Analysis and Numerical Simulations of a Stochastic SEIQR Epidemic System with Quarantine-Adjusted Incidence and Imperfect Vaccination

This paper considers a high-dimensional stochastic SEIQR (susceptible-exposed-infected-quarantined-recovered) epidemic model with quarantine-adjusted incidence and the imperfect vaccination. The main aim of this study is to investigate stochastic effects on the SEIQR epidemic model and obtain its thresholds. We first obtain the sufficient condition for extinction of the disease of the stochastic system. Then, by using the theory of Hasminskii and the Lyapunov analysis methods, we show there is a unique stationary distribution of the stochastic system and it has an ergodic property, which means the infectious disease is prevalent. This implies that the stochastic disturbance is conducive to epidemic diseases control. At last, computer numerical simulations are carried out to illustrate our theoretical results.

In fact, the main meaning of the research of infectious disease dynamics is to make people more comprehensively and deeply understand the epidemic regularity of infectious disease; then more effective control strategies are adopted to provide better theoretical support for the prevention and control of epidemics. To this end, many mathematical biology workers considered more realistic factors in the course of the study, such as population size change, migration, cross infection, and other practical factors. In the course of epidemics and outbreaks of infectious diseases, people always take various measures to control the epidemic in order to minimize the harm of epidemic diseases. Quarantine is one of the important means to prevent and control epidemic diseases; it has been used to control contagious diseases with some success. Specifically, during the severe acute respiratory syndrome (SARS) outbreak in 2002, remarkable results were also achieved. Among them, mathematical models have been used to investigate their impact on the dynamics of infectious diseases under quarantine [18][19][20][21][22], which attracts deep research interest of many mathematicians and biologists. Recently, Hethcote et al. [21] considered an SIQR (susceptible-infected-quarantined-recovered) model with quarantine-adjusted incidence. The system can be expressed as follows: where the total population size is given bỹ( ) = ( ) + ( ) + ( ) + ( ), Λ is the inflow rate corresponding to birth 2 Computational and Mathematical Methods in Medicine and immigration, and is the outflow rate corresponding to natural death and emigration. Since the quarantine process, using the standard incidence ( /̃) , the contact rate /w ith the quarantined fraction /̃does not occur. Hence the standard incidence is replaced by /(̃− ) (it is called quarantine-adjusted incidence); here is the transmission coefficient between susceptible individuals and infected individuals. is the quarantine rate of infected individuals, is the recovery rate of quarantined individuals, and 1 and 2 stand for the rate of disease-related death of infected and quarantined individuals, respectively. is the recovery rate of infected individuals. Furthermore, all the parameters are positive and the regioñ= {( , , , ) | ≥ 0, ≥ 0, ≥ 0, ≥ 0, + + + ≤ Λ/ } is a positively invariant set of system (1). In the regioñ, they established the basic reproduction number 0 , which determines disease extinction or permanence, where Meanwhile, they analyzed the global dynamics of system (1) and derived the equilibria (including the disease-free equilibrium and the endemic equilibrium) and their global stability. In addition, the parameter conditions for the existence of a Hopf bifurcation are obtained.
In the real world, with the development of modern medicine, vaccination has become an important strategy for disease prevention and control in addition to quarantine, and numerous scholars have investigated the effect of vaccination on disease [23][24][25][26][27][28][29][30]. For another, many infectious diseases incubate inside the hosts for a period of time before becoming infectious, so it is very meaningful to consider the effect of the incubation period. Motivated by the aforementioned work, this paper considers an SEIQR (susceptibleexposed-infected-quarantined-recovered) epidemic model with imperfect vaccination, which is described by the following system: where the total population size is given by ( ) = ( )+ ( )+ ( ) + ( ) + ( ), (0 ≤ < 1) is the vaccine coverage rate, (0 ≤ ≤ 1) is the vaccine efficacy, and is the rate at which the exposed individuals become infected individuals. Other parameters are the same as in system (1). Now we assume that all the parameters are positive constants here except that , are nonnegative constants. Clearly, the region = {( , , , , ) | ≥ 0, ≥ 0, ≥ 0, ≥ 0, ≥ 0, + + + + ≤ Λ/ } is a positively invariant set of system (3). For system (3), the basic reproduction number is and it has the following properties: (1) When 1 ≤ 1 holds, system (3) has a unique diseasefree equilibrium 0 = ( 0 , 0, 0, 0, 0 ) = (Λ/( + ), 0, 0, 0, Λ/ ( + )) which is globally asymptotically stable in the region . That means the epidemic diseases will die out and the total individuals will become the susceptible and recovered individuals.
Computational and Mathematical Methods in Medicine 3 In this paper, we are mainly concerned with two interesting problems as follows: (P1) Under what parameter conditions, will the disease die out?
(P2) Under what conditions, will system (5) have a unique ergodic stationary distribution?
Throughout this paper, let (Ω, F, {F} ≥0 , P) be a complete probability space with a filtration {F } ≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while F 0 contains all P-null sets). Further ( ) ( = 1, 2, 3, 4, 5) is defined on the complete probability space.

Global Positive Solution
To investigate the dynamical behaviors of a population system, we first concern the global existence and positivity of the solutions of system (5). Proof. The following proof is divided into two parts.

Extinction
In this section, we mainly explore the parameter conditions for extinction of the disease in system (5). Before proving the main results, we first give a useful lemma as follows.

Stationary Distribution and Ergodicity
Ergodicity is a significant property in a population system. Recently, it attracts deep research interest of numerous 6 Computational and Mathematical Methods in Medicine scholars [52,53]. In this section, based on the theory of Hasminskii et al. [54] and the Lyapunov analysis methods, we study the conditions of the existence of an ergodic stationary distribution. Assume ( ) as a time-homogeneous Markov process in E ⊂ R , which is described by the stochastic differential equation and here E stands for a -dimensional Euclidean space. The diffusion matrix takes the following form: Assumption 5. Assume that there is a bounded domain ⊂ E with regular boundary Γ such that ⊂ E ( is the closure of ), satisfying the following properties: (i) In the domain and some neighborhood thereof, the smallest eigenvalue of the diffusion matrix̃( ) is bounded away from zero.
(ii) If ∈ E \ , the mean time at which a path issuing from reaches the set is finite, and sup ∈Θ E < ∞ for every compact subset Θ ⊂ E .
Lemma 6 (see [54]). When Assumption 5 holds, then the Markov process ( ) has a stationary distribution (⋅). Furthermore, when (⋅) is a function integrable with respect to the measure , then Remark 7. To demonstrate Assumption 5(i) [55], it suffices to demonstrate that is uniformly elliptical in any bounded domain ; here namely, there exists a positive number such that To verify Assumption 5(ii) [56], it suffices to demonstrate that there exist some neighborhood and a nonnegative 2function such that ∀ ∈ E \ , ( ) < 0.
Using Lemma 6, we can get the following main results.
Remark 9. From Theorem 8, we see that if * > 1 holds, then system (5) has a unique ergodic stationary distribution.
It is worthwhile noting that if = 0 ( = 1,2,3,4,5), the expression of * coincides with the basic reproduction number 1 of system (3). This shows that we generalize the results of system (3). For another, this theorem also shows that the disease can resist a small environmental noise to maintain the original persistence.

Conclusions and Simulations
This paper studies the stochastic SEIQR epidemic model with quarantine-adjusted incidence and imperfect vaccination and obtains two thresholds which govern the extinction and the spread of the epidemic disease. Firstly, the existence of a unique positive solution of system (5) with any positive initial value is proved. Then, from Theorems 4 and 8, the sufficient conditions for extinction of the disease and existence of ergodic stationary distribution of the stochastic system are derived by using the theory of Hasminskii and the Lyapunov analysis methods, which means the infectious disease is prevalent. This implies that the stochastic disturbance is conducive to epidemic diseases control. Now we summarize the main conclusions as follows: ) and̃ * = (2 (1 − ) ( + ))/(( + 1 + + + 2 3 /2)( + ) 2 ∧ ( 2 2 2 /2)) < 1 hold, then the infected individuals go to extinction almost surely.
To illustrate the results of the above theorems, we next carry out some numerical simulations by the Matlab software. Let us consider the following discretization equations of system (5) and here , ( = 1, 2, 3, 4, 5; = 1, 2, . . . , ) stands for (0, 1) distributed independent random variables and time increment Δ > 0. satisfies the condition in Theorem 8; we can obtain that system (5) has a unique stationary distribution (⋅) and it has ergodic property. Figure 2 shows that the solution of system (5) swings up and down in a small neighborhood. According to the density functions in Figures 2(b)-2(f), we can see that there exists a stationary distribution. As expected, Figure 2 confirms our results of Theorem 8.
In which means the disease will persist. As expected, Figure 3(a) shows the disease persists in real life. Synchronously, in Figure 3(b), take 1 = 0.15, 2 = 1,