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Dynamics Analysis of an Epidemiological Model with Media Impact and Two Delays.

1. Introduction

The investigation of dynamics of epidemiological models has been of importance in improving our understanding of disease control [1-6]. Media education has been an important control strategy for the emerging and reemerging epidemics, such as HIV/AIDS [7], SARS [2], Ebola virus disease (EVD), Middle East Respiratory Syndrome (MERS), which can not only alert the general public to the hazard from the infectious diseases but also educate the people about the requisite preventive measures such as wearing protective masks, vaccination, voluntary quarantine, and avoidance of congregated places. The extensive media education will bring about reducing the frequency and probability of potentially contagious contacts among the well-informed people [2-4]. In order to describe the impact of media on the diseases, Cui and his coauthors [3] used the transmission rate of the form [beta] exp(-[alpha]I) in SEI model with logistic growth, where [beta] is the transmission rate before media alert and I denotes the number of infected individuals. This work provided a theoretical basis that a Hopf bifurcation can occur for weak media impact (small values of a) while the model may have up to three endemic equilibria for strong media impact (large values of a). Liu and Cui [4] proposed the transmission rate taking the form [beta] - [[beta].sub.1] I/(m + I) to capture the impact of media on disease spread, where [[beta].sub.1] [less than or equal to] [beta] represents the reduced maximum value of the transmission rate when I approaches infinite and m reflects the reactive velocity of media coverage and individuals to the epidemic disease. For more details concerning the application of this transmission rate, we refer the reader to recent works [5, 6].

On the other hand, the individuals infected by infectious disease may develop symptoms after an incubation period [8], such as Hepatitis B virus (HBV), Hepatitis C virus (HCV), the human tuberculosis (TB), and Herpes simplex virus type 2 (HSV-2). The average latency period after the genital acquisition of HSV-2 is approximately 4 days [9], and latent tuberculosis may take months, years, or even decades to become infectious. Moreover, it has been found clinically that numerous diseases may make the recovered individuals suffer from a relapse of symptoms, including HBV [10], HCV [11], the majority of TB due to incomplete treatment [12], and genital HSV-2 [13, 14]. Recently, many epidemiological models incorporating both latency and relapse have been extensively investigated and many good results have been obtained (e.g., [15-19]). However, there are few investigation on both latency and relapse delays in the epidemiological models with media impact.

Suppose that the total population N(t) at time t is divided into four disjoint epidemic subclasses: susceptible S(t), latent/exposed E(t), infectious I(t), and temporarily recovered R(t), respectively And A denotes recruitment rate of susceptible class S(t), [mu] is natural death rate, v indicates the death rate due to the disease, and [eta] represents the recovered rate for infectious class I(t) due to natural recovery or treatment. As pointed by Cui et al. in [5], media can effectively reduce the contact rates among the population to a limited level. Hence, it may be more realistic to use the transmission rate [beta] - [[beta].sub.1] I(t)/(m + I(t)) compared to [beta] exp(-mI(t)). By incorporating media impact into the bilinear incidence rate, we now consider the incidence rate function as follows:

f(S(t), I(t)) = ([beta] - [[beta].sub.1] I(t)/m + I(t). (1)

In this work, we assume that the latency and relapse periods are constants, denoted by [[tau].sub.1] and [[tau].sub.2], respectively. Hence, the probabilities [P.sub.1] (t) and [P.sub.2] (t) of remaining in the latent class and the temporarily recovered class are the step-functions taking the forms

[mathematical expression not reproducible]. (2)

This suggests that all individuals remain in latent class for a constant period [[tau].sub.1] and in temporarily recovered class for a constant period [[tau].sub.2]. One further assumes that the disease has been in the population for at least a time of t > [tau] := max{[[tau].sub.1], [[tau].sub.2]}.

Following closely the ideas of [5, 8, 15, 18] and incorporating media impact, we consider the following integro-differential epidemic model with latency and relapse delays:

[mathematical expression not reproducible]. (3)

Differentiating the second and the fourth equations of (3), we derive the delay model

[mathematical expression not reproducible], (4)

where the term [mathematical expression not reproducible] indicates the individuals surviving in the latent period [[tau].sub.1] and entering into infectious class at time t and the term [mathematical expression not reproducible] represents the individuals surviving in temporarily recovered period [[tau].sub.2] and entering into infectious class at time t. The initial conditions for model (4) are given by

[mathematical expression not reproducible]. (5)

Here, let [phi] := [([[phi].sub.1], [[phi].sub.2], [[phi].sub.3], [[phi].sub.4]).sup.T] [member of] C, which denotes the Banach space C([-[tau], 0], [R.sup.4.sub.+0]) of continuous functions mapping the interval [-[tau], 0] into [R.sup.4.sub.+0], equipped with the uniform norm defined by [mathematical expression not reproducible]. In consideration of the continuity of the initial conditions, one requires

[mathematical expression not reproducible]. (6)

Our main aim of this study is concerned with investigating the global dynamics of model (4) and the impact of media on the disease spread. The basic structure of this paper is as follows. In the next section, we study the existence and the local stability of equilibria of (4). Section 3 carefully addresses the permanence of (4). In Section 4, global stability analysis of (4) is carried out. Finally, a discussion section ends this paper.

2. The Equilibria

2.1. The Existence of Equilibria. Throughout this paper, denote k := [mu] + v + [eta].

Lemma 1. Any solution of model (4) with the initial conditions (5) and (6) is unique, positive, and bounded on [0, + [infinity]). Moreover, the biologically feasible region

[OMEGA] = {(S, E, I, R) [member of] [R.sup.4.sub.+] : S + E + I + R [less than or equal to] A/[mu]} (7)

is a positive invariant with respect to (4).

Proof. From the fundamental theory of functional differential equations [20], (4) admits a unique solution [(S(t), E(t), I(t), R(t)).sup.T] satisfying the initial conditions (5) and (6).

Firstly, one shows that the solution S(t) is positive, [for all]t [greater than or equal to] 0. If not, we assign [t.sub.1] > 0 to be the first time such that S([t.sub.1]) = 0, which implies that S(t) > 0 for t [member of] [0, [t.sub.1]). Thus, we must have dS([t.sub.1])/dt = A > 0 from the first equation of (4). Then there is a sufficiently small constant [member of] > 0 such that S(t) < 0 holds for t [member of] ([t.sub.1] - [epsilon], [t.sub.1]). This leads to a contradiction. So S(t) > 0 for t [greater than or equal to] 0. Secondly, I(t) > 0 also holds [for all]t [greater than or equal to] 0. In fact, if I(t) would lose its positivity and [t.sub.2] > 0 were the first time such that I([t.sub.2]) = 0, then I(t) > 0 for t [member of] [0, [t.sub.2]). Solving the third equation of (4) on [0, [t.sub.2]] gives

[mathematical expression not reproducible]. (8)

Due to S([rho]), I([rho]) > 0 [for all][rho] [member of] [0, [t.sub.2]), the right hand side of the above equality is positive, which yields that I ([t.sub.2]) > 0, contracting to I ([t.sub.2]) = 0. Thirdly, since the second equation of (4) is equivalent to the second equation of (3) and S(t), I(t) > 0, it follows from the second equation of (3) that E(t) > 0 [for all]t [greater than or equal to] 0. Similarly, we can obtain that R(t) > 0 [for all]t [greater than or equal to] 0. The positivity of solutions is proved.

Finally, the boundedness of the solutions is shown. Since dN(t)/dt = A - [mu]N(t) - vI(t) [less than or equal to] A - [mu]N(t), we get that [lim sup.sub.t [right arrow] [infinity]] N(t) [less than or equal to] A/[mu]. This suggests that S(t), E(t), I(t), R(t) are bounded on [0, + [infinity]). Hence, the feasible region [OMEGA] is a positive invariance that attracts all solutions of (4) in [R.sup.4.sub.+]. The proof is completed. ?

Theorem 2. Model (4) admits a unique endemic equilibrium (EE) [E.sup.*] ([S.sup.*], [E.sup.*], [I.sup.*], [R.sup.*]) if [R.sub.0] > 1, and there always exists a disease-free equilibrium (DFE) [E.sub.0] ([S.sub.0], 0, 0, 0).

Proof. Model (4) always has a disease-free equilibrium [E.sub.0] = ([S.sub.0], 0, 0, 0), where [S.sub.0] = A/[mu]. Applying the theory of the next generation matrix in [21], we derive that

[mathematical expression not reproducible] (9)

is the basic reproduction number of (4), which stands for the average number of new infections brought out by a typical infectious individual during the whole infectious period [22].

Let (S, E, I, R) be any positive equilibrium (if it exists), then

[mathematical expression not reproducible]. (10)

Solving the second and the fourth equations of(10) yields that

[mathematical expression not reproducible]. (11)

From the first and the third equations of (10), eliminating f(S, I) leads to

[mathematical expression not reproducible]. (12)

Denote [mathematical expression not reproducible]. Because of S = (A - aI)/[mu] > 0, one has I < A/a. Now, one turns to studying the following equation:

[phi] (I) := f (A - aI/[mu], I) - aI = 0, I [member of] [0, A/a]. (13)

By f(S,0) = f(0,I) = 0, we get that [phi](0) = 0, [phi](A/a) = -A < 0. Applying [partial derivative]f([S.sub.0], 0)/[partial derivative]S = 0 gives

[phi]' (0) = a/[mu] x [partial derivative]f ([S.sub.0], 0)/[partial derivative]S + [partial derivative]f ([S.sub.0], 0)/[partial derivative]I = a ([R.sub.0] - 1) > 0, (14)

whence one obtains [phi](I) > 0 when [R.sub.0] > 1 and I is sufficiently small. One thus deduces that (13) admits a positive real root, denoted by [I.sup.*]. This also suggests that model (4) at least admits positive equilibrium [E.sup.*] ([S.sup.*], [E.sup.*], [I.sup.*], [R.sup.*]) from (11).

In fact, [E.sup.*] is proved to be a unique EE. From (13), it follows that a = f([S.sup.*], [I.sup.*])/[I.sup.*]. Due to [beta] [greater than or equal to] [[beta].sub.1], we can examine that [partial derivative]f([S.sup.*], [I.sup.*])/[partial derivative]S > 0 and

[mathematical expression not reproducible], (15)

which leads to

[mathematical expression not reproducible], (16)

which implies that [phi](I) strictly decreases at any positive equilibrium [E.sup.*]. Note that [phi](I) is continuously differentiable on [0, A/a]. Assume that (13) has more than one positive root; then there must exist certain one positive equilibrium [E.sup.+]([S.sup.+], [E.sup.+], [I.sup.+], [R.sup.+]) such that [phi]'([I.sup.+]) [greater than or equal to] 0, resulting in a contraction. Thus, [I.sup.*] is unique and so is [E.sup.*] if [R.sub.0] > 1.

2.2. The Local Stability of Equilibria. In the following sections, one uses the notations [mathematical expression not reproducible].

Theorem 3. For model (4), the DFE [E.sub.0] is locally asymptotically stable if [R.sub.0] < 1 but unstable if [R.sub.0] > 1. Moreover, the EE [E.sup.*] is locally asymptotically stable if [R.sub.0] > 1.

Proof. The characteristic equation of model (4) at some equilibrium is calculated as

[mathematical expression not reproducible]. (17)

(1) By [partial derivative]f([S.sub.0], 0)/[partial derivative]S = 0, evaluating (17) at [E.sub.0]([S.sub.0], 0, 0, 0) yields

[mathematical expression not reproducible]. (18)

Clearly, an eigenvalue of (18) is [[lambda].sub.1] = -[mu] < 0, and the remaining ones satisfy

[mathematical expression not reproducible]. (19)

Suppose that [R.sub.0] > 1. From (19), we directly get

[mathematical expression not reproducible], (20)

which means that (19) has at least one positive root. That is, [E.sub.0] is unstable if [R.sub.0] > 1. Suppose that [R.sub.0] < 1. Assign

[mathematical expression not reproducible]. (21)

Using (19) yields [PSI]([lambda]) = 1. Let [lambda] = x + yi (x, y [member of] R) be any root of (19). If x [greater than or equal to] 0, one has

[mathematical expression not reproducible], (22)

contradicting with (21), and thus x < 0. So [E.sub.0] is locally asymptotically stable if [R.sub.0] < 1.

(2) From (17), the characteristic equation of model (4) at [E.sup.*]([S.sup.*], [E.sup.*], [I.sup.*], [R.sup.*]) reads

[mathematical expression not reproducible]. (23)

Apparently, [[lambda].sub.1] = -[mu] < 0 is an eigenvalue of (23). Furthermore, assume that (23) has another root [[lambda].sub.2] with Re[[lambda].sub.2] [greater than or equal to] 0, and then

[mathematical expression not reproducible]. (24)

From (15) and [mathematical expression not reproducible], we can deduce that

[mathematical expression not reproducible]. (25)

Recall that the left side of (24) satisfies [absolute value of [[lambda].sub.2] + k] [greater than or equal to] k. This contradicts with (24). It follows from Theorem 9.17.4 in [23] that (23) does not admit any root with a nonnegative real part. So [E.sup.*] is locally asymptotically stable if [R.sub.0] > 1. This completes the proof.

3. Permanence

In order to study the permanence of model (4), we first discuss its uniform persistence when [R.sub.0] > 1 by the persistence theory for infinite dimensional systems [24].

Definition 4. Denote [??] as the interior of [OMEGA]. If there is a constant [??] > 0 independent of initial values in [??], such that [lim inf.sub.t [right arrow] [infinity]] S [greater than or equal to] [??], [lim inf.sub.t [right arrow] [infinity]] E [greater than or equal to] [??], [lim inf.sub.t [right arrow] [infinity]] I [greater than or equal to] [??], and [lim inf.sub.t [right arrow] [infinity]] R [greater than or equal to] [??], then model (4) is uniformly persistent in [??].

In the sequel, some notations and terminology are introduced. Denote [PHI](t), t [greater than or equal to] 0, as the family of solution operators with respect to (4). Consider Y [member of] C with the uniform norm [parallel][phi][parallel]. Let us define the [omega]-limit set as [omega](y) := {z [member of] Y | there is a sequence [t.sub.n] [right arrow] [infinity] as n [right arrow] [infinity] with [lim inf.sub.t [right arrow] [infinity]] [PHI]([t.sub.n])y = z}. The semigroup [PHI](t) is referred to as being asymptotically smooth, if for any bounded subset U of Y, for which [PHI](t)U [subset] U [for all]t [greater than or equal to] 0, there is a compact set A such that d([PHI](t)U, A) [right arrow] 0 as t [right arrow] [infinity]. Set

[mathematical expression not reproducible]. (26)

It can be seen that [Y.sub.0] = Y/[Y.sup.0] = [partial derivative]Y, where [partial derivative]Y represents the boundary of Y.

Lemma 5 (see [24], Theorem 4.2). Let the following conditions be satisfied:

(i) [Y.sup.0] is open and dense in Y with [Y.sup.0] [union] [Y.sub.0] = Y and [Y.sup.0] [intersection] [Y.sub.0] = [empty set].

(ii) The solution operators [PHI](t) satisfy [PHI](t) : [Y.sup.0] [right arrow] [Y.sup.0], [PHI](t) : [Y.sub.0] [right arrow] [Y.sub.0].

(iii) [PHI](t) is point dissipative in Y.

(iv) [[gamma].sup.+](U) is bounded in Y if U is bounded in Y.

(v) [PHI](t) is asymptotically smooth.

(vi) [mathematical expression not reproducible] is isolated and has an acyclic covering M, where [B.sub.b] is the global attractor of T(t) restricted to [mathematical expression not reproducible].

(vii) [for all] [M.sub.i] [member of] M, [W.sup.s] ([M.sub.i]) [intersection] = [empty set] holds, where [W.sup.s] denotes the stable set.

Then [THETA](t) is a uniform repeller with respect to [Y.sup.0]; that is, there exists a constant [??] > 0 such that, [for all]y [member of] [Y.sup.0], [lim inf.sub.t [right arrow] [infinity]]([PHI](t)y, [Y.sub.0]) [greater than or equal to] [??].

Theorem 6. Model (4) is permanent provided that [R.sub.0] > 1.

Proof. From (26), one examines that (i) and (ii) clearly hold. And (iii)-(iv) immediately follow from Lemma 1. It is seen that B = {[[??].sub.0]} (here, [[??].sub.0] := ([S.sub.0], 0, 0, 0)) is isolated, which implies that the covering is simply M = {[[??].sub.0]}. Since no orbit connects [[??].sub.0] to itself in [Y.sub.0], we have that M is acyclic. Thus, (vi) is checked out.

Now, we prove that [W.sup.s]([[??].sub.0]) [intersection] [Y.sup.0] = [empty set], where [W.sup.s] ([[??].sub.0]) = {y [member of] [Y.sup.0] : [lim.sub.t [right arrow] [infinity]] [PHI](t)y = [[??].sub.0]}. Assume by contradiction that there is a solution y = (S, E, I, R) [member of] [W.sup.s]([[??].sub.0]) [intersection] [Y.sup.0], such that

[mathematical expression not reproducible]. (27)

From Lemma 1, we know that S, I > 0 for t [greater than or equal to] 0. Choose a Lyapunov function

[mathematical expression not reproducible]. (28)

From (27), it follows that there exists [T.sub.0] > 0 such that U([T.sub.0]) > 0. And the time derivative of U(t) along the solutions of (4) reads

[mathematical expression not reproducible]. (29)

By (27), we can use L'Hospital's rule, yielding

[mathematical expression not reproducible]. (30)

If [R.sub.0] > 1, by (9), then the function U(t) is not decreasing when t is large enough. For the given above [T.sub.0], one thus gets U(t) [greater than or equal to] U([T.sub.0]) for t [greater than or equal to] [T.sub.0]. Note that U([T.sub.0]) > 0 prevents (E, I, R) from converging to (0, 0,0) as t [right arrow] [infinity], This contradicts to S [right arrow] [S.sub.0]. For the dissipative system (4), uniform persistence is equivalent to permanence, completing the proof.

In an epidemiological sense, uniform persistence of model (4) implies that there are always infectious individuals if the disease is initially present and [R.sub.0] > 1.

4. Global Stability

We are now in a position to study the global asymptotic behaviors of model (4).

Theorem 7. The DFE [E.sub.0] of model (4) is globally asymptotically stable in [OMEGA] if [R.sub.0] [less than or equal to] 1.

Proof. We still consider the Lyapunov function U(t) defined in (29). Since f([S.sub.0], I)/I decreases with I and S [less than or equal to] [S.sub.0], applying L'Hospital's rule one obtains that

[mathematical expression not reproducible] (31)

holds [for all]S, I > 0. From (29), the derivative of U(t) along the solutions of (4) reads

[mathematical expression not reproducible] (32)

if [R.sub.0] [less than or equal to] 1. Hence, when [R.sub.0] [less than or equal to] 1, there is a singleton {[E.sub.0]}, as the maximal compact invariant set in {(S, E, I, R) [member of] [OMEGA] : dU(t)/dt = 0}. Applying LaSalle's invariance principle immediately leads to the global asymptotic stability of [E.sub.0] in [OMEGA], completing the proof.

Let us introduce the well-known Volterra-type function, H(x) = x-1-ln x, x > 0. Obviously, H(x) is positive-defined and reaches the global minimum at x = 1 with H(1) = 0.

Theorem 8. The EE [E.sup.*] of model (4) is globally asymptotically stable in [??] if [R.sub.0] > 1.

Observe that the variables E and R in model (4) do not appear in the following subsystem:

[mathematical expression not reproducible]. (33)

From Lemma 1 and Theorems 2, 3, and 6, one can derive the uniqueness, positivity, and boundedness of the solutions of (33). The region [PI] = {(S, I) [member of] [R.sup.2.sub.+] : S + I [less than or equal to] A/[mu]} is a positive invariant with (33) and denoting its interior by n. Furthermore, (33) has a locally asymptotically stable trivial equilibrium [[epsilon].sub.0] = ([S.sub.0], 0) if [R.sub.0] < 1, and it is unstable if [R.sub.0] > 1. There exists one unique locally asymptotically stable nontrivial equilibrium [[epsilon].sub.0]([S.sup.*], [I.sup.*]) if [R.sub.0] > 1. Subsystem (33) is permanent when [R.sub.0] > 1. In order to prove Theorem 8, we first explore the global stability of [[epsilon].sup.*] of (33) based on its permanence.

Lemma 9. The equilibrium [[epsilon].sup.*] of (33) is globally asymptotically stable in [??] if [R.sub.0] > 1.

Proof. Construct the following Lyapunov function V(t) = [W.sub.1](t) + [W.sub.2](t) + [W.sub.3](t), where

[mathematical expression not reproducible]. (34)

It follows from Theorem 6 that the variables and are sufficiently bounded and bounded away from 0. This ensures the boundedness of V(t) for t [greater than or equal to] 0, and thus V(t) is well defined. Together with the properties of H(x), we find that V(t) [greater than or equal to] 0 with global minimum 0 at [epsilon].sup.*]. By [mathematical expression not reproducible], differentiating [W.sub.1] (t) along the solutions of (33) yields

[mathematical expression not reproducible]. (35)

And the time derivatives of [W.sub.2](t) and [W.sub.3](t) along the solutions of (4), respectively, read

[mathematical expression not reproducible]. (36)

From (35)-(36), we therefore have

[mathematical expression not reproducible]. (37)

Note that

[mathematical expression not reproducible]. (38)

Further, it can be found that

[mathematical expression not reproducible]. (39)

We can show that f(S, I) is nondecreasing and concave down with respect to since

[mathematical expression not reproducible], (40)

which ensure that

[mathematical expression not reproducible]. (41)

From the properties of H(x), one deduces that dV(t)/dt [less than or equal to] 0, and the strict equality holds if and only if [mathematical expression not reproducible]. It follows from Theorem 5.3.1 in [25] that the solutions of system (33) converge to [omega], the maximal invariant set in [dV(t)/dt = 0}. Accordingly, we obtain [omega] = ([S.sup.*], [I.sup.*]). LaSalle's invariance principle for delay differential systems (see, e.g., [26]) indicates that [[epsilon].sup.*] is globally asymptotically stable in n. The proof is completed.

Proof of Theorem 8. Let (S, E, I, R) be a positive solution of model (4) with initial conditions (5) and (6). Applying Lemma 9, one derives that [mathematical expression not reproducible]. By L'Hospital's rule, it follows from the second and the fourth equations of (3) that

[mathematical expression not reproducible]. (42)

The fact that [E.sup.*] is local stable when it exists, implies that it is also globally asymptotically stable in [??] if [R.sub.0] > 1. We finish the proof.

5. Concluding Remarks

In this paper, we propose SEIR epidemic model with media impact, which incorporates latent and relapse delays. One focuses on analyzing the permanence and global stability of model (4). In detail, we show that (4) is permanent when [R.sub.0] > 1. Based on the permanence obtained, we carry out global stability analysis of the equilibria by proper Lyapunov functionals. It is found that (4) is still a threshold dynamical system.

Now several numerical simulations are demonstrated to check Theorems 7 and 8. For the purpose of controlling the spread of genital HSV-2, for instance, we implement media propaganda and education. Choose A = [mu] = 1/7300, v = 0, [eta] = 3.5, and [[tau].sub.2] = 27, as used in Blower et al. [27]. Since the average incubation period after the genital acquisition of HSV-2 is approximately 4 days (range, 2 to 12 days) [9], we may take [[tau].sub.1] = 4. When [beta] = 0.0001, [[beta].sub.1] = 0.00005, and m = 10, we then get that [R.sub.0] = 0.0077. Figure 1 shows us that the disease-free equilibrium [E.sub.0] of (4) is globally asymptotically stable for two different initiate conditions (S(0),E(0),I(0),R(0)) = (0.69,0.05,0.3,0.05) (green line) and (S(0), E(0), 1(0), R(0)) = (0.8,0.05, 0.01,0.05) (blue line). And, given that [beta] = 0.02, [[beta].sub.1] = 0.01, and m = 10, direct calculation gives [R.sub.0] = 1.5308. From Theorem 8 one knows that the unique endemic equilibrium [E.sup.*] is globally asymptotically stable as shown in Figure 2 for the different initiate conditions used in Figure 1. We observe that there is no sustained oscillatory solution and thus media control and the two delays do not have fundamental influence on the qualitative behaviors of model (4).

Additionally, since d[R.sub.0]/d[[tau].sub.1] < 0 and d[R.sub.0]/d[[tau].sub.2] < 0, it is an advantage for controlling the disease spread to increase both latent and relapse delays. In practice, the latent period may be hard to change, but the likelihood of symptomatic recurrence [14] and the frequency of subclinical (asymptomatic) viral shedding [13] can be substantially reduced under suppressive therapy rather than episodic treatment, such that the relapse period (delay) can be lengthened.

Note that media education does not change the basic reproduction number [R.sub.0] [4]. However, the greater the reactive velocity of media coverage and individuals (i.e., the smaller the value of m), the endemic level [I.sup.*] will be controlled to a much lower level, seeing the blue lines shown in Figure 3 (where we change the values of m and [[beta].sub.1] but keep the same initiate condition (S (0), E(0), I(0), R(0)) = (0.8, 0.05, 0.01, 0.05) and the reminding parameters values are the same with Figure 2). On the contrary, if media departments and the public do not respond timely to the epidemic, the effect of media propaganda on the disease transmission is almost the same with the case with no media impact (i.e., [[beta].sub.1] = 0, seeing the red line in Figure 3). Hence, timely response of media coverage and individuals plays a more key role in controlling the epidemic.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publishing of this paper.

Acknowledgments

The work was partially supported by the Open Research Fund Program of Institute of Applied Mathematics Yangtze University (no. KF1507), Basic Subject of Scientific Research and Development Fund of Yangtze University (no. 2014JCY001), and Key Scientic Research Project of Higher Education Institutions of Henan Province (16A110005).

References

[1] W. O. Kermack and A. G. McKendrick, "A contribution to the mathematical theory of epidemics," Proceedings of the Royal Society of London, Series A: Mathematical, Physical and Engineering Sciences, vol. 115, no. 772, pp. 700-721, 1927.

[2] R. S. Liu, J. H. Wu, and H. P. Zhu, "Media/psychological impact on multiple outbreaks of emerging infectious diseases," Computational and Mathematical Methods in Medicine, vol. 8, no. 3, pp. 153-164, 2007.

[3] J. A. Cui, Y. H. Sun, and H. P. Zhu, "The impact of media on the control of infectious diseases," Journal of Dynamics and Differential Equations, vol. 20, no. 1, pp. 31-53, 2008.

[4] Y. Liu and J.-A. Cui, "The impact of media coverage on the dynamics of infectious disease," International Journal of Biomathematics, vol. 1, no. 1, pp. 65-74, 2008.

[5] J.-A. Cui, X. Tao, and H. P. Zhu, "An SIS infection model incorporating media coverage," The Rocky Mountain Journal of Mathematics, vol. 38, no. 5, pp. 1323-1334, 2008.

[6] J. M. Tchuenche, N. Dube, C. P. Bhunu, R. J. Smith, and C. T. Bauch, "The impact of media coverage on the transmission dynamics of human influenza," BMC Public Health, vol. 11, supplement 1, article S5, 2011.

[7] S. Broder and R. C. Gallo, "A pathogenic retrovirus (HTLV-III) linked to AIDS," The New England Journal of Medicine, vol. 311, no. 20, pp. 1292-1297, 1984.

[8] P. van den Driessche, L. Wang, and X. F. Zou, "Modeling diseases with latency and relapse," Mathematical Biosciences and Engineering, vol. 4, no. 2, pp. 205-219, 2007.

[9] D. W. Kimberlin and D. J. Rouse, "Genital herpes," The New England Journal of Medicine, vol. 350, no. 19, pp. 1970-1977, 2004.

[10] A. Marzano, S. Gaia, V. Ghisetti et al., "Viral load at the time of liver transplantation and risk of hepatitis B virus recurrence," Liver Transplantation, vol. 11, no. 4, pp. 402-409, 2005.

[11] A. Y. Kim, J. Schulze zur Wiesch, T. Kuntzen et al., "Impaired hepatitis C virus-specific T cell responses and recurrent hepatitis C virus in HIV coinfection," PLoS Medicine, vol. 3, no. 12, article e492, 2006.

[12] M.-L. Lambert, E. Hasker, A. Van Deun, D. Roberfroid, M. Boelaert, and P. Van der Stuyft, "Recurrence in tuberculosis: relapse or reinfection?" The Lancet Infectious Diseases, vol. 3, no. 5, pp. 282-287, 2003.

[13] A. Wald, J. Zeh, G. Barnum, L. G. Davis, and L. Corey, "Suppression of subclinical shedding of herpes simplex virus type 2 with acyclovir," Annals of Internal Medicine, vol. 124, no. 1, part 1, pp. 8-15, 1996.

[14] M. Reitano, S. Tyring, W. Lang et al., "Valaciclovir for the suppression of recurrent genital herpes simplex virus infection: a large-scale dose range-finding study," Journal of Infectious Diseases, vol. 178, no. 3, pp. 603-610, 1998.

[15] P. van den Driessche and X. Zou, "Modeling relapse in infectious diseases," Mathematical Biosciences, vol. 207, no. 1, pp. 89-103, 2007.

[16] K. L. Cooke and P. van den Driessche, "Analysis of an SEIRS epidemic model with two delays," Journal of Mathematical Biology, vol. 35, no. 2, pp. 240-260, 1996.

[17] S. Q. Liu, S. K. Wang, and L. Wang, "Global dynamics of delay epidemic models with nonlinear incidence rate and relapse," Nonlinear Analysis. Real World Applications, vol. 12, no. 1, pp. 119-127, 2011.

[18] R. Xu, "Global dynamics of an SEIRI epidemiological model with time delay," Applied Mathematics and Computation, vol. 232, pp. 436-444, 2014.

[19] L. L. Liu, J. Wang, and X. Liu, "Global stability of an SEIR epidemic model with age-dependent latency and relapse," Nonlinear Analysis: Real World Applications, vol. 24, pp. 18-35, 2015.

[20] J. Hale, Theory of Functional Differential Equations, Springer, Heidelberg, Germany, 1977.

[21] P. van den Driessche and J. Watmough, "Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission," Mathematical Biosciences, vol. 180, no. 1, pp. 29-48, 2002.

[22] R. A. Ross, The Prevention of Malaria, John Murray, London, UK, 2nd edition, 1911.

[23] L. E. El'sgol'ts and S. B. Norkin, Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Academic Press, New York, NY, USA, 1973.

[24] J. K. Hale and P. Waltman, "Persistence in infinite-dimensional systems," SIAM Journal on Mathematical Analysis, vol. 20, no. 2, pp. 388-395, 1989.

[25] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, NY, USA, 1993.

[26] J. R. Haddock and J. Terjeki, "Liapunov-Razumikhin functions and an invariance principle for functional-differential equations," Journal of Differential Equations, vol. 48, no. 1, pp. 95-122, 1983.

[27] S. M. Blower, T. C. Porco, and G. Darby, "Predicting and preventing the emergence of antiviral drug resistance in HSV2," Nature Medicine, vol. 4, no. 6, pp. 673-678, 1998.

http://dx.doi.org/10.1155/2016/1598932

Lianwen Wang, (1) Yong Li, (2,3) and Liuyong Pang (4)

(1) School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

(2) School of Information and Mathematics, Yangtze University, Jingzhou 434023, China

(3)Institute of Applied Mathematics, Yangtze University, Jingzhou 434023, China

(4)Department of Mathematics, Huanghuai University, Zhumadian 463000, China

Correspondence should be addressed to Yong Li; gxbyl@163.com

Received 15 July 2015; Revised 7 December 2015; Accepted 21 December 2015

Academic Editor: Giovanni Garcea

Caption: Figure 1: The DFE [E.sub.0] of model (4) is globally asymptotically stable for different initiate conditions when [R.sub.0] = 0.0077, where we take (A, [mu], [tau], [[tau].sub.1], [[tau].sub.2], [beta], [[beta].sub.1], m) = (1/7300, 1/7300,0,3.5,4, 27,0.0001,0.00005,10).

Caption: Figure 2: The EE [E.sup.*] of model (4) is globally asymptotically stable when [R.sub.0] = 1.5308, where [beta] = 0.02, [[beta].sub.1] = 0.01, and initiate conditions and the reminding parameters values are kept the same as Figure 1.

Caption: Figure 3: The impact of media on the genital HSV-2 under the same initiate condition (S(0), E(0), I(0), R(0)) = (0.8,0.05,0.01,0.05) except for the values of m and [[beta].sub.1].
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Title Annotation:Research Article
Author:Wang, Lianwen; Li, Yong; Pang, Liuyong
Publication:Mathematical Problems in Engineering
Article Type:Report
Date:Jan 1, 2016
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