Research Article | Open Access
Dynamical Behaviors of a Stochastic SIQR Epidemic Model with Quarantine-Adjusted Incidence
We study the dynamics of a stochastic SIQR epidemic disease with quarantine-adjusted incidence in this article. In order to find the sufficient conditions for the ergodicity and extermination of the model, we construct suitable stochastic Lyapunov functions and find the results of the stochastic SIQR epidemic model. From the results, we find that when the white noise is relatively large, the infectious diseases will become extinct; this also shows that the intervention of white noise will play an important part in controlling the spread of the disease.
Recently, owing to the negative impact of infectious diseases on population growth, understanding the dynamic behavior of these diseases and predicting what will happened have become an important research topic (see e.g., [1–8]). Therefore, the establishment of mathematical models has become an important method to study the properties of infectious diseases. For more contagious diseases such as smallpox, measles, plague, mumps, and Ebola, the most direct and effective methods of interference are to isolate those who have already been infected, in order to decrease transmissions to susceptible individuals. From then on, one of the famous disease models, SIQR (see ), has been established, which can be described as follows: In this model, one assumed that the infection is given a permanent immunization after recovery. is the susceptible individual. When these people are infected with the disease, some enter compartment, which will be infected; other people can quickly and completely recover and access compartment. In addition, when the susceptible individual enters compartment, it may be quarantined directly to enter compartment. Before they recover, they all will go into compartment. Here, the total population of the model varies, because vulnerable parts of the population can be received through birth or immigrants and people will die of natural and disease deaths. Besides, in this model, the incidence given by is the quarantine-adjusted incidence. The total contacts of a susceptible person using this form of incidence are , so that, during quarantine process, the total number of contacts per day remains at . From the model, the parameters can be summarized in the following list: is the influx of people into the susceptible person’s compartment. d is the natural death rate of compartments , and . is transmission coefficient from compartment to compartment . is the recovery rate of infectious individuals. is the isolation rate from to . is the recovery rate of isolated individuals. is the disease-caused death rate of infectious individuals. is the disease-caused death rate of isolated individuals.
Assume that all parameters are nonnegative parameters. In particular, and are positive constants.
In model (1), the quarantine reproduction number is , which determines whether the disease occurs. If , system (1) has a unique disease-free equilibrium and is globally asymptotically stable in invariant set , where . This reveals that the disease will die out and all people are susceptible to it. If and , then is locally asymptotically stable in the region and system (1) has only a positive endemic equilibrium which we can find in , where In addition, for some parameter values the Hopf bifurcation may occur.
In real life, disease systems are often affected by white noise (see [10–17]). So it is important to include the effect of stochastic perturbation in estimation of parameters. In many cases, stochastic systems can better describe the spread of infectious diseases (see [16–18]). For instance, the stochastic model can account for the stochastic infectious contact during latent and infectious period . Compared with the deterministic system, this is more practical (see [10, 20–25]). Reference  cleared that the stochastic systems are more adapted to the problem of extinction of the disease. Paper  showed that the unique equilibrium in a deterministic model may disappear in corresponding stochastic system due to stochastic fluctuations .
There are so many methods to introduce stochastic perturbations in this system. From biological perspective, random effects can be expressed in Itô or Stratonovich stochastic integrals . From mathematical perspective, random effects are manifested directly in input parameters which are assumed to have specific probability distributions such as uniform, beta, exponential distribution, and gamma . In this paper, we used the stochastic SIQR epidemic model introduced by the approach of Hethcote et al. ; following this approach, we can establish a SDE SIQR epidemic model with quarantine-adjusted incidence.
Think the effects of random fluctuations out, we assume that the fluctuations in the environment are represented as a parameter change to random variable and we assume that random fluctuations are linear perturbations corresponding to the rate of change for each population. Then corresponding to model (1), we can establish the stochastic model: where are independent standard Brownian motions and represent the intensities of the white noise. The other coefficients are the same as system (1).
In this part, we will give some theoretical knowledge about stochastic differential equations. First of all, let be -dimensional standard Brownian motion which is defined on the complete probability space adapted to the filtration and let be -dimensional Itô’s process on with the stochastic differential equation  Let ; then is again Itô’s process with the stochastic differential equation  which can be defined as follows: where .
Lemma 1. The Markov process has a unique ergodic stationary distribution if there exists a bounded domain with regular boundary and if () there is a positive number such that , () there exists a nonnegative -function that LV is nonpositive for any  .
2. Existence and Uniqueness of Positive Solution
In this section, we will study the existence and uniqueness of positive solutions in system (3), which is also the premise of studying the long-term behavior of the model.
Theorem 2. For any initial value , there is a unique positive solution of system (3) for and the solution will maintain in with probability 1.
Proof. Since the coefficients of system (3) satisfy the local Lipschitz condition , then for any initial value , there is a unique local solution on , where is the explosion time . To find that this solution is global, we only need to prove that a.s. Since the following argument is similar to that of , here we simply have to prove the difference with it. Let which make every component of sufficiently large in interval . For every integer , define the stopping timewhere we set throughout this article. Through the definition, we can get that increases with . Let ; then a.s. Next, we have to prove that a.s. If the assertion is against this, then there will be a pair of parameters and so that . Therefore, here exists an integer so that Define a -function as follows:Using Itô’s formula, one yields that is a positive constant and other parts of proof of Theorem 3, which we can get from Mao , here are omitted. Therefore, we have completed the proof.
3. Ergodic Properties
In this section, we define
Theorem 3. Assume that ; there exists a stationary distribution and the ergodicity holds for any initial value in system (3).
Proof. Let where is a sufficiently small constant. In the set , one can find that the following conditions hold: where satisfying here In order to be more intuitive, we divide into the following five regions: The diffusion matrix of (3) is given by Choosing , we can get Thus condition () in Lemma 1 holds.
Now, we can define where are the positive constants to be determined. Using Itô’s formula, one gets Let Then Thus we can get In addition, we can define combined with (17), we can easily find that here Moreover, is also a continuous function . Therefore must have a minimum point  which is inside . Then we define a nonnegative -function : as follows: Combined with the calculation method of stochastic differential, we get Next we consider the following five cases. Let , where is positive constant.
Case 1 (if ). We get combined with (13), we have Case 2 (if ). We can obtain Thus one can see that combined with (12) and (17), for any sufficiently small , we get Case 3 (if ). We find that combined with (14), we have Case 4 (if ). One can see that combined with (15), we get Case 5 (if ). We can obtain according to (16), one can see that Obviously, from (30), (33), (35), (37), and (39), we can easily find that for a sufficiently small , Thus we can easily find that () in Lemma 1 is satisfied. From Lemma 1, we get that model (3) has a uniquely stationary distribution and satisfies ergodicity. Hence, the proof of Theorem 3 has been carried out.
4. Disease Extinction
In this section, we will investigate the extinction of disease under the following conditions:() and , where ,().
Theorem 4. If or holds, the disease will die out exponentially with probability 1; that is,
Proof. By Itô’s formula, one can easily find thatIntegrating (43) from to , one can obtain that Let We can obtain Next, let , then one can get
Case 1. Suppose that holds; we can get thus we haveBy the strong law of large numbers (see ), we get that Both sides simultaneously take maximum of (49) and combining with (50), we have this means that . In other words, the disease will tend to zero exponentially with probability 1.
Case 2. If holds, this conclusion is also correct. In fact, by the condition, we can get thus we have Both sides simultaneously take maximum of (53) and combining with (50), we get By system (3), one can easily obtain that when , then and . This completes the proof.
Remark 5. From condition , we can easily find that if and white noise is small, the disease will be extinct. From condition we get that if white noise is large enough, the disease will also be extinct. Otherwise, the manifestation of the disease is uncertain, which does not happen in deterministic system (1). Furthermore, we notice that in is smaller than the basic reproduction number of system (1).
5. Examples and Numerical Simulations
In this section, we will give some numerical examples to illustrate our main results by using Milstein’s Higher Order Method .
Example 1. We choose the parameter values in system (3) as follows: By calculation, we can get . That is to say, the conditions of Theorem 3 are satisfied. In Figure 1, we choose the different initial values to illustrate that wherever and start from, the density functions of and converge to the same functions, respectively. In Figure 2, selecting different iterations, one can easily find that the density functions of , and also converge to the same functions, respectively. Hence, Figures 1 and 2 verify Theorem 3 very well so that there exists a unique ergodic stationary distribution of system (3). In Figure 3, the blue line and red line are almost the same. This strongly illustrates ergodicity.
Example 2. In order to obtain the extinction of the model, we give numerical simulations under which one of the conditions of and hold. In addition, neither nor holds; we also give numerical simulations to verify that in this case, the disease may be persistent in Figure 6 or extinct in Figure 7.
In Figure 4, the parameters in (3) are chosen by By calculation, we know that and . This means that holds. In Figure 5, we keep all the parameters unchanged but increase to . Note that and , which means that holds in Theorem 4. Figures 4 and 5 verify Theorem 4 very well so that if or holds, will tend to zero exponentially with probability one.
In Figure 6, by changing , , and and not changing other parameters in Figure 4, we calculate and and , which means that both and in Theorem 4 do not hold. From the line in Figure 6, we can easily find that persist. In Figure 7, we also change some parameters and other parameters are unchanged in Figure 4. By calculation, we have and , which also means that both and in Theorem 4 do not hold. From Figure 7, it is shown that become extinct. Thus, combining Figure 6 with Figure 7, one can easily find that both and do not hold; the direction of is uncertain and may persist or become extinct.
(a) Beta = 0.4
(b) Beta = 0.4
(c) Beta = 0.4
(d) Beta = 0.4
(a) Beta = 0.4
(b) Beta = 0.4
(c) Beta = 0.4
(d) Beta = 0.4
(a) Beta = 0.4
(b) Beta = 0.4
(c) Beta = 0.4
(d) Beta = 0.4