Abstract
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.
1. 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 [1–8]. 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: and studied the dynamical behavior of the model, where sheep population is classified into five compartments: the susceptible young sheep , the susceptible adult (or sexually mature) sheep , the vaccinated sheep , the exposed sheep , and the infectious sheep . and are the input number of young sheep and the natural birth rate of sheep, and is the extent of the birth being delayed. and are the young sheep natural mortality rate and the elimination rate of adult sheep. and are the transfer rate from young sheep to adult sheep and exposed sheep to infected sheep. is the output number of young sheep, is the vaccination rate, is sheep-to-sheep transmission rate, is the ineffective vaccination rate, and 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, 9–28]. In particular, the stochastic epidemic models have been extensively studied [3, 10–25]. 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, 9–28]. 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 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 brucellosis model, we let , and then it is appropriate to model as a Markov process; thus, from [15] and model (1), we can get the following properties when , the conditional mean and the conditional covariance
Then, we derive the following stochastic form of system (1)
where , and are the standard one-dimensional independent Brownian motions and 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.
2. Preliminaries
Throughout this paper, let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is increasing and right continuous, while contains all -null sets); are defined on this complete probability space, and also let .
In general, consider the -dimensional stochastic differential equation with initial value . denotes an -dimensional standard Brownian motion defined on the complete probability space . denotes the family of all nonnegative functions defined on such that they are continuously twice differentiable in and once in . The differential operator of equation (5) is defined by [16].
If acts on a function , then where By Itô’s formula, if , then
Next, we present a result about the existence of stationary distribution (see Has’minskii [17]).
Let be a homogeneous Markov process in ( denotes -dimensional Euclidean space) and be described by the following stochastic differential equation:
The diffusion matrix is defined as follows:
Lemma 1. The Markov process has a unique ergodic stationary distribution if there exists a bounded domain with regular boundary and
: there is a positive number such that .
: there exists a nonnegative -function such that is negative for any . Then
for all , where is a function integrable with respect to the measure .
3. Main Results
3.1. 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).
Theorem 2. For any initial value , there is a unique solution of system (4) on , and the solution will remain in with probability one.
Proof. Since the coefficients of system (4) satisfy the local Lipschitz condition, then for any initial value , there is a unique local solution on , where is the explosion time [16]. To show this solution is global, we only need to prove that a.s. To this end, let be sufficiently large such that every component of lying within the interval . For each integer , define the stopping time as follows:
Throughout this paper, we set (as usual denotes the empty set). It is easy to see that is increasing as . Let , then a.s. In what follows, we need to verify a.s. If this assertion is violated, there is a constant and an such that . As a result, there exists an integer such that
Define a -function V: by
Using It’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 is positive constant which is independent of , and , we can get
Integrating both sides (20) from 0 to and taking expectations, then we can obtain
Set for by (13), . Notice that for every , there is at least one of , and that equal either or . Hence, , and are no less than
Consequently, where donates the minimum of and . In view of (21) and (23) we have where is the indicator function of . Let leads to the contradiction
Therefore, we must have a.s.
3.2. 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 , then system (4) has a unique stationary distribution and it has the ergodic property.
Proof. In view of Theorem 2, we have obtained that for any initial value , there is a unique global solution .
The diffusion matrix of system (4) is given by
Choose ; one can get that
Then the condition in Lemma 1 is satisfied.
Construct a -function in the following from where is a constant satisfying , and satisfies the following condition where
It is easy to check that where . Furthermore, is a continuous function. Hence, must have a minimum point in the interior of . Then we define a nonnegative -function as follows:
Making use of It’s formula, we have
Using the inequality leads to where is defined in (32).
Similarly
Then where
We can also get
Hence, by (37)-(45), we obtain
Thus, we can construct a compact subset such that the condition in Lemma 1 holds. Define the bounded closed set where are sufficiently small constants satisfying the following conditions: where , and are positive constants which can be seen from (60), (68), (70), (72), (74), and (76), respectively. Note that for sufficiently small . For convenience, we divide into ten domains
Next, we will show that on , which is equivalent to proving it on the above ten domains.
Case 1. If , one can get that where
According to (48), we have
Case 2. If , we have
where is defined in (33).
In view of (49), we can obtain that for sufficiently small , for any
Case 3. If , one can see that
We obtain that
Case 4. If , one can see that
In view of (51), we can obtain that for sufficiently small , for any
Case 5. If , one can see that
We can obtain that for sufficiently small , for any
Case 6. If , one can see that where
By (53), we conclude that .
Case 7. If , one can see that where
Together with (54), we can deduce that .
Case 8. If , one can see that where which together with (55) implies that .
Case 9. If , we obtain where
By (56), we can conclude that on .
Case 10. If , it follows that where
Combining with (57) yields on .
Obviously, in Lemma 2.1 is satisfied. According to Lemma 2.1, we can obtain that system (4) is ergodic and has a unique stationary distribution.
Remark 4. Theorem 3 reveals that system (4) has a unique ergodic stationary distribution if . Note that the expression of coincide with the threshold of the deterministic system (1) if there is no stochastic perturbation. This shows that we generalize the result of the deterministic system.
3.3. Extinction of the Disease
As it is well known, one of the main concern of epidemiology is how we regulate the disease dynamics in order to eradicate the disease in the long term. Moreover, in [10], Allen et al. proposed and studied several types of stochastic epidemic models and pointed out that the stochastic models should suit the question of disease extinction better. Hence, in this section, we shall establish some sufficient conditions for extinction of the disease in stochastic model (4).
Theorem 5. Let be the solution of system (4) with any initial value . If , then the disease will extinct exponentially with probability one, i.e., moreover
Proof. Applying It’s formula to , we have
Integrating the above inequality from 0 to , and the fact that [16], yields
For any , and almost such that
Remark 6. Theorem 5 suggests that the disease will become extinct if .
4. Numerical Examples
In this section, we will give two numerical examples to illustrate the main theoretical results obtained in this paper. The numerical simulation method can be found in [9, 22, 23]. The following is a corresponding discrete equations of system (4): where are the Gaussian random variables which follow standard normal distribution , and are intensities of white noises.
Example 1. We take parameters as , , , , , , , , , and , , , , , , , and . It is clear that conditions of Theorem 3 are satisfied; by calculating, we have the basic reproduction number
The histogram and the smoothing curves of the probability density functions of are given in Figure 1.
Example 2. We take parameters as , , , , , , , , and , , , , , , , and . It is clear that conditions of Theorem 5 are satisfied.
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The curves on the persistence of and extinction of for stochastic model (4) are given in Figure 2, where the initial value is .
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].
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5. 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 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.
Acknowledgments
This project is supported by the National Natural Science Foundation of China (Grant No. 11861063) and the National Natural Science Foundation of Xinjiang (Grant No. 2021D01C067) .