Abstract
This paper proposes a stochastic SIVS epidemic system with nonlinear saturated infection rate under vaccination and investigates the dynamics predicted by the model. By using Itô’s formula and Lyapunov methods, we first study the existence and uniqueness of global positive solution. Then we investigate the stochastic dynamics of the system and obtain the thresholds which govern the extinction and the spread of the epidemic disease. Results show that large stochastic noises can lead to the extinction of epidemic diseases; that is, stochastic disturbances can suppress the outbreak of epidemic diseases. Finally, we carry out a series of numerical simulations to demonstrate the performance of our theoretical findings.
1. Introduction
Infectious diseases pose a serious threat to public health around the world. Therefore, the study of epidemic diseases has been part of the burning issues of scientists. Mathematically, after the pioneering work of Kermack and McKendrick on the epidemic regularity of the Black Plague in London by the famous SIR model [1], mathematical models have been used extensively by scientists to study the spread and control of diseases [2–7]. There are many ways to suppress the spread of disease, for instance, cutting off transmission routes, paying attention to food hygiene, and vaccination. Vaccination is an effective method of preventing infectious diseases and many scientists have explored the effect of vaccination on diseases [8–17]. For instance, Li and Ma [8] established an SIS epidemic model with vaccination. The model has the following form:
The incidence rate in model (1) is bilinear, which means that the number of people infected by a patient within a unit time is proportional to the total number of susceptible individuals in the environment. However, the number of susceptible individuals to contact with a patient within a unit time is limited, and many scholars have noted that the saturated incidence is more accurate to describe the spread of epidemic [18–21]. Besides, the WHO reports that licensed vaccines are currently available to prevent or contribute to the prevention and control of twenty-five preventable infections. But the effect of vaccination is not absolute. In other words, some people who are vaccinated still have the risk of being infected [22, 23]. Moreover, epidemic diseases may be subject to uncertain environmental disturbances, such as fluctuations of birth rate, death rate, and infection rate. These phenomena can be characterized by stochastic processes. Numerous scholars have introduced stochastic interferences into differential systems, and the stochastic dynamics of such systems were investigated [24–36].
Motivated by the above works, in this paper, we suppose the following:(i), (usually ), respectively, stand for the average contact rate of an infective individual explored to a susceptible and a vaccinated individual per unit time at time . indicates the immune efficiency is , indicates the immunization was completely ineffective, and indicates the immunity is effective but not completely effective. Authors in [8] discussed the case .(ii)The infection rates from infective individual to a susceptible and a vaccinated individual are and , respectively. Here are constants.(iii)Environmental interference mainly affects the rate of infection, and we have , Then model (1) becomeswhere , , and , respectively, stand for the density of susceptible and infective and vaccinated individuals at time ; is a constant input of new numbers into the population; means a fraction of vaccinated newborn; represents the natural death rate of , , and ; is the proportional coefficient of vaccinated susceptible; is the recovery rate of ; stands for the rate of losing their immunity for vaccinated individuals; and is the disease-caused death rate of infectious individuals .
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). Function is a Brownian motion defined on the complete probability space , and the intensity of is . For an integrable function on , we define
2. The Existence and Uniqueness of Global Positive Solutions
Due to physical meaning, variables , , and in model (2) should remain nonnegative for . We next prove that this is actually the case and, furthermore, the positive solution is unique.
Lemma 1. For any initial value , model (2) has a unique local positive solution in interval , where is the explosion time.
Proof. For any initial value , the coefficients of system (2) are locally Lipschitz continuous. Thus model (2) has a unique local positive solution in interval , and we complete the proof.
Theorem 2. For any initial value , model (2) has a unique positive solution on with probability 1.
Proof. By Lemma 1, we only need to prove that a.s. To this end, let be a sufficiently large constant such that , , and all lie in . For each , define the stopping time Easily, is a monotonically increasing function when . Let , and thus a.s. Now we need to prove a.s.; otherwise, there exist two constants and such that . Thus, there is an integer such thatDefine a -function : Applying Itô’s formula to stochastic differential system (2) yields where By (2), we haveThus Therefore, for any and each , we have Sowhere is a positive constant.
ThenIntegrating (12) from to and taking expectation on both sides, we haveLet , and from inequality (4) we can see that . Note that, for every , there exists at least one of , , and that equals either or . As a result, we haveApplying (13) and (14), we get where is the indicator function of .
When , we have This is a contradiction. So and we complete the proof.
Without losing generality, in this paper, we always assume that
3. Extinction
Theorem 3. Let be the solution of model (1) with initial value . If one of the following conditions holds: (i), , and ,(ii), , and ,(iii), , and ,(iv), , and ,then epidemic disease goes extinct; that is, Moreover, we have
Proof. Applying Itô’s formula to , we haveIntegrating from to and dividing both sides of (18) by , one haswhere are real-valued continuous martingales.
Thus By the strong law of large numbers [37], we have Case 1. When , , and , applying (19) we haveTaking the limit superior on both sides of (23), we have Thus Case 2. When , , and , applying (19) we haveTaking the limit superior on both sides of (26), we have Thus Case 3. When , , and , applying (19) we haveTaking the limit superior on both sides of (29), we have Thus Case 4. When , , and , applying (19) we haveTaking the limit superior on both sides of (32), we have Thus Applying (8) we have Applying L’Hospital rule, one hasApplying (36) and , there exist two arbitrarily small constants such that when , we have and so we have Letting and applying the arbitrariness of and , we obtain Similarly, we have Then, applying (36), we have This completes the proof.
4. Persistence in Mean
Theorem 4. Let be a solution of model (2) with initial value . If condition holds, then epidemic disease is persistence in mean; that is,
Proof. Integrating from to and dividing both sides of (2) by , we haveChoose , and then, applying Itô’s formula to , we haveIntegrating from to and dividing both sides of (44) by , one haswhere are local continuous martingales.
Thus By the strong law of large numbers, we have From (45), we obtainwhere Hence, we have and as and
Taking the limit inferior on both sides of (49), we haveOn the other hand, choose , and, applying (44), we haveIntegrating from to and dividing both sides of (52) by , one haswhere are local continuous martingales.
Thus By the strong law of large numbers, we have Applying , we have
From (53), we obtainwhere Applying (51) yields , and then .
Taking the limit superior on both sides of (58), we haveThis completes the proof.
5. Conclusions and Numerical Simulations
This paper investigates a stochastic epidemic system with nonlinear incidence rate under vaccination. We first demonstrate the existence and uniqueness of global positive solution of model (2). Then we study the persistence in mean and extinction of the stochastic SIV system. Our result shows that large noises can lead to the extinction of epidemic diseases. In addition, notice that the basic reproduction number for epidemic disease to die out is negatively related to the intensity of interference, which indicates that stochastic impact can suppress the epidemic diseases. Compared with the existing work in [8, 10], the model constructed in this paper also considered the efficiency of vaccination. Authors in [8, 10] considered the case that the average contact rate of an infective individual explored to a vaccinated individual per unit time at time is , and we study the more general case , which is more consistent with the actual situation. In addition, since the number of susceptible and vaccinated individuals to contact with a patient within a unit time is limited, in this paper, we employ a saturated incidence to describe the spread of an epidemic; that is, ,
We next give some numerical simulations to support our results. Considering the following discrete equations: where and , obeys the Gaussian distribution . To this end, we set
Under the setting for plotting Figures 1(b)–1(e), we have
(a)
(b)
(c)
(d)
(e)
(f)
Figure 1(a) is the time sequence diagram of model (2) with noises and . In this case, the disease in model (2) is persistent. Figures 1(b)–1(e) are the time sequence diagram of model (2), satisfying conditions (i)–(iv) in Theorem 3, respectively. In Figure 1(b), , , and this satisfies condition (i) of Theorem 3. In Figure 1(c), , , and this satisfies condition (ii) of Theorem 3. In Figure 1(d), , , , and this satisfies condition (iii) of Theorem 3. In Figure 1(e), , , , and this satisfies condition (iv) of Theorem 3. From Figures 1(b)–1(e), the disease in model (2) is extinct. Comparing Figure 1(a) with Figures 1(b)–1(e), we find that random interference can lead to the extinct of disease. This is in line with our conclusion in Theorem 3.
In Figure 1(f), In this case, , and the disease in model (2) is persistence in mean, and this is in line with our conclusion in Theorem 4.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (11371230, 11501331, and 11561004), the SDUST Research Fund (2014TDJH102), Shandong Provincial Natural Science Foundation, China (ZR2015AQ001, BS2015SF002), Joint Innovative Center for Safe And Effective Mining Technology and Equipment of Coal Resources, the Open Foundation of the Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques, Gannan Normal University, China, and SDUST Innovation Fund for Graduate Students (SDKDYC170225).