let C 1 = β I * + α 2 β I * 2 f 2 ( S * , I * ) and C 2 = β S * + α 1 β S * 2 f 2 ( S * , I * ) , then

J ( E * ) = ( − μ − C 1 − C 2 0 0 C 1 C 2 − ( δ + γ + μ + μ 1 ) 0 0 0 δ − ( ρ + μ + μ 2 ) 0 0 γ ρ − μ ) .

Therefore the characteristic equation of system (1) at E * is

( λ + μ ) [ λ + ( ρ + μ + μ 2 ) ] { ( λ + μ + C 1 ) [ λ + ( δ + γ + μ + μ 1 ) − C 2 ] + C 1 C 2 } = 0.

Obviously, λ 1 = − μ < 0 , λ 2 = − ( ρ + μ + μ 2 ) < 0 and the other two eigenvalues are determined by the following quadratic equation

λ 2 + [ C 1 + μ + ( δ + γ + μ + μ 1 ) − C 2 ] λ   + ( C 1 + μ ) [ ( δ + γ + μ + μ 1 ) − C 2 ] + C 1 C 2 = 0

⇒ λ 2 + a 1 λ + a 2 = 0 ,

where

a 1 = C 1 + μ + [ ( δ + γ + μ + μ 1 ) − C 2 ] , a 2 = C 1 ( δ + γ + μ + μ 1 ) + μ [ ( δ + γ + μ + μ 1 ) − C 2 ] .

By utilizing Routh-Hurwitz criteria, we know that the system is stable if a 1 , a 2 > 0 and unstable if a 1 , a 2 < 0 . From the second equation of (4), we obtain ( δ + γ + μ + μ 1 ) > C 2 , thus all eigenvalues have negative real parts. The endemic equilibrium E * is locally asymptotically stable.

Now we confirm the global stability at the equilibrium E * when R 0 > 1 . The first two equations of system (1) do not contain Q and R, so we consider the following Lyapunov function in the positive quadrant of the two-dimensional plane SI.

W 2 ( t ) = S − S * − ∫ S * S l ( S * , I * ) l ( x , I * ) d x + I * Ψ ( I I * ) ,

where l ( S , I ) = β S f ( S , I ) , Ψ ( x ) = x − 1 − ln x , x > 0 . Clearly, attains its global minimum at and. Besides, the function has the global minimum at and. Then, for any and for any. Consequently, with equality holding if and only if for all. And, So the derivative function of is given by

Due to

we have, , thus and if and only if and. By the Liapunov-Lasalle theorem, all solutions starting in the positive quadrant of SI-plane with approach at. In this case, the differential equation for Q has the limiting equation which implies as, and similarly, the limiting equation for R is so that as. Therefore, by Lemma 2.3, the endemic equilibrium is globally asymptotically stable in the region for the system (1). This completes the proof.

In order to study the dynamics of stochastic models, the primary question to be considered is whether the solution is global and nonnegative existence. Although the coefficients of the model (2) satisfy the local Lipschitz condition, it’s not enough to prove that the solution does not explode within a finite time for any given initial value. Hence in this section, we will show that the solution of model (2) is positive and global.

Theorem 4.1. For any given initial value

, there exists a unique solution of system (2) on, which is in with probability one.

Proof. Since the system (2) has locally Lipschitz continuous coefficients, then for any initial value, system (2) has a unique local solution on, where is the explosion time. To verify this solution is global, we only need to show that. Now define the stopping time as

Set (denotes the empty set). Obviously, if we can show, then. Assume that this statement is false, then there exists a constant such that.

Define a -function by

Applying Itô’s formula, for all, we obtain

where

Since for all, and, we have

Hence,

From the definition of, it follows that. Therefore,

Letting in (5) and then taking the expectation on both sides of (5), we have that

which is a contradiction and we confirmed. This completes the proof.

Remark 4.2. The region is almost surely positive invariant of stochastic model (2), refer to [12] . In addition, from biological consideration, we next focus on the disease dynamics of model (2) in the bounded set.

One of the most concerning issues in epidemiology is how to establish the threshold condition for the extinction and persistence of the disease. The target of this section is to study the extinction and persistence of the disease. First of all, we define corresponding random threshold as follows:

Theorem 4.3. Let be a solution of system (2) for any given initial value.

1) If and, then

2) If, then

which means that tends to zero exponentially a.s., i.e. the disease dies out with probability 1. Furthermore,

Proof. Define Lyapunov function, by Itô’s formula, we get that

where.

Suppose 1) holds. Noting that is monotone increasing for and, we have

Then,

Integrating both sides of the above inequality from 0 to t and dividing by t, we obtain

where. By the strong law of large numbers for local martingales [17] , we derive that. Since, Equation (10) becomes

We obtain the desired assertion (6).

If 2) holds, from Equation (9), we get

Since, Equation (11) becomes

We obtain the desired assertion (7). And so

From the first two equations of system (2), there is

Integrating both sides of (13) from 0 to t and dividing by t, we have

Therefore,

where. Clearly, and from (12), we have

Therefore the assertion (8) holds. The conclusion is proven.

Next, the conditions for the persistence of the disease are presented.

Theorem 4.4. Suppose that, then the solution of system (2) is persistent in the mean for any given initial value. Moreover,

where

Proof. Since, we have

Then

Integrating both sides of (19) from 0 to t, there is

From (14), we have

where. Obviously,

. By Lemma 2.2 and, we deduce that

This is the required inequality (15), and from (14), we have

Therefore,

the inequality (16) is valid. From the third equation of system (2), we have

Then,

where. It follows from the strong law of large numbers for local martingales that, hence (17) holds for

The last equation of system (2) gives

Then,

So we have

This is the required inequality (18).

In this section, we numerically simulate solutions of the models by using the Milstein’s method [18] to confirm main results. We compare the threshold parameters of the deterministic model and stochastic model to illustrate the effect of white noise on the system. The model (2) can be rewritten as the following discrete equation:

where and, are the Gaussian random variables. Similarly, the model (1) can also be written in the above form. We just need to delete the disturbance term and will not repeat it here.

Example 4.3.1 For the deterministic system (1), we choose the initial value and the parameter values, , , , , , , , , ,. By Matlab software, we get and find that the class I, Q and R tend to 0, which means that the disease dies out (see Figure 1(a)). Theorem 3.2 is illustrated. Then, let and, , , , , , , , , , to draw Figure 1(b). It shows that the disease becomes endemic and. The condition of theorem 3.3 is satisfied.

Example 4.3.2 For the stochastic system (2), we choose the initial value and the parameter values, , , , , , , , , , , , ,. By calculation, and. Hence, the condition 1) of theorem 4.3 is satisfied. In Figure 2(a), the class I exponentially decays to zero which indicates the extinction of the disease. Next, we let parameter and others are the same as above. In this case,. Therefore, the condition 2) of theorem 4.3 is satisfied and the disease dies out (Figure 2(b)). Finally, let, , , , and keep the other parameters, we get. According to theorem 4.4, all classes of the system (2) are persistent and are shown in Figure 2(c).

Example 4.3.3 Now, we reselect the parameters , , , , , , , , , , , , , , and give a set of comparison charts of simulation results. In Figure 3, , the class S, I, Q and R of deterministic model all exist, which means that the disease break out, but after adding white noise, except for the class S, the others tend to be 0. It reveals that the random fluctuations can suppress disease prevail.