Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 891765, 8 pages http://dx.doi.org/10.1155/2013/891765
Research Article Stochastic Extinction in an SIRS Epidemic Model Incorporating Media Coverage Liyan Wang, Huilin Huang, Ancha Xu, and Weiming Wang College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, China Correspondence should be addressed to Weiming Wang;
[email protected] Received 2 December 2013; Accepted 12 December 2013 Academic Editor: Kaifa Wang Copyright Β© 2013 Liyan Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We extend the classical SIRS epidemic model incorporating media coverage from a deterministic framework to a stochastic differential equation (SDE) and focus on how environmental fluctuations of the contact coefficient affect the extinction of the disease. We give the conditions of existence of unique positive solution and the stochastic extinction of the SDE model and discuss the exponential π-stability and global stability of the SDE model. One of the most interesting findings is that if the intensity of noise is large, then the disease is prone to extinction, which can provide us with some useful control strategies to regulate disease dynamics.
1. Introduction Recent years, a number of mathematical models have been formulated to describe the impact of media coverage on the dynamics of infectious diseases [1β10]. Mass media (television, radio, newspapers, billboards, and booklets) has been used as a way of delivering preventive health messages as it has the potential to influence peopleβs behavior and deter them from risky behavior or from taking precautionary measures in relation to a disease outbreak [7, 11, 12]. Hence, media coverage has an enormous impact on the spread and control of infectious diseases [2, 3, 9]. On the other hand, for human disease, the nature of epidemic growth and spread is inherently random due to the unpredictability of person-to-person contacts [13], and population is subject to a continuous spectrum of disturbances [14, 15]. In epidemic dynamics, stochastic differential equation (SDE) models could be the more appropriate way of modeling epidemics in many circumstances and many realistic stochastic epidemic models can be derived based on their deterministic formulations [16β28]. In [10], Liu investigated an SIRS epidemic model incorporating media coverage with random perturbation. He assumed that stochastic perturbations were of white noise type, which were directly proportional to distance susceptible π(π‘), infectious πΌ(π‘), and recover π
(π‘) from values of endemic
equilibrium point (πβ , πΌβ , π
β ), influence on the ππ(π‘)/ππ‘, ππΌ(π‘)/ππ‘, ππ
(π‘)/ππ‘, respectively. In fact, besides the possible equilibrium approach in [10], there are different possible approaches to introduce random effects in the epidemic models affected by environmental white noise from biological significance and mathematical perspective [28β30]. Some scholars [17, 28, 30, 31] demonstrated that one or more system parameter(s) can be perturbed stochastically with white noise term to derive environmentally perturbed system. In [10], the author proved that the endemic equilibrium of the stochastic model is asymptotically stable in the large. Therefore, it is natural to ask how environmental fluctuations of the contact coefficient affect the extinction of the disease. In this paper, we will focus on the effects of environmental fluctuations on the diseaseβs extinction through studying the stochastic dynamics of an SIRS model incorporating media coverage. The rest of this paper is organized as follows. In Section 2, based on the results of Cui et al. [2] and [10], we derive the stochastic differential SIRS model incorporating media coverage. In Section 3, we give the conditions of existence of unique positive solution and the stochastic extinction of the SDE model. In Section 4, we provide some examples to support our research results. In the last section, we provide a brief discussion and the summary of main results.
2
Abstract and Applied Analysis
2. Model Derivation and Related Definitions 2.1. Model Derivation. Let π(π‘) be the number of susceptible individuals, πΌ(π‘) the number of infective individuals, and π
(π‘) the number of removed individuals at time π‘, respectively. Based on the work of Cui et al. [2] and [10], we consider the SIRS epidemic model incorporating media coverage as follows: π½πΌ ππ = Ξ β ππ β (π½1 β 2 ) ππΌ + ππ
, ππ‘ π+πΌ π½πΌ ππΌ = (π½1 β 2 ) ππΌ β (π + πΌ + π) πΌ, ππ‘ π+πΌ
(1)
where Ξ is the recruitment rate, π represents the natural death rate, π is the loss of constant immunity rate, πΌ is the diseases induced constant death rate, and π is constant recovery rate. π½1 is the usual contact rate without considering the infective individuals and π½2 is the maximum reduced contact rate due to the presence of the infected individuals. No one can avoid contacting with others in every case, so it is assumed that π½1 > π½2 . The half-saturation constant π > 0 reflects the impact of media coverage on the contact transmission. The function πΌ/(π + πΌ) is a continuous bounded function which takes into account disease saturation or psychological effects. For model (1), the basic reproduction number (2)
is the threshold of the system for an epidemic to occur. Model (1) has a disease-free equilibrium πΈ0 = (Ξ/π, 0, 0) and the endemic equilibrium if π
0 > 1. The disease-free equilibrium is globally asymptotically stable if π
0 β€ 1 and unstable if π
0 > 1. The endemic equilibrium is globally asymptotically stable if π
0 > 1. These results of model (1) were studied in [10]. If we replace the contact rate π½1 in model (1) by π½1 + π(ππ΅/ππ‘), where ππ΅/ππ‘ is a white noise (i.e., π΅(π‘) is a Brownian motion), model (1) becomes as follows: π½2 πΌ ) ππΌ + ππ
] ππ‘ + πππΌ ππ΅ (π‘) , π+πΌ
π½2 πΌ ) ππΌ β (π + πΌ + π) πΌ] ππ‘ + πππΌ ππ΅ (π‘) , π+πΌ ππ
= (ππΌ β (π + π) π
) ππ‘. (3)
Obviously, the stochastic model (3) has the same diseasefree equilibrium πΈ0 = (Ξ/π, 0, 0) as model (1). Throughout this paper, let (Ξ©, F, P) be a complete probability space with a filtration {Fπ‘ }π‘βR+ satisfying the usual conditions (i.e., it is right continuous and increasing while F0 contains all P-null sets). Define a bounded set Ξ as follows: Ξ = {(π, πΌ, π
) β R3+ : 0 < π + πΌ + π
0,
ππ
= ππΌ β (π + π) π
, ππ‘
ππ = [Ξ β ππ β (π½1 β
2.2. Related Definitions. Consider the general π-dimensional stochastic differential equation
Ξ } β R3+ . π
(4)
1 σ΅¨ σ΅¨ lim sup log σ΅¨σ΅¨σ΅¨π₯ (π‘, π₯0 )σ΅¨σ΅¨σ΅¨ < 0 a.s.; π‘ββ π‘
(9)
(v) exponentially π-stable if there is a pair of positive constants πΆ1 and πΆ2 such that for all π₯0 β Rπ , σ΅¨ σ΅¨ σ΅¨π σ΅¨π E (σ΅¨σ΅¨σ΅¨π₯ (π‘, π₯0 )σ΅¨σ΅¨σ΅¨ ) β€ πΆ1 σ΅¨σ΅¨σ΅¨π₯0 σ΅¨σ΅¨σ΅¨ πβπΆ2 π‘
on π‘ β₯ 0.
(10)
3. Dynamics of the SDE Model (3) In what follows, we first use the method of Lyapunov functions to find conditions of existence of unique positive solution of model (3). 3.1. Existence of Unique Positive Solution of Model (3). In this subsection, we show the existence of the unique positive global solution of SDE model (3). Theorem 2. Consider model (3), for any given initial value (π(0), πΌ(0), π
(0)) β Ξ; then there is a unique solution (π(π‘), πΌ(π‘), π
(π‘)) on π‘ β₯ 0 and it will remain in R3+ with probability one. Proof. The proof is almost identical to Theorem 2 of [33], but for completeness we repeat it here. Let (π(0), πΌ(0), π
(0)) β Ξ. Summing up the three equations in (3) and denoting π(π‘) = π(π‘) + πΌ(π‘) + π
(π‘), we have ππ (π‘) = (Ξ β ππ (π‘) β πΌπΌ (π‘)) ππ‘.
(11)
Abstract and Applied Analysis
3
Then, if (π(π ), πΌ(π ), π
(π )) β R3+ for all 0 β€ π β€ π‘ almost surely (briefly a.s.), we get (Ξ β (π + πΌ) π (π )) ππ β€ ππ (π ) β€ (Ξ β ππ (π )) ππ a.s. (12) Hence, by integration, we check
Ξ Ξ + (π (0) β ) πβππ . π π
Ξ π2 Ξ (π½1 + π½2 + ) := π. π π
ππ (π, πΌ, π
) β€ πππ + π (πΌ β π) ππ΅ (π ) ,
π‘β§ππ
β«
0
for all π β [0, π‘] a.s. (14)
ππ = inf {π‘ β [0, ππ ] : π (π‘) β€ π or πΌ (π‘) β€ π or π
(π‘) β€ π} . (15) Then
ππ (π (π ) , πΌ (π ) , π
(π ))
β€β«
π‘β§ππ
0
π‘β§ππ
πππ + π β«
0
(22) (πΌ (π ) β π (π )) ππ΅ (π ) .
Taking the expectations of the above inequality leads to Eπ (π (π‘ β§ ππ ) , πΌ (π‘ β§ ππ ) , π
(π‘ β§ ππ )) β€ π (π (0) , πΌ (0) , π
(0)) + ππ‘.
π = lim ππ
Eπ (π (π‘ β§ ππ ) , πΌ (π‘ β§ ππ ) , π
(π‘ β§ ππ ))
+ E [I(ππ >π‘) π (π (π‘ β§ ππ ) , πΌ (π‘ β§ ππ ) , π
(π‘ β§ ππ ))]
πβ0
= inf {π‘ β [0, ππ ] : π (π‘) β€ 0 or πΌ (π‘) β€ 0 or π
(π‘) β€ 0} . (16) Define a πΆ2 -function π : R3+ β R+ by (17)
By the ItΛoβs formula, for all π‘ β₯ 0, π β [0, π‘ β§ ππ ], we obtain ππ = β +
1 1 1 ππ + ππΌ ππ ππ β 2 π (π ) πΌ (π ) 2π(π ) 1 1 1 ππ
+ ππΌ ππΌ β ππ
ππ
π
(π ) 2πΌ(π )2 2π
(π )2
(18)
π½2 ππΌ π2 2 + (π + πΌ2 ) π+πΌ 2
π2 2 β€ 3π + 2π + πΌ + π½1 πΌ + π½2 π + (π + πΌ2 ) . 2
β₯ E [I(ππ β€π‘) π (π (ππ ) , πΌ (ππ ) , π
(ππ ))] , where Iπ΄ is the indicator function of π΄. Note that there is some component of (π(ππ ), πΌ(ππ ), π
(ππ )) equal to π; therefore, π(π(ππ ), πΌ(ππ ), π
(ππ )) β₯ β log(ππ/Ξ) > 0. Thereby Eπ (π (π‘ β§ ππ ) , πΌ (π‘ β§ ππ ) , π
(π‘ β§ ππ )) β₯ β log (
ππ ) P (π β€ π‘) . Ξ (25)
P (π β€ π‘) β€ β
π (π (0) , πΌ (0) , π
(0)) + ππ‘ . log (ππ/Ξ)
(26)
Let π β 0; we obtain, for all π‘ β₯ 0, P(π β€ π‘) = 0. Hence, P(π = β) = 1. As ππ β₯ π, then ππ = π = β a.s. which completes the proof of the theorem.
where
π½ πΌ2 ππ
Ξ πΌπΌ β 2 β β π½1 π β β π+πΌ π π π
(24)
Combining (23) with (25) gives, for all π‘ β₯ 0,
β πΏπππ + π (πΌ (π ) β π (π )) ππ΅ (π ) ,
πΏπ = 3π + 2π + πΌ + π½1 πΌ +
(23)
On the other hand, in view of (14), we have π(π(π‘ β§ ππ ), πΌ(π‘ β§ ππ ), π
(π‘ β§ ππ )) > 0. It then follows that
= E [I(ππ β€π‘) π (π (π‘ β§ ππ ) , πΌ (π‘ β§ ππ ) , π
(π‘ β§ ππ ))]
ππ ππΌ ππ
) β log ( ) β log ( ) . Ξ Ξ Ξ
(21)
which implies that
Since the coefficients of model (3) satisfy the local Lipschitz condition, there is a unique local solution on [0, ππ ), where ππ is the explosion time. Therefore, the unique local solution to model (3) is positive by the ItΛoβs formula. Now, let us show that this solution is global; that is, ππ = β a.s. Let π0 > 0 such that π(0), πΌ(0), π
(0) > π0 . For π β€ π0 , define the stop-times
π (π, πΌ, π
) = β log (
(20)
(13)
Then, 0 < Ξ/(π + πΌ) < π(π ) < Ξ/π a.s., so, Ξ 3 (π (π ) , πΌ (π ) , π
(π )) β (0, ) π
πΏπ β€ 3π + 2π + πΌ +
Substituting this inequality into (18), we see that
Ξ Ξ + (π (0) β ) πβ(π+πΌ)π π+πΌ π+πΌ β€ π (π ) β€
By (14) we assert that (π(π ), πΌ(π ), π
(π )) β (0, Ξ/π) for all π β [0, π‘ β§ ππ ] a.s. Hence
(19)
From Theorem 2 and (14), we can conclude the following corollary. Corollary 3. The set Ξ is almost surely positive invariant of model (3); that is, if (π(0), πΌ(0), π
(0)) β Ξ, then P((π(π‘), πΌ(π‘), π
(π‘)) β Ξ) = 1 for all π‘ β₯ 0.
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Abstract and Applied Analysis
3.2. Stochastic Extinction of Model (3). In this subsection, we investigate stochastic stability of the disease-free equilibrium πΈ0 = (Ξ/π, 0, 0) in almost sure exponential and exponential π stability by using the suitable Lyapunov function and other techniques of stochastic analysis. The following theorem gives a sufficient condition for the almost surely exponential stability of the disease-free equilibrium πΈ0 = (Ξ/π, 0, 0) of model (3). 2
π½12 /2π,
Theorem 4 (almost sure exponential stability). If π > then disease-free πΈ0 = (Ξ/π, 0, 0) of model (3) is almost surely exponentially stable in Ξ. Proof. Define a function π by Ξ π (π, πΌ, π
) = log ( β π + πΌ + π
) . π
=
(27)
+
π2 π π2 π π2 π ππ ππΌ + ππ ππ
+ ππΌ ππ
ππππΌ ππππ
ππΌππ
=
1 2
2((Ξ/π) β π + πΌ + π
) 2
π½2 β 2π2 π Ξ β π (0) + πΌ (0) + π
(0)) + 1 2 π‘ + πΊ (π‘) , π 2π (30)
where πΊ(π‘) is a martingale defined by πΊ(π‘) = 2π β«0 π ππ΅(π ). In virtue of Corollary 3, the solution of model (3) remains in Ξ. It then follows that (31)
a.s. (32)
which is the required assertion.
(ππ ππ + ππΌ ππΌ)
We now consider the concept of exponential π-stability. The following lemma gives sufficient conditions for exponential π-stability of stochastic systems in terms of the Lyapunov functions (see [32]).
ππ ππΌ
2π½2 ππΌ2 π+πΌ
β (π + πΌ) πΌ β (π + 2π) π
) ππ‘ 2π2 π2 πΌ2 2
ππ‘ +
2πππΌ ππ΅ (Ξ/π) β π + πΌ + π
= (2π½1 π β 2π2 π β π) ππ‘ + 2π ππ΅ π β
β€ log (
Lemma 5 (see [32]). Suppose that there exists a function π(π§, π‘) β πΆ2 (Ξ©) satisfying the following inequalities:
Γ (βΞ + ππ + 2π½1 ππΌ β
((Ξ/π) β π + πΌ + π
)
Ξ β π (π‘) + πΌ (π‘) + π
(π‘)) π
π½2 β 2π2 π 1 Ξ lim sup log ( β π + πΌ + π
) β€ 1 2 0 and πΎπ (π = 1, 2, 3) is positive constant. Then the equilibrium of mode (3) is exponentially π-stable for π‘ β₯ 0. When π = 2, it is usually said to be exponentially stable in mean square and the the equilibrium is globally asymptotically stable. From the above Lemma, we obtain the following theorem.
2π½ ππΌ2 1 ( 2 + πΌπΌ + 2ππ
) ππ‘ (Ξ/π) β π + πΌ + π
π + πΌ
β€ (2π½1 π β 2π2 π β π) ππ‘ + 2π π ππ΅, (28)
Theorem 6 (exponential π-stability). Let π β₯ 2. If the conditions π
0 = π½1 Ξ/π(π + πΌ + π) < 1 and π
0π := π
0 + ((π β 1)Ξ2 π2 /2π2 (π + πΌ + π)) < 1 hold, the disease-free equilibrium πΈ0 = (Ξ/π, 0, 0) of model (3) is πth moment exponentially stable in Ξ.
Abstract and Applied Analysis
5
Proof. Let π β₯ 2 and (π(0), πΌ(0), π
(0)) β Ξ; in view of Corollary 3, the solution of model (3) remains in Ξ. We define the Lyapunov function π(π, πΌ, π
) as follows:
π = π1 (
π Ξ 1 β π) + πΌπ + π2 π
π , π π
(35)
where π1 > 0 and π2 > 0 are real positive constants that are to be chosen later. It is easy to check that inequalities (33) are true. Furthermore, by the ItΛoβs formula, it follows from π, πΌ, π
β (0, Ξ/π) that πβ1 π½ ππΌ2 Ξ β π) (Ξ β ππ β π½1 ππΌ + 2 + ππ
) π π+πΌ
πΏπ = βπ1 π(
+πΌ +
1 (π β 1) π2 π2 πΌπ + π2 ππ
πβ1 (ππΌ β (π + π) π
) 2
π΄2 = π + πΌ + π β β(
π½1 Ξ 1 β 2 (π β 1) π2 Ξ2 ) πΌπ π 2π
β π2 π (π + π) π
π + π2 πππΌπ
πβ1 . (36)
π΄ 3 = π2 (π (π + π) β π (π β 1) π) . (39) In view of π
0 + ((π β 1)Ξ2 π2 /2π2 (π + πΌ + π)) < 1, we have π+πΌ+πβ(π½1 Ξ/π)β(1/2π2 )(πβ1)π2 Ξ2 > 0. Hence, we chose π sufficiently small and π1 , π2 are positive such that π΄ 1 , π΄ 2 , π΄ 3 > 0. According to Lemma 5 the proof is completed.
ππ+1 = ππ + (Ξ β πππ β π½1 ππ πΌπ + + πππ πΌπ βΞπ‘ππ +
Using the fact that
(
Ξ β π) π
πΌβ€
πΌπ+1 = πΌπ + (π½1 ππ πΌπ β
π
πβ1 Ξ 1 π( β π) + π1βπ πΌπ , π π π
πβ2 π πβ2 Ξ Ξ 2 β π) πΌ2 β€ π( β π) + π(2βπ)/2 πΌπ , π π π π
π
πβ1 πΌ β€
π½1 Ξ 1βπ π2 Ξ2 (π β 1) (2βπ)/2 π ) π1 β π2 ππ1βπ π + π π2
In this section, as an example, we give some numerical simulations to show different dynamic outcomes of the deterministic model (1) versus its stochastic version (3) with the same set of parameter values by using the Milstein method mentioned in Higham [34]. In this way, model (3) can be rewritten as the following discretization equations:
πβ2 Ξ 1 + 2 π (π β 1) π1 π2 Ξ2 ( β π) πΌ2 2π π
(
π½1 Ξ 1 β 2 (π β 1) π2 Ξ2 π 2π
4. Numerical Simulations and Dynamics Comparison
π πβ1 π π½ πΞ Ξ Ξ β€ βπ1 ππ( β π) + 1 1 ( β π) πΌ π π π
πβ1
π½1 Ξ (π β 1) π2 Ξ2 (π β 1) ) π) π1 , + π 2π2
Corollary 7 (globally asymptotically stable). If the conditions π
0 = π½1 Ξ/π(π + πΌ + π) < 1 and π
0π := π
0 + (Ξ2 π2 /2π2 (π + πΌ + π)) < 1 hold, the disease-free equilibrium πΈ0 = (Ξ/π, 0, 0) of model (3) is globally asymptotically stable in Ξ.
π½ ππΌ2 (π½1 ππΌ β 2 β (π + πΌ + π) πΌ) π+πΌ
β (π + πΌ + π β
π΄ 1 = (ππ β (
Under Lemma 5 and Theorems 6, we have in the case π = 2 the following corollary.
πβ2 Ξ 1 + π (π β 1) π1 π2 π2 πΌ2 ( β π) 2 π πβ1
where
π β 1 π 1 1βπ π ππ
+ π πΌ , π π
π2 π πΌ (π2 β 1) Ξπ‘, 2 π π π
π
π+1 = π
π + (ππΌπ β (π + π) π
π ) Ξπ‘,
Ξ = 1, π
Ξ β π) β π΄ 2 πΌπ β π΄ 3 π
π , π
(40)
where ππ , π = 1, 2, . . . , π, are the Gaussian random variables π(0, 1). For the deterministic model (1) and its stochastic model (3), the parameters are taken as follows:
we get
πΏπ β€ βπ΄ 1 (
π2 π πΌ (π2 β 1) Ξπ‘, 2 π π π
π½2 ππ πΌπ2 β (π + πΌ + π) πΌπ ) Ξπ‘ π + πΌπ
+ πππ πΌπ βΞπ‘ππ + (37)
π½2 ππ πΌπ2 + ππ
π ) Ξπ‘ π + πΌπ
(38)
π = 0.03,
π = 0.01,
πΌ = 0.1,
π½1 = 0.02, π = 0.05,
π½2 = 0.018, π = 10. (41)
6
Abstract and Applied Analysis 14 12 10 8 6 4 2 0
0
10
20
30
40
50 t
60
70
80
90
100
S I R
Figure 1: The paths of π(π‘), πΌ(π‘), and π
(π‘) for the deterministic model (1) with initial values (π(0), πΌ(0), π
(0)) = (9, 1, 0). The parameters are taken as (41) (π
0 = 7.407).
35
25
30
20
25 15
20 15
10
10 5
5 0
0
10
20
30
40
50 t
60
70
80
90
100
0
0
10
20
30
40
50 t
60
70
80
90
100
S I R
S I R
(a) π = 0.1
(b) π = 0.02
Figure 2: The paths of π(π‘), πΌ(π‘), and π
(π‘) for the stochastic model (3) with initial values (π(0), πΌ(0), π
(0)) = (9, 1, 0). The parameters are taken as (41) (π
0 = 7.407).
(1) The Endemic Dynamics of the Deterministic Model (1). For the deterministic model (1), π
0 = π½1 Ξ/π(π + πΌ + π) = 7.407 > 1; thus, it admits a unique endemic equilibrium πΈβ = (8.1035, 9.7664, 12.2080) which is globally stable for any initial values (π(0), πΌ(0), π
(0)) β Ξ according to [10] (see, Figure 1).
(2) The Stochastic Dynamics of Model (3). For the corresponding stochastic model (3), we choose π = 0.1; then, we have 0.01 = π2 > π½12 /2π = 0.007. Thus, from Theorem 4, we can conclude that for any initial value (π(0), πΌ(0), π
(0)) β Ξ, disease-free πΈ0 = (Ξ/π, 0, 0) of model (3) is almost surely exponentially stable in Ξ (see Figure 2(a)).
Abstract and Applied Analysis To see the disease dynamics of model (3) more, we decrease the noise intensity π to be 0.02 and keep the other parameters unchanged. Then, we have 0.0004 = π2 < π½12 /2π = 0.007. Therefore, the condition of Theorem 4 is not satisfied. In this case, our simulations suggest that model (3) is stochastically persistent (see Figure 2(b)).
5. Concluding Remarks In this paper, we propose an SIRS epidemic model with media coverage and environment fluctuations to describe disease transmission. It is shown that the magnitude of environmental fluctuations will have an effective impact on the control and spread of infectious diseases. In a nutshell, we summarize our main findings as well as their related biological implications as follows. Theorem 4 and [10] combined with numerical simulations (see Figures 1 and 2) provide us with a full picture on the dynamics of the deterministic model (1) and stochastic model (3). In [10], the authors showed that the deterministic model (1) admits a unique endemic equilibrium πΈβ which is globally asymptotically stable if its basic reproduction number π
0 > 1 (see Figure 1). If the magnitude of the intensity of noise π is large, that is, π2 > π½12 /2π, the extinction of disease in the stochastic model (3) occurs whether π
0 is greater than 1 or less than 1 (see Figure 2(a)). While the magnitude of the intensity of noise π is small, one of our most interesting findings is that disease may persist if π
0 > 1, (see Figure 2(b)). Needless to say, both equilibrium possible approach and parameter possible approach in the present paper have their important roles to play. Obviously, our results in the present paper may be a useful supplement for [10].
Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments This research was supported by the National Science Foundation of China (61373005, 11201344, and 11201345) and Zhejiang Provincial Natural Science Foundation (LY12A01014).
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