Stationary Distribution and Extinction of a Stochastic Brucellosis Model with Standard Incidence

In this paper, we proposed a stochastic SVEI brucellosis model with stage structure by introducing the effect of environmental white noise on transmission dynamics of brucellosis. By Has'minskii theory and constructing suitable Lyapunov functions, we established sufficient conditions on the existence of ergodic stationary distribution for the considered model. Moreover, we also established sufficient condition for extinction of the disease. Finally, two examples with numerical simulations are given to illustrate the main results of this paper.


Introduction
Brucellosis, which is recognized as a major public health problem, is a serious and economically devastating zoonosis which can infect animals, such as sheep, cattle, pig, and dogs. The disease is caused by bacteria of the genus Brucella, of which there are six species: B. abortus, B. melitensis, B. suis, B. ovis, B. canis, and B. neotomae [1]. Brucella can survive for long periods in dust, dung, water, slurry, aborted fetuses, soil, meat, and dairy products. In animals, brucellosis can be infected by contact with the infected animals (direct way of infection) and by contact of polluted environment (indirect way of transmission), and the disease mainly affects reproduction and fertility and reduces survival of newborns [2]. Brucellosis also can infect human being; the main transmission sources of human brucellosis include exposure to a contaminated environment by infected animals, direct contact with infected animals, and the ingestion of fresh milk or dairy products prepared from unpasteurized milk and unheated meat and animal liver [3]; there is no recorded cases of the infection between humans. Most of the human brucellosis cases are infected by Brucella melitensis (which is infected in sheep and goats), accounting for 84.5% of the total cases [4]. In humans, disease-related mortality is negligible, but the illness can last for several years [5]. Therefore, the key to solve the problem of this public health problem is the elimination of animal brucellosis.
It is worth noting that, mathematical models are widely used not only to study the transmission dynamics of brucellosis, but also to study the epidemiological characteristics of brucellosis [1][2][3][4][5][6][7][8]. Recently, the authors in [6] presented a sheep brucellosis model with immigration and proportional birth, considering both direct and indirect transmission. In [7], the authors proposed a multigroup SEIRV dynamical model with bidirectional mixed cross infection between cattle and sheep and investigated the influence of cross infection of mixed feeding on the brucellosis transmission. In [8], the authors proposed the following deterministic brucellosis transmission model:    3 Computational and Mathematical Methods in Medicine μ 1 and μ 2 are the young sheep natural mortality rate and the elimination rate of adult sheep. η 1 and η 2 are the transfer rate from young sheep to adult sheep and exposed sheep to infected sheep. d is the output number of young sheep, δ is the vaccination rate, β is sheep-to-sheep transmission rate, ε is the ineffective vaccination rate, and c is the elimination rate caused by brucellosis. All the parameters are assumed to be positive.
However, the epidemics in the real world are often disturbed by some uncertain factors, such as environmental white noises. Therefore, it is difficult to describe these epidemic dynamics by using determined differential equation [9]. Thus, the deterministic models has some limitations in mathematical modeling of epidemics, and it is quite difficult to fitting data perfectly and to predicting the future dynamics of the epidemic system. In the past years, there has been a lot of researchers who are interested in the stochastic dynamical models [3,[9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]. In particular, the stochastic epidemic models have been extensively studied [3,[10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. For example, in [3], the authors proposed and studied a periodic stochastic brucellosis model and obtained some conditions on the existence of nontrivial positive periodic solution of the model. In [11], the authors studied a stochastic SIRS epidemic model with standard incidence rate and partial immunity and obtained sufficient conditions on the extinction and existence of a stationary probability measure for the disease of the system. In [12], the authors studied a kind of stochastic SEIR epidemic model with standard incidence and obtained sufficient conditions for the existence of stationary distribution and the extinction of the disease in the system. In [24], the authors discussed a stochastic SIRS epidemic model with logistic growth and nonlinear incidence and obtained sufficient conditions on the ergodic stationary distribution and extinction of the considered model.
On the other hand, there are different approaches used in the literature to introduce random perturbations into population models, both from a mathematical and biological perspective [3,[9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]. In this paper, in light of the above analysis and reasons, we consider the stochastic perturbations for deterministic system (1) and we employ the approach used in Mao et al. [26] and assume that the parameters involved in the model always fluctuate around some average value due to continuous fluctuations in the environment. This approach is reasonable and well justified biologically [27,28]. By this approach, we study a stochastic S 1 S 2 VEI brucellosis model with standard incidence, and we assume that the environmental white noise affects the natural mortality rate, the elimination rate, transfer rate, and transmission rate. In order to obtain the stochastic S 1 S 2 VEI brucellosis model, we let XðtÞ = ðS 1 ðtÞ, S 2 ðtÞ, VðtÞ, EðtÞ, IðtÞÞ T , and then it is appropriate to model X = ðS 1 , S 2 , V, E, IÞ T as a Markov process; thus, from [15] and model (1), we can get the following properties when 0 ≤ Δt ≪ 1, the conditional mean Δt, Δt, and the conditional covariance

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Then, we derive the following stochastic form of system (1) where B 1 ðtÞ, B 2 ðtÞ, B 3 ðtÞ, B 4 ðtÞ, and B 5 ðtÞ are the standard one-dimensional independent Brownian motions and σ 2 i > 0ði = 1, 2, 3, 4, 5Þ are the intensity of the white noises.
The main purpose of the paper is to obtain the conditions for the existence of ergodic stationary distribution and extinction of the disease for model (4).
This paper is organized as follows. In Section 2, we present some preliminaries which will be used in the following analysis. In Section 3, we show that there is a unique global positive solution of system (4). In Section 4, we prove the existence of ergodic stationary distribution for system (4) under certain conditions. In Section 5, we establish sufficient conditions for the disease extinction.

Preliminaries
Throughout this paper, let ðΩ, F, ℙÞ be a complete probability space with a filtration fF t g t≥0 satisfying the usual conditions (i.e., it is increasing and right continuous, while F 0 contains all ℙ -null sets); B i ðtÞði = 1, 2, 3, 4Þ are defined on this complete probability space, and also let ℝ d In general, consider the d-dimensional stochastic differential equation with initial value xð0Þ = x 0 ∈ ℝ d . BðtÞ denotes an n-dimensional standard Brownian motion defined on the complete probability space ðΩ, F, fF t g t≥0 , ℙÞ. C 2,1 ðℝ d × ½t 0 ,∞ ; ℝ + Þ denotes the family of all nonnegative functions Vðx, tÞ defined on ℝ d × ½t 0 ,∞ such that they are continuously twice differentiable in x and once in t. The differential operator L of equation (5) is defined by [16].
Let XðtÞ be a homogeneous Markov process in E d (E d denotes d-dimensional Euclidean space) and be described by the following stochastic differential equation: The diffusion matrix is defined as follows: for all x ∈ E d , where f ð·Þ is a function integrable with respect to the measure π.

Existence and Uniqueness of the Positive Solution.
In studying the dynamical behavior of an epidemic model, the first importance is whether the solution is global and positive. Hence, in the following theorem, we will study the existence and uniqueness of the global positive solution, which is a prerequisite for researching the long-term behavior of model (4).
Proof. Since the coefficients of system (4) satisfy the local Lipschitz condition, then for any initial value ðS 1 ð0Þ, S 2 ð0Þ, Vð0Þ, Eð0Þ, Ið0ÞÞ ∈ ℝ 5 + , there is a unique local solution ðS 1 ðt Þ, S 2 ðtÞ, VðtÞ, EðtÞ, IðtÞÞ on ½0, τ e Þ, where τ e is the explosion time [16]. To show this solution is global, we only need to prove that τ e = ∞ a.s. To this end, let n 0 > 0 be sufficiently 5 Computational and Mathematical Methods in Medicine large such that every component of X 0 lying within the interval ½ð1/n 0 Þ, n 0 . For each integer n > n 0 , define the stopping time as follows: Throughout this paper, we set inf ∅ = ∞ (as usual ∅ denotes the empty set). It is easy to see that τ n is increasing as n ⟶ ∞. Let τ ∞ = lim n⟶∞ τ n , then τ ∞ ≤ τ e a.s. In what follows, we need to verify τ ∞ = ∞ a.s. If this assertion is violated, there is a constant T > 0 and an ε ∈ ð0, 1Þ such that ℙfτ ∞ ≤ Tg > ε. As a result, there exists an integer n 1 ≥ n 0 such that Define a C 2 -function V: Using Itô's formula, we have where By applying the following invariant set of model (1) which is obtained in [8] and from the following inequalities and also cancel the items less than zero, so we have Since ζ is positive constant which is independent of S 1 , S 2 , V, E, I, and t, we can get Integrating both sides (20) from 0 to T∧τ ε and taking expectations, then we can obtain Set Ω ε = fτ ε ≤ tg for n ≥ n 1 by (13), PðΩ n Þ ≥ ε. Notice that for every ω ∈ Ω ε , there is at least one of S 1 ðτ ε , ωÞ, S 2 ð τ ε , ωÞ, Vðτ ε , ωÞ, Eðτ ε , ωÞ, and Iðτ ε , ωÞ that equal either n or 1/n. Hence, S 1 ðτ ε , ωÞ, S 2 ðτ ε , ωÞ, Vðτ ε , ωÞ, Eðτ ε , ωÞ, and Iðτ ε , ωÞ are no less than Computational and Mathematical Methods in Medicine Consequently, where a∧b donates the minimum of a and b. In view of (21) and (23) we have where 1 Ω ðωÞ is the indicator function of Ω n . Let n ⟶ ∞ leads to the contradiction Therefore, we must have τ ∞ = ∞ a.s.

Stationary
Distribution and Ergodicity. The difference between model (1) and the stochastic model is that the stochastic model does not have the endemic equilibrium. Hence, we cannot study the persistence of the disease by studying the stability of the endemic equilibrium and turn to check out the existence and uniqueness of the stationary distribution for the system (4) which implies the persistence of the disease in some sense. In this section, based on the theory of Has'minskii [17], we verify that there is an ergodic stationary distribution, which reveals the persistence of the disease. Define a parameter Theorem 3. Assume that R s 0 > 1, then system (4) has a unique stationary distribution Πð·Þ and it has the ergodic property.

5
, and M > 0 satisfies the following condition where Making use of Itô's formula, we have Using the inequality a + b + c + d + e ≥ 5 ffiffiffiffiffiffiffiffiffiffiffi abcde 5 p , a, b, c, d, e > 0 leads to where λ is defined in (32). Similarly Computational and Mathematical Methods in Medicine where B = sup We can also get Hence, by (37)-(45), we obtain Thus, we can construct a compact subset D such that the condition A 2 in Lemma 1 holds. Define the bounded closed where ε i > 0ði = 1, 2, 3, 4, 5Þ are sufficiently small constants satisfying the following conditions: where F, G, H, J, K, and L are positive constants which can be seen from (60), (68), (70), (72), (74), and (76), respectively. Note that for sufficiently small ε i , i = 1, 2, 3, 4, 5. For 9 Computational and Mathematical Methods in Medicine convenience, we divide ℝ 5 + \ D into ten domains Next, we will show that LVðS 1 , S 2 , V, E, IÞ ≤ −1 on ℝ 5 + \ D, which is equivalent to proving it on the above ten domains.
Remark 7. In this paper, we consider the stochastic perturbations for deterministic model (1) and derived model (4). Thus, model (4) can be specialized as models (1). Hence, model (4) can be seen as a general model compared to model (1), and the theoretical results obtained in this article can be seen as the extensions and supplements of the model and the theoretical results obtained in [8].

Conclusion
In this paper, firstly, we have considered the stochastic perturbations for deterministic system (1) and established corresponding stochastic system (4). Secondly, under the condition R s 0 > 1 and applying the theory of stochastic differential equations, Has'minskii theory, Ito's formula, and Lyapunov function method, we obtained some sufficient conditions on the existence of ergodic stationary distribution of model (4). We also established sufficient conditions on the extinction of the disease. Finally, two examples are presented to validate the main results of this paper. The results obtained in this paper suggest that stochastic perturbations have remarkable effects on the disease in model (4). Especially, from the numerical simulations, we can see that, under the stochastic perturbations, the disease in the stochastic system will become extinct more quickly than the corresponding deterministic one.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no competing interests.