Epidemics exert considerable influence on human life. Controlling and eradicating infectious diseases are ongoing problems that have received increasing attention from numerous authors [1] [2] [3] [4] [5] [6] . Various factors, such as vaccination, time delay, impulse and so on, are used to construct mathematical models and seek effective ways to eliminate infectious diseases. Time delay, which has great biologic meaning in epidemic systems, is often used [7] [8] [9] [10] . Many scholars have also paid close attention to the effects of the temporary disease immunity of epidemic models, i.e., a fleeting immunity to a disease after recovery before becoming susceptible again. The phenomenon is common during the transmission of many epidemic diseases, such as influenza, Chlamydia trachomatis, Salmonella and so on. Thus, in current paper, we introduce temporal delays to make epidemic models more realistic and interesting.
Epidemic models are inevitably subject to environmental noise. Most epidemic models are driven by white noise, and many results have been achieved in this area [11] [12] [13] [14] [15] [16] [17] [18] [19] . However, under severe environmental perturbations, such as avian influenza, severe acute respiratory syndrome, volcanic eruptions, earthquakes, and hurricanes, the continuity of solutions may be broken; accordingly, a jump process should be introduced to prevent and control diseases [20] [21] [22] [23] .
In the present work, we consider the effects of vaccination and temporary immunity on a stochastic susceptibleinfectious-recovered (SIR) epidemic model driven by Lévy noise. A generalized nonlinear incidence rate f (S(t))g(I(t)) is also introduced. Based on the above factors, we formulate the delayed vaccinated SIR epidemic model as follows:
( Λ − µS(t) − pS(t) − βf (S(t))g(I(t)) + pS(t − τ 1 )e −µτ 1 + γ I(t − τ 2 )e −µτ 2 ) dt − f (S(t − ))g(I(t − ))(σ 1 dB 1 (t) + ∫ Y γ (u)Ñ(dt, du)), dI(t) = ( βf (S(t))g(I(t)) − (µ + γ )I(t) ) dt + f (S(t − ))g(I(t − ))(σ 1 dB 1 (t) + ∫ Y γ (u)Ñ(dt, du)),
where S(t), I(t), and R(t) are the numbers of susceptible, infectious, and recovered populations, respectively. Λ represents the constant recruitment of susceptible individuals, µ is the natural death rate of populations, β is the transmission rate, γ represents the recovery rate, p denotes the proportional coefficient of the vaccinated for the susceptible, τ 1 represents the validity period of the vaccination and τ 2 is the length of the immunity period. All parameters are assumed to be positive constants.
Herein, σ 2 i (t) (i = 1, 2) is the intensity of white noise and σ i > 0 (i = 1, 2); and B i (t) (i = 1, 2) is the standard Brownian motions, which are defined on a complete probability space (Ω, F , P) with filtration {F t } t∈R+ satisfying the usual conditions [21] . N is a Poisson counting measure with compensatorÑ and characteristic measure λ on a measurable subset Y of (0, ∞) which satisfies λ(Y) < ∞; λ is assumed to be a lévy measure, such thatÑ(dt, du) = N(dt, du) − λÑ(du)dt; γ : Y × Ω → R is bounded and continuous with respect to λ and is B(Y) × F t -measurable, where B(Y) is a σ -algebra with respect to the set Y, F t is a sub σ -algebra of F , and F is a σ -algebra of subsets of a given set Ω [22] . In this paper, B i and N are assumed to be independent of each other.
As the first two equations do not depend on the last equation in system (1.1), therefore, we only consider the equations as follows:
(1.
2)
The initial conditions of model (1.2) are 
According to the phenomena observed in nature, such as bee colonies, we assume that the self-regulating competitions within the same species are strictly positive, that is,
We also establish the following assumptions on functions f (S) and g(I):
Assumption (H2) f (S) is a continuously differentiable function and monotonically increasing on R + , f (0) = 0. A constant l > 0 exists, such that m l ≜ inf 0<S≤l
Assumption (H3) g(I) is twice continuously differentiable, and
I is monotone decreasing on R + , g(0) = 0, and g ′ (0) > 0.
The aim of this paper is to prove the existence and uniqueness of a global positive solution. The extinction and persistence in the mean of the system are also discussed.
In this section, we list some notations, definitions and lemmas. First, we denote
where f (t) is a continuous and bounded function defined on [0, +∞).
Then, we give the Itô's formula for general stochastic differential equations. Define the n-dimensional stochastic equation [24] :
with initial value X (t 0 ) = X 0 . Here, f (t, X ) = (f 1 (t, X ), f 2 (t, X ), . . . , f n (t, X )) is a n-dimensional vector function, (g(t, X )) n×l is a n × l matrix function and B 1 (t) = (B 1 (t), B 2 (t), . . . , B l (t)) is a l−dimensional standard Brownian motion defined on the probability space (Ω, F , P). Define the differential operator L associated with Eq. (2.1)
If function V (t, X ) ∈ C 2,1 (R n × R; R), then we have
Thus, the Itô's formula is listed as follows Lemma 2.1 ([24] ). Let X (t) satisfy Eq. (2.1) and function V (t, X ) ∈ C 2,1 (R n × R; R). Then
, . . . , ∂V (t,X ) ∂xn ).
(2) If positive constants λ 0 , T and λ ≥ 0 exist such that
for all t ≥ T , then, ⟨x⟩ * ≥ λ/λ 0 a.s. Proof. According to the local Lipschitz condition of system (1.2), we obtain that for any initial value X 0 = (S(0), I(0)) ∈ R 2 + , a unique local solution (S(t), I(t)) exists on [−τ , τ e ), herein, τ e represents the explosion time. To prove that the solution is global, it is required to obtain that τ e = ∞ a.s. Then, we suppose that k 0 ≥ 1 is sufficiently large such that S(0) and I(0) lie within the interval [1/k 0 , k 0 ]. For each integer k > k 0 , we define the stopping time τ k = inf{t ∈ [−τ , τ e ] : S(t)̸ ∈(1/k, k), or I(t)̸ ∈(1/k, k)}. Then, τ k increases as k → ∞. Denote τ ∞ = lim k→+∞ τ k , thus τ ∞ ≤ τ e . In the following, we need to show that τ ∞ = ∞. If not, there are constants T > 0 and ε ∈ (0, 1) satisfying P{τ ∞ < ∞} > ε. Thus, an integer
where a is a constant that will be given later. By virtue of Itô's formula, we derive
Here, LV :
. On the other hand, we achieve that
Thus,
Then applying Taylor formula to function ln(1 − t) here t = f (S) S g(I)γ (u) and Assumption (H1) to ϕ 1 , we have that
Here δ ∈ (0, 1) is an arbitrary number. Similarly, we can obtain that
Therefore, we achieve that
Taking integral on the above inequality from 0 to τ k ∧ T , then
Let Ω k = {τ k ≤ T }, then P(Ω k ) ≥ ε. For each ω ∈ Ω k , S(τ k , ω), or I(τ k , ω) equals either k or 1/k, and
Thus,
where 1 Ω k is the indicator function of Ω k . Letting k → ∞, we obtain the contradiction.
The proof is completed.
Next, we prove that Γ is a positively invariant set of system (1.2). 
, I(t)) / ∈ Γ }. We need to show that P(τ < t) = 0 for all t > 0.
Notice that P(τ < t) ≤ P(τ n < t), then we have to prove lim sup n→+∞ P(τ n < t) = 0. Define the function 
Taking integral and expectation on both sides of (3.7) and by virtue of Fubini Theorem, then we derive
Applying Gronwall Lemma, we obtain that
for all s ∈ [0, t ∧ τ n ]. Thus, E(W (X (t ∧ τ n ))) ≤ W (X 0 )e η(t∧τn) ≤ W (X 0 )e ηt , t ≥ 0.
(3.8)
In consideration of W (X (t ∧ τ n )) > 0 and some component of X (τ n ) being less than or equal to 1 n , we achieve that
(3.9)
By (3.8) and (3.9), we obtain that
for all t ≥ 0. Therefore,
The proof is completed.
In this section, to discuss the extinction of the disease, we define
and for the sake of simplicity, we denote ⟨x(t)⟩ = 1 t ∫ t 0 x(s)ds, then the following theorem is obtained. 
Proof. Applying Itô's formula, we derive that
g(I(t)) 
on the other hand, we have then,
Therefore, (4.5) and here, φ(t) =
According to (4.5), we obtain that
In addition,
Then, we have that By virtue of the condition (2) and (4.6), we achieve that
Moreover, by (4.6), we have that
According to the condition (1) and (4.8), we obtain that lim sup
That is lim t→∞ I(t) = 0. Moreover, we have that
The conclusion is proven.
2(µ+γ )(µ+p(1−e −µτ 1 )) 2 hold, then for an arbitrary solution (S(t), I(t)) of system (1.2), we have lim t→∞ I(t) = 0, which implies that the disease is extinct.
Now we are in the position to discuss the persistence in the mean of the disease and before that some notations are presented in the following.
For the convenience, we denotê
) , 
Proof. Since lim I→0 g(I)
, then a constant ε > 0 exists satisfying g(I) > (g ′ (0) − ε)I for all 0 < I ≤ ε. By virtue of (4.6), we have that
Considering lim t→∞ F (t) t = 0, then for an arbitrary ζ > 0, there exists a T 1 = T 1 (ω) > 0 and a set Ω k such that F (t) t ≤ ζ and P(Ω k ) ≥ 1 − ζ for all t ≥ T 1 , ω ∈ Ω k . LetT = max{T , T 1 }, then according to Lemma 2.2 and Theorem 3 in Ref. [22] , we achieve that
On the other hand, by (4.2) and (4.5), we obtain that
As 0 < S + I ≤ N 0 , then we derive that −∞ < ln I(t) < ln(N 0 ). Thus,
By virtue of the conclusion lim t→∞ φ(t) = 0 and the arbitrariness of ε, we obtain that
In addition, by virtue of (4.5) and (5.6), we have that
In the following , we prove lim inf t→∞ ⟨S(t)⟩ ≥ I * .
By assumptions (H2) and (H3), a constant T 2 ≥ T 1 > 0 exists satisfying
for all t ≥ T 2 .
Using the first equation of model (1.2), we achieve that 
The desired result is obtained.
Remark 2. From Theorem 5.1, we obtain that ifR 0 > 1 +σ 1 2 Λ 2 2µ 2 (µ+γ ) , and choose A 1 = min{I * ,Î * }, then the solution (S(t), I(t)) of system (1.2) with an initial condition (1.3) is persistent in the mean. Moreover, denote A 2 = max{I * ,Ĩ * }, we can also obtain the condition for the permanence in the mean of the system, that is,
In this section, we will perform some numerical simulations to illustrate our theoretical results by Euler numerical approximation [26] .
(1) Choose the parameter values in model (1.2) as follows: 2m N 0 (µ + γ ) = 0.008, and the condition (1) of Theorem 4.1 is satisfied. Thus, the disease I goes to extinction with probability one and Fig. 1 confirms it. Moreover, by Fig. 1 , we achieve that the disease is extinct whereas the corresponding deterministic system is persistent because of the effect of the jump noise. In Fig. 2 , the intensities of the noises are σ 1 = 0.08 and γ (u) = 0.2, then we obtain that R 0 = 1.4651 > 1 +σ 2) goes to extinction with probability one. The red lines, the green lines and the blue lines are solutions of system (1.2), the corresponding deterministic system and the system with white noise, respectively.
The disease I of system (1.2) is persistent with probability one. The red lines, the green lines and the blue lines are solutions of system (1.2), the corresponding deterministic system and the system with white noise, respectively. and the condition (1) of Theorem 5.1 holds. Therefore, the disease I is persistent with probability one.
(2) Let µ = 0.26 and the other parameters are the same as those in Fig. 1 . Considering different values of τ 1 and τ 2 , then we can observe the effects of the validity period of the vaccination τ 1 to system (1.2) (See Fig. 3) .
From Figs. 1-3, we achieve that the intensity of Lévy noise γ (u) and the validity period of the vaccination τ 1 can greatly influence the extinction and persistence of the disease I.
A stochastic delay epidemic model is proposed with vaccination and generalized nonlinear incidence rate. The effects of Lévy jumps are considered in this model. The existence of a unique global solution is proven. Sufficient conditions that guarantee diseases to be extinct and persistent in the mean are given. From the analysis and discussion, we derive that noise intensity and the validity period of vaccination greatly influence the extinction and persistence of diseases. Finally, some interesting issues merit further investigations. In this paper, we obtain sufficient conditions for the persistence and extinction of the model. However, whether the threshold value could be derived is an interesting issue. In addition, if we also consider the effects of non-autonomous environment to the proposal of epidemic model, how will the properties change? We will investigate these questions in our future work.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.